On the mathematical representation of nonlinearity

Emanuel Gluskin, " On the represe ntation .." arXi v:0807.0966v2 [ nlin.SI], corr. posted 24 Oct., 2008 . 1 On the mathematical representatio n of nonlinearity Emanuel Glus kin Electrical Engineering Department, Academic Technological Institute, 52 Golomb St., Holon 58102, Israel, and Electrical Engineering Department, Ben-Gurion University, Beer-Sheva 84105. Abstract : The suggestion of writing, for some problems, nonlinear state equations not a s d x /d t = F ( x , u , t ), but a s d x /d t = [ A( t , x )] x + [B( t , x )] u ( t ), which is more "constructive", is considered supporte d by arguments related to: the axiomatization of system theory, the classification of switched circuits as linear and nonlinear, the use of nonlinea r sampling f or measuring f requency without clock, the consideration of an ensemble of c olliding particles i n which the establishing of the chaotic movements is seen as a result of the same kind of nonlinearity a s in electronic switc hing circuits, the modeling of a liquid medi um a s a " system" who se st ructure is directly inf luenced by the "input", and some other problems. 1. In troduction It is suggested i n the remarkable pedagogical work [ 1] to start the laborat ory s tudy of electrical circuits by electrical engineering s tudents from nonlinear, and not linear, circuits, and the rationale for teaching nonlinear circuits before ones linear is articulated in the capital source [2] too. Obv iously, a n acquai ntance with nonlinear circuits before linea r ones does not encourage one to just grammatically define /see "nonlinear" as "not a linear one" (w hich some s tudents could erroneously unde rstand as "not a good one" ), but as something constructively defined and de scribed . This position of [ 1,2] seems t o us to be important not only pe dagogically, but also axiomatically and we deal with t he c onstructive definiti on of nonlinearity, suggesting some a ssociated notati ons relevant to s ystem classification, and develop an outlook, via system concepts, on some physical phenomena. The physical aspects contribute to bet ter understandi ng of the system concepts a nd to making basic system theory become a part of one's general education. Singular systems are one of our mai n focuses, and among s uch systems, switched systems are men tioned most of ten. Definitions of linear and non linear switched systems are unus ually close b ecause the ve ry fact of switc hing has to be c onsidered first of a ll, a nd the nonlinearity is the n directly see n f rom the nature of the ti me- functions that control the s witching. In the nonlinear case, this "nature" i s a map x ( t )  t* of the state-variables { x p (t )} of a s ystem on the s witching insta nces { t k } that are important system parameters . This is inevitably done i n the str uctural terms of the elements being switched. This approach is started in [3-5], but contrary to the original line of [ 3] which is m ore oriented towards a designer of switched circuits, here we are concerned with logical basics, a nd cover points that s hould be more interesting for a theoretician. Some se emingly si mple c oncepts had to be revisited; in particular, that of input (port) functi on. Without noting the freedom that mode rn switched s ystems give to the Emanuel Gluskin, " On the represe ntation .." arXi v:0807.0966v2 [ nlin.SI], corr. posted 24 Oct., 2008 . 2 input c onnections, and thus to the very conce pt of " input", on e can have diff iculty in seeing possible nonlinearity even of some very simple systems/equations. The main equational frame of the work is as follows. Usually, nonlinea r state-equations are written as ( t is time and x is the vect or of state-variables) d x /d t = F ( x , u , t ) (1) (dim F = dim x ), but we use equations in the form that is c loser t o that of linear state- equations (for inst ance many classical Switc hed Capacitor Circuits/Filters are described as (2)), d x /d t = [A( t )] x + [B( t )] u ( t ) , (2) i.e. as d x /d t = [A ( t , x )] x + [B ( t , x )] u ( t ) . (1a) The nonlinearity in (1 a) is thus see n via the influ ence of x (and we c an c all such a system an " x -sy stem") on the "structure" understood as in linear sy stems, which makes a di fference with respect to (1) whe re an alytical dependencies are on the fir st place. Of course, t here is no suggestion of being detached f rom (1) i n any case. For instance the e xample of Sec tion 4.3 with a hardlimiter nonlinearity would require (see also [3]) [A( x )] to have a pole, which i s unusual. Howe ver in the very important applications for switched s ystems, w hen a nonlinear system "jumps" a t som e-time instants defined by x ( t ), from one LTI system to another, form (1a) certainly has an advantage in understanding the system. There a lso ca n be nonsingular systems, as e.g. in the physical analogy in Section 5.1, when (1a ) is more adequa te than (1). In pa rticular, a linearization i s obtained in this a nalogy very na turally, as the transfer from ( 1a) to (2) by substituting in (only) the matrices a certain approximating vector function x o ( t ) instead of x ( t ). Starting f rom Section 3, we shall often speak only about [ A]. If u ≠ 0, then similar things are meant to a pply to [B]. However, for nonlineari ty of (1a) it is also sufficient that only [B] depends on x ; some s uch systems of the second or der are considere d in [6,7] (d x /d t = Px + B ( x , t ), in the notations of [6]), but only as regards limit cycles. The nonlinear non-autonomous systems have a very good interpret ation here ; a system is nonlinear i f it's struct ure is inf luenced by the input . This influence c an be indirect, expressed a s [A( x ( u ))], or direct, expressed as [ A( u )]. In the la tter case, we speak about " u -systems" tha t are treated in Section 2. The l atter s ection completes also some other knowledge about nonlinear equations that we actually need here. The main line starts in Sec tion 3 a nd 4 from x -systems t hat are most c ommon nonlinear realization. Section 5 shows tha t our system a pproach can be useful for some physical problems. Sections 6 is devoted to consideration of another important singular operation, -- sampling , and f or the nonlinear case an analogy to the procedure defining Lebesgue's integral is considered. Section 7 extends the topic of nonlinear sampling to an application to spectrum analysis. Section 8 overviews the main results and conclusions adding some final comments. Appendix 1 is a quite pragmatic completion to the discussion o f s witched system s in the main text, and Appendix 2 is devoted to some purely logical points tha t should interest a theoretician. Emanuel Gluskin, " On the represe ntation .." arXi v:0807.0966v2 [ nlin.SI], corr. posted 24 Oct., 2008 . 3 For s ome problems b elow it is useful to somew hat generalize t he Sma ll Theorem of [4] in which onl y the scaling te st of linearity u  k u ( k is constant) i s mentioned. This theorem says t hat if with increase in k starting f rom zero, a switching appears or disappears (or, in a pe riodic proce ss, the density of the switching instants is c hanged), then the system is nonl inear. The proof uses the facts th at a ny switching nec essarily causes a singularity, and tha t two wave forms (shapes) having different numbers (or densities) of the points of singularity cannot be proportional each other, i .e. u  k u cannot be f ollowed by x  k x , as linearity requires. It i s possible, as the point of the generalization, to a lso involv e the linear test of additivity , by noting tha t if u 1 + u 2 causes a new switching that was not caused by e ither u 1 or u 2 , the n the system i s nonlinear. Indeed, if neither of the associated x 1 and x 2 includes a singularity at an instant, then their sum also cannot inc lude it. This mea ns that u 1(2)  u 1 + u 2 c annot be followed, under the the orem's condition, by x 1(2)  x 1 + x 2 , i.e. the system is nonlinear. It is a lso important f or relevance of the theorem t hat switching is a clea rly registerable phenomenon/operation. The po int of measurement is important both axiomatically and prac tically (Appendix 2), a nd according to our actua l percept ion of nonlinearity as something constructively defined, just "not a linear one" is not yet "nonlinear". 2. The " u-systems" and the concept of " given input" It i s the flexibility and the structural freedom existing in the des ign of modern electronics switc hing s ystems (e.g. Fig. 2 in [4] include s such a possibility), which cause us to conside r " u - systems ", in which t he input c an dir ectly influence the structure: d x /d t = [A( u )] x + [B( u )] u ( t ) , (3) or d x /d t = [A( u )] x . (4) For such an equation, the map u → x that i s realized by the assoc iated system is nonlinear . Indeed, the scaling te st of linearity (let us for simplicity always take zero initial conditions), i.e. the map k u → k x ( k is constant) (5) is not allowed by (3) or (4), obviously. One ca n find that this seems to contradict with the fact th at s ince u ( t ) is given, [A( u )] and [ B( u )] are a lso given, as [A( t )] and [B ( t )] in (2), and though it is just a particular case of ( 4), it is not easy for man y t o accept that the equation ( ) ( ) 0 dx a t x t dt + = (6) can be either linear or nonlinea r, depending, re spectively , on whether a ( t ) is a fixed function defined by the very system's structure (i.e. given by the produc er of the system) or an externally defined input of the system. Since we see in the definition of input an axiomatic point, let us conside r this seeming terminology difficulty in detail. Emanuel Gluskin, " On the represe ntation .." arXi v:0807.0966v2 [ nlin.SI], corr. posted 24 Oct., 2008 . 4 When a fixed time-function th at defines the structure of a s ystem arises in a linear system operator, say a 1 ( t ) in the following operator-sum ˆ ( ) p t p p d M a t dt ∑ = (7) (think e. g. about the equation ˆ ( ) ( ) t M x t u t = ), then this is, usually , a function def ined by t he producer of th e device, a nd it is the s ame in any w ork-state. Such a s ystem i s linear. Because of the complete f ixation of a 1 ( t ) , interpreting the system's action as the map a 1 ( t ) → x ( t ) would be unnatural. If, however, a 1 ( t ) is an input of the sy stem, th en such a function h as t o be seen as taken from a set o f the pe rmitted function , i .e. i t belongs to an available for the (human) ope rator region i n some function space, and not to only one of its points. This is required by both the operational purposes in which the input ha s to be changed and by the necessity to perf orm the input scaling test ( 5), or t he additive tes t of linearity (and thus a linear space is included). Though in each particular experiment, the input func tion is given, we have to consider its possible changes, and, in particular, s uch expressions as ku ( t ), with an arbitrary k . This is the disquieting sense in which t he input func tion is " given". (Letting first u ( t ) to be a n essentially positive, known f unction desc ribing the sta te of one's bank account, take the n in ku ( t ), k as 1/2 or -2.) It be comes clear that though u is, in principle, know n, [A ( u )] in (3) is not a s "given" as [A( t )] i n (2). (Space with its operations ca nnot be re placed by a point/function.) If the scaling factor, or the input-function waveform ca n be changed, then, in fact, t he system is not c losed, i.e. it is not completely known. This agrees with the c lassical and most natural view of "input" as a kind of "interface" of a system with the external world, not as an internal fixed characteristic of the system. However, a s a rule, when considering (6), one does not ask what the known function a ( t ) is, a nd one's im mediate reaction is: " Everybody k nows t hat this is a linear equation! ". S uch a react ion, obviously, f ollows from one i magining (6 ) to be the particular case of ( ) ( ) ( ) dx a t x t u t dt + = (6a) with the input u ( t ) t aken as z ero. One simply al ways assumes that the right-hand side of an equation i s the reserved place for the "input" . Thus a ( t ) in ( 6a) and (6) is not seen by one as any input. However one has to remember that: 1. " Right-hand sid e" is n ot a mathema tical, but a physiological concept, and s uch, -- however t raditional, -- a v iew o f t he equa tional s tructure has no relation to mathematical rigor. 2. Mode rn electronics s ystems can hav e num erous unusual ports, and thus the input function(s) can a ppear in different equation terms. T his is not as in the old mechanical problems of the peri od when differential equations a ppeared, and when the outlook on such equation was established. 3. It is insufficient to know tha t a ( t ) is given; one ha s t o know what is the role of this function in the real system. No gener al reason exists why a ( t ) in (6) can not be an Emanuel Gluskin, " On the represe ntation .." arXi v:0807.0966v2 [ nlin.SI], corr. posted 24 Oct., 2008 . 5 input, and why in (6a) both u ( t ) and a ( t ) c an not be inputs. Note that i t can be, in particular, that a ( t ) = u ( t ) in (6a). Solving (6), 0 ( ) ( ) (0) t a d x t x e λ λ − ∫ = , one sees that the map a ( t ) → x ( t ) is not addit ive (not linear), but multiplicative. Thus, if a ( t ) is the input, (6) is a nonlinear equation. One noti ces that this is a standard nonl inearity in the sense of our basic e quations. Indeed, while seeing in each case a ( t ) as u ( t ), one can either see (6) as a case of (4), ( ) dx a t x dt = − , or, with a minor change in writing, ( ) dx x a t dt = − , as a case of (1a), in which [A] = 0 and [B( x )] = - x . See also [6]. Returning to the operator of type (7), w e can say that if such an operat or i ncludes functions-coefficients that by themselves (via some input) influence the functions on which the ope rator is intended to act, then this ope rator is nonl inear. Thus, in the proper system context, a mat hematical operator form ally having a line ar form can become nonlinear. Of course, it does not y et follow f rom t he axiomatic importance of c lassifying the u -systems as nonlinear that any nonlinear effect that can be obtaine d in x -systems can be also obtained in a u -system. Example of a circuit with u -nonlinearity is given in [33]. There is anot her interesting case of nonlinea rity to be separately noted. It can occur (see Section 4.3) in a nonlinear s witched s ystem that for s ome value s of systems parameters a nonl inear term be comes a known time-function. Namely, the equation ˆ ( )( ) ( ) ( ) Lx t f x k t ξ + = (8) where ( ) k t ξ is the input and f ( x ) is a nonlinear function, can became (because of a saturation of f (.), see Section 4.3) for some values of the parameters included in the linear operator ˆ L ˆ ( )( ) ( ) ( ) Lx t t k t ζ ξ + = (8a) where ( ) ( ( )) t f x t ζ ≡ is a known tim e-function tha t relates, j ust as f ( x ) in (8), to the system structure, and not to the i nput. T he left-hand side is now an affine , i.e. a nonlinear f orm by x (the test of linearity does not pa ss because of the f ixed term), and thus also for this specific range of the parameters of ˆ L the equation remains nonlinear and no drastic change in the associated physical system occ urs. O f c ourse, t he i ssue is delicate because physically the t erm ( ) t ζ is obtained (Sec tion 4. 3 f or de tails) only for some nonz ero, sufficiently la rge k . Nevertheless, the known function appears not as a part of the input. Emanuel G luskin, " On the representation .." arXiv :0807.0966v2 [nlin.SI], corr. post ed 24 Oct., 2008 . 6 Though the system behind the example of Section 4.3 is practica lly important, the function ( ) f x leading to the affine nonlinearity, is rather specifi c, and we shall not include the affine case in the m ain classification below, considering it a s pa rt of the x - nonlinearity. However, one se es here too how important it is to know/see whether or not a known function belongs to the input. 3. Linear an d nonlinear sw itchings Switching systems, linear a nd nonlinear, are a very importa nt background for the present study. Foll owing [3], we e xclude in these s ystems a ny " analytical nonlinearity", leaving the possibility of nonlin earity to arise only bec ause of the switchings. Considering any switched system, -- either linear or nonlinear, -- as d x /d t = [A( t , t* )] x + [B( t , t* )] u ( t ) , (9) where t* is the set of s witching poi nts { t k } tha t defi ne the system operation, i.e. at which the elements are changed, we have the case of linearity (2) if t* = t* ( t ) , (10) i.e. if t* is defined by known functions (usually e xternal generators), or, simply, is prescribed , and the case of nonlinearity (1a) if t* = t* ( x ) , (11) i.e. if t* is defined by initially unknown functions that have to be found when solving the system. The case of t* = t* ( u ) (11a) is also nonlinear. Thus, for the s witched s ystems, (10) comp actly represe nts (2), (11) represent s (1a), and (11a) represents (3). Respectively to (10), (11) and (12) we speak about " t -", " x -", and " u -" SS. In the research scheme introduced by [3], for (10) a nd (11) t * (.) is defined by the functions (ei ther known/prescribed, or not) tha t are inputs of t he comparators that determine the level-crossings at which the pulses are generated to trigger the swit ches. Figure 1 schematically illustrates this. Emanuel G luskin, " On the representation .." arXiv :0807.0966v2 [nlin.SI], corr. post ed 24 Oct., 2008 . 7 Compar ator f 1 ( t ) f 2 ( t ) t* f ( t , t* ) Swit ched (at t* ) elements Swit ch oper ated (at t* ) Fig. 1: A switching subs y stem somewhere i nside the given system. S tarting from t he i ntersect ions of the s hapes of f 1 ( t ) and f 2 ( t ), wh ich define t* , we come to a func tion f ( t , t* ) measured on the switched unit. If both f 1 ( t ) and f 2 ( t ) are prescribed we have a linear (LTV) s ystem, and if at least one of these functions is one of the state-variables to be found, then the system is nonlinear; t* = t* ( x ). The case of t* ( u ) is non linear too. (For a u -syste m, it can occur that changing a scaling fac tor may not influence t* for e very input waveform; then the gene ralization of the Small Theorem mentioned in Section 1, associated w ith considering additive changes in u , can be important.) In order to s ee how t* = { t k } c an be incl uded within the system’s s tate-equations, assume that one cr eates a switching system by replacing a capacitor C i n a linear time-invariant (L TI) ci rcuit described using ma trix [ A(C)], by a swit ched unit C 1 ↔ C 2 . Assume that at some instant t 1 , known/prescribed, or not , C 1 is replaced by C 2 . Considering tha t the coefficients i n Kirchhoff's equations, which define [A] are given as some ins tantaneous time-functions, one ca n simply r eplace in [A(C)] the pa rameter C by the time-function of the type C( t ; t 1 , … ) = … C 1 u( t 1 - t ) + C 2 u( t - t 1 ) + … , (12) where u( t ) is the step func tion (u(z) = 0 for z<0, and 1 for z>0), obtai ning i nstead of [A(C)] some matrix of the type [A(C 1 , C 2 , t , t* )] , ( t* ⊃ ⊃ ⊃ ⊃ t 1 ) (13) According to the sai d about equation (3), such a switc hed system may be ei ther LTV or nonli near, depending on whethe r or not the functions whose mutual crossings define t* are known; i.e. depending on whether or not t* is known. If the sw itched system is nonli near, then one can, perhaps, obtain stabilization of oscillations in it (a lim it cycle), or chaos (an attr actor). If the system is linear, then some nee ded linear response t o a port-input can be obtaine d, and the system c an be well controlled, e.g., in the sense of its power consumption. Emanuel G luskin, " On the representation .." arXiv :0807.0966v2 [nlin.SI], corr. post ed 24 Oct., 2008 . 8 This is a simple and very gene ral way of classifying s witched sy stems a s line ar or nonlinear, and i s suitable for a designer who actually chooses the f unctions a t the inputs of the comparators. See the circuit examples in [3]. Our final scheme of classification of switched systems is s hown in Fig. 2. This is, essentially, also a classification of more general singular systems, e.g. sampling systems, or systems with singular passive elements. t* ( . ) N L- SS t* ( x ), t* ( u ) LT V-S S t* ( t ) SS Fig. 2: The g eneral scheme for classification of swi tched system (SS) . 4. Sw itching instants as fun ctionals. Let us return to Fi g. 1. If we take in i t f 1 ( t ) as x 1 ( t ) and (for simplicity) f 2 ( t ) a s a constant level D , then the a ssociated t k can be written using the i nverse function, as x 1 -1 ( D ). However t he inverse function (certainly existi ng i n the vicinity of a level- crossing of x 1 ( t )), is a local one and it is not a very s uitable concept here. In general, the notation t* ( x ) j ust means that t* is defined b y x , and i t is not necessary here that some analytical dependence of a t k on an x k be given. Indee d, the switching instants are, finally, some numerical val ues, and, using the terminol ogy of the calculus of variations, the map x ( t )  t* is of the type of a functi onal , i.e. a map of a function on a num ber (or a vector-function on a set of numbers). The simplest way of def ining a t k ∈ t* is through the concept of le vel-crossin g of a time-function and the use of comparators as illust rat ed in Fig. 1. For insta nce, x ( t ) = 0 (i.e. f 1 ( t ) is x ( t ), a nd f 1 ( t ) ≡ 0) can be the equation for a t k in the nonli near cas e. In any c ase, we a ssume that existence of a map x ( t )  t* (or u ( t )  t* ) should be a general rule for any nonlinear switched circuit. Emanuel G luskin, " On the representation .." arXiv :0807.0966v2 [nlin.SI], corr. post ed 24 Oct., 2008 . 9 4.1. Two examples using scalar equation form In [8] the linear-form equation ( a , b , c -- constants) t c t bx dt dx a dt x d ω sin ) ( 2 2 = + + (14) with the “forced” condition of the mirror-type reflection of x ( t ) from the time axes: (d x /d t )( t k + ) = - (d x /d t )( t k − ) , t k : x ( t k ) = 0 , (15) is c onsidered. Obviously, t k are included via { t - t k } in the solution of (8); x ( t ) = F ( t ,{ t - t k }) with a known function F (.,{.}), and for any certa in p , t he c ondition x ( t p ) = 0 becomes F ( t p ;{ t p – t k }) = 0, i.e. all t k may be thus found. System (14,15) i s nonlinear because zeros belonging to the function to be found are used. The computer e xperiment [8] indeed has shown that for some ranges of t he parameters a , b and c , the sequential dev elopme nt of the process leads to a ch aotic x ( t ) , w hich is a feature of a nonlinear system. Another c haotic e quation, this t ime with a nonlinea r singular chara cteristic , which is a more standard case, is described in [9]. In the equation 2 2 ( ) sin( 2 ) ( ) d x x t t s x dt + = + (16) s ( x ) is the piece wise linear m ap defined a s max {-1, 5 x } if x < 0, a nd as min {5 x , 1} if x > 0, i .e. the insta nts of singularit y at which an LTI equat ion i s switched are def ined by crossings by the function 5 x the levels -1 and 1. Introducing the unknown " switching" (singularity) instants { t k } = { t k ( x )}, defined by the relevant level-crossings of x ( t ), we can write s ( x ) as some f ({ t - t k ( x )}), having 2 2 ( ) sin( 2 ) ({ }) k d x x t t f t t dt + = + − (16a ) where f ({.} is, in principle, known. It is f ound in t he intervals of t he type t k < t < t k+1 , (note that t k ( x ) = t k ( x )) which are defined by either (whe n s ( x ) = x , i.e. | x | <1/5) the equation 2 2 sin( 2 ) d x t dt = (16b) or (when | s ( x )| = 1, i.e. | x | >1/5) by the equations 2 2 ( ) 1 sin( 2 ) d x x t t dt + = ± + . (16c,d) Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 10 The t* ( x )-nonlinearity is obvious, directly following from the si ngularity of s ( x ), even though it is not eas y here to write an intermediate c onstructive sol ution of (16) in terms of { t k }. H owever, in the e xample of Section 4.3 such a constructive solution is not difficult. 4.2. Switching "between linear systems" Assume t hat all of the e lements of [ A] that a re changed, are changed at the same instants. Of c ourse, one can assume that when only some (e.g. one ) of the elements of [A] are changed, al l the other elements undergo zero changes (i.e. s witching to the same value) at the same instants . Then, suc h functions as u( t 1 - t ) and u( t - t 1 ), appeari ng i n the matrix elements of type (12), can be taken from the matrix, and (12) is generalized to [A( t , t k )] = [A] 1 u( t k - t ) + [A] 2 u( t - t k ) + … , (17) where [A] 1 , [A] 2 , … are some fi xed (because the switc hing is b etween LTI ele ments) numerical matrices. Assume now that al l the matrix-coefficients in (17) are taken (repeated in some way) from the set composed of only two fixed matrices, [A] a and [A] b , i.e. [A( t , t k )] = [A] a u( t k - t ) + [A] b u( t - t k ) (18) which is [A] a  [A] b , and then back ([A] b  [A] a ): [A( t , t k+1 )] = [A] b u( t k+1 - t ) + [A] a u( t - t k+1 ) (18a) (say, (18) for a ll-even, and (18a) f or a ll-odd indices) . That is , [A] alternates the two "states", [A] a and [A] b , according to some specific criteria for t k (.). In such a c ase, instead of s peaking a bout the swit ching of elements in a certain LTV or nonlinear system, one can speak about switching " between two LTI systems", each corresponding to one of the two given numerical matrices, [A] a or [A] b . This kind of terminology, though without writi ng (17) or (18), i.e. not e xplicitly introducing t* (. ) in the equations, but indeed creating two complete L TI subsystems and switc hing from one to anot her, is us ed in work [10] devoted to the generat ion of a chaotic process. t* = t* ( x ) in [10]. However this is obtained not via the level- crossings of some of the x k ( t ) but via s ome constraints on the norms of x ( t ) which require the calculation of some associated measures. Unfortunately, in stressing the li nearity of t he two LTI subsystems, work [ 10] does not mention the nonlinea rity of the whole system. Howe ver, from our pos itions, it i s immediately seen from { t k } = { t k ( x )} that the system is nonlinear. Thus, re garding such works as [ 8-10] our classification of the nonlinea rity of switched systems seem s to be heuristically usef ul. W e conti nue, however, with other important examples. Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 11 4.3. An integral equation related to the case of a "physical switching" inside a singular passive element Usually, determination (calculation) of { t k } i s done only at the very final stage of solving a problem where the m ap x  t* takes place. A rele vant example is given in [12-14]; it can be formulated in terms of the differential electrical circuit equation, 1 sign[ ( )] ( ) ( ) di L A i t i t dt f t dt C ∫ + + = , (19) where f ( t ) is a T -periodi c given function, whic h is equi valent to the integral equa tion of the type (using more common mathematical notations) ( ) ( ) ( ) [ ( )] t x t t h t sign x d ϕ µ λ λ λ −∞ ∫ = − − (20) where h ( t ) is the current shock response (Green's function) of the oscillatory subcircuit to which the pa rameters L and C of (19) relat e, and ϕ (t) = ( h ∗ f )( t ) is a known T -periodic function having zero a verage, whos e zerocrossing fea tures are important. This equation play s an important rol e in a sim ple version of a nonlinear theory of fluorescent lamp circuits. See [12-14] and t he wor ks quoted there. It c an be proved that for µ small enough x ( t ) in (20) is a T -periodic zerocrossing function, hav ing the s ame density of the zero-c rossing s a s ϕ ( t ). Using F ourier series for the time-function ξ ( t ) ≡ sign[ x ( t )], one obtains from (20) for such µ ( ) ( ) ({ }) k x t t t t ϕ µζ = − − (21) with known function ζ ({.}), i.e. ( ) ( , { } ) k x t F t t = (21a) with F (.,{.}) known, but t k themselves still unknown. One observes from (21) tha t the zerocrossing feat ures of x ( t ) require [ 14,13] that d ζ /d t be limited. Starting from µ = 0, and increa sing µ , we observe, in the general case, "movement" of the ze ro-crossings of x ( t ) f rom thei r initial positions defined by ϕ ( t ); the range f or this "movement" i s defined by appe arance of some touc hing of the time-axes by x ( t ). There is an interesting c ase [ 5,13,14] when t k are unmoved (constant) w ithin these bounds for µ . In t his case, x ( t ) given by (21a) is already completely known. However, in the general case, (21a), is just a stage of the solution. The problem of sol ving (19) is thus redu ced to t he a lgebraic p roblem of determination of t k . The equa tions of the type x ( t k ) = 0 (22) which define t k , become the constructive (usually transcendent) equations of the type Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 12 F ( t k ,{ t p }} = 0, (23) and all such equations together define all of the t k . In view of (21), (23) is ( ) ({ }) , k k p t t t k ϕ µζ = − ∀ , (23a) while for the mentioned case of t k "unmoved", bot h sides of the latter equalities are zero. Equation (23) is an example showing how the map x  t* can be realized. This f ormal simplicity of the solut ion of (20) in terms of the zero-crossing s even can create the impression that (20) is not an integral equation at all, rather a set of some algebraic equations. T his see ms to be paradoxical since we star ted from differential equation (19). However for the investigation of the possibility of x ( t ) having the zerocrossing features of ϕ ( t ), t he integral features of (20), ma inly the boundedness of d ζ /d t [14], a re i mportant. In other words, the dynamic s of the system is ex pressed in ( 23a ) in terms of the structural stability of the zer ocrossing fe atures of x ( t ). Finally, the case when A sign[ i ( t )] in (19) becomes a known f unction that does not, however, belong to the input, the nonlinearity of the e quation is kept by the affine form of the left-hand side; certainly, the physical system is a lso not c hanged essentially in this case; the fluorescent lamp remains on its place. Among other relevant examples, the use of t* ( x )-nonlinearity to s tabilize the amplitude o f parametric oscillati ons can be menti oned; see [12] and referenc es given there. 5. Some phy sical systems We turn now to some physical systems where the suggestion t o see the nonlinearity as an influence of x on the structure, and the conce pt of the t* ( x )-nonlinearity are useful. Hopefully, such arguments can be helpful in making system theory principles a part of one's general education. We consider first an analytical " x -system", and in Section 5.2 a t* ( x )-system. 5.1. Linearization of a nonli near sy stem; modeling of a physical system whose very structure is as sociated wi th t he "input-o utput" tra nsferal of a signal, i.e. is dependent on " x ". The suggestions of s eeing in (1a) a direct connection of the structure of the system with its nonlinearity, and the possibl e transfe r to a linea rized s ystem of type (24), where followed for a problem of vorticed liquid flow in [20]. Starting from an analogy to d x /d t = [A( x ( t ))] x , we shall pass on to an equation of the type d x /d t = [A( x o ( t ))] x , (24) i.e. the autonomous form of (2). Even when not knowi ng an ything about the Navier- Stokes equati on, but obs erving the flow, one understands in view of (1a) that since the velocity vector field is both the solution vector-function and the structure of the liquid "system", this system is Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 13 nonlinear . Thus, f or instance, possible c haotic movement (i.e. turbulence) can be understood as a kind of chaos obtained in a nonlinear system. Assume that a perturbation of a f low (that initially may be, for simplicity, laminar) is m ade at a point in the stream. This perturbation i s expressed in some vortices introduced in to the flow, whi ch are carried by the fl ow as a n input "sig nal", and, at the same ti me, a re a lso a part of the fl ow. Thus, following the line of thought dictated by (1a), we see the " input-output signal" as a part of the very structure of the "system" . Since it is c lear that the " signal" propagating with the f low i s being somewhat spread and changed, it is also clear t hat the t ransfer function of the standard type, which would expre ss t ransfer of the " signal" as it is , i.e. only with a time delay t x (the time needed for the perturbation to come to the point of obse rvation x , taken in the direction of the flow), H o ( s ) ~ exp{- t x s }, (25) is non-realizable. The work-hypothesis of [20] is th at if the perturbation is c hanged weakly, the true H(s) has, however, to be close to H o ( s ). Using H( s ) = (1+ st x /n) - n , (26) (but, of course, not (1- st x /n) n which enhances, as one sees taking s →∞ , high frequencies a nd requires negative viscosity in (27 )) a s an a pproximation to (25), [20] shows that this s ystem function agrees with a l inearized Navier-Stokes equation , while n is the sim ultaneously obtained Reynolds number in which x is the "parameter" of the distance. Consider the Navier-Stokes equation [20-22] in which, f ollowing [20], we om it the term with the gradient of pressure: ( ) v v v v t ν ∂ + ∇ = ∆ ∂     (27) where v  is the veloci ty ve ctor field (i.e. v x , v y , and v z , a re our state-variables) and ν is the ki nematic (di vided by the densit y of the liquid) viscosit y. In the 1D -case (27) becomes the known Burger's equation u t + uu x = au xx used for demonstrating the specificity of some hydro- and aero-dynamic phenomena. The ignora nce of the pressure ( p ) is, of course, unacce ptable for general physical analysis (the c omplete dynamic vector equation for incompressible liquid , 1 ( ) v v v v p t ν ρ ∂ + ∇ = ∆ + ∇ ∂     includes f our unknowns, and the condition for incompressibility, 0 v ∇ =  , has to be added), but v x , v y , and v z are the parameters of x = { v  , p } w hich relate to the immediately seen "structure" of th e liquid sys tem, and thus our compa rison below of (28) with (1a) is le gitimized, even t hough [ ( )] A v Σ  be low is not the full physical [A( x )]. Equation (27) can be easily rewritten in the spirit of (1a), as [ ( )] [ ] v A v v v t α ∂ = − + ∂     (28) Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 14 where the nonlinear matrix-operator 0 0 [ ( )] 0 0 [ ]( ) 0 0 v A v v I v v ∇     = ∇ = ∇     ∇        includes the unit matrix [ I ], and the scalar differential operator x y z v v v v x y z ∂ ∂ ∂ ∇ ≡ + + ∂ ∂ ∂  , and 2 2 2 2 2 2 [ ] [ ] [ ]( ) I I x y z α ν ν ∂ ∂ ∂ = ∆ ≡ + + ∂ ∂ ∂ is a linear matrix operator. Introducing 0 0 [ ( )] [ ( )] [ ] [ ]( ) 0 0 0 0 v A v A v I v v v ν α ν ν ν Σ − ∇ + ∆     = − + = − ∇ + ∆ = − ∇ + ∆     − ∇ + ∆         , we can rewrite (28) as [ ( )] v A v v t Σ ∂ = ∂    . (28a) Matrix [A Σ ] represents the system's "structure" and is of the type [A( x )] in (1a). The linearization of (27) means in [20] ( ) o v v v v t ν ∂ + ∇ = ∆ ∂     (29) where o v  is the given av erage velocity of the flow . In state-space terms, this linearization means the repl acement in (28) of [ ( )] A v  by [ ( )] o A v  , or in (28a) [ ( )] A v Σ  by [ ( )] o A v Σ  : [ ( )] [ ] [ ( )] o o v A v v v A v v t α Σ ∂ = − + = ∂       , which is similar to (24). Orienting the x - axis in the direction of v o  , taking int o acc ount t hat for the " inertial part" of the perturbation of v  (i.e. the purely del ayed " input signal" ), d enoted as v i  , the left-hand side of (29) is precisely zero, and writing ( ) ( , ) o i o v v v x v t x t ε = + − +    , Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 15 where ( , ) x t ε  is the small deviation occurring in the introduced perturbation v i  af ter it has passed the distance x , [20] derives from (29) the following equation for ( , ) x t ε : 2 2 ( ) ( , ) i o o v x v t x t v t x x ε ε ν ∂ − ∂ ∂ + = ∂ ∂ ∂    . (30) The Laplace transform of (30) by t is then done, turning this PDE to a simply solved ODE with derivation by only x . The assumpt ion of the t ransfer f unction (26) is f inally shown in [20] to be reasonable, in ag reement with the linearization of the Navier-Stokes equation, and ' n ' appears to be an analogy of Reynold's number, including x as the distance parameter. As the m atter of fact, -- it was this r esearch which ori ginally l ed the a uthor to the opinion that in some problems form ( 1a) may be genera lly be tter than (1). However, the approach to x -systems via t* -systems also has some certain advantages, and we continue with singular systems. 5.2. An ensemble of colliding bal ls : "thermalization" as a result of th e t*(x) nonlinearity It may be obs erved that for an ensemb le of many c olliding particles, the chaotic movement of the particles is obtained b ecause of the nonlinearity of the "switching " type. Indeed, the ins tants when the strikes (collisions) of th e particles appear depend on the traject ories r i ( t ) of the particles, which are our sta te-variables, i.e. this is a t* ( x )-nonlinearity. This consideration even leads us to the general assumption that: Any ensemble of colliding particles where chaotic distribution of the movement is obtained is a nonlinear system w ith a kind of nonlinearity which is cl ose t o that found in nonlinear switched systems . Of course, cha os a ppears here as a pos sible indication of nonlinearity whose existence should be shown directly. If the above assumption is c orrect, the n, in particular, t he tendenc y to thermodynamic equilibrium (" thermalization"), occurri ng in many syste ms, is a nonlinear t* ( x )-process. In order to better see the t* ( x )-nonlinea rity of the ensemble of colliding particles, let us consider the particles as small rigid balls whose collisions are momentary. The spatial position r i ( t ) of a ball number ‘ i ’ is defined by it s initial position and the δ ( t ) -type f orces that arise in t he c ollisions of the ball wit h other balls at the instants t i,j that are roots of equations of the type r i ( t ) - r j ( t ) = 0 , j ≠ i . These f orces ma y be written as F ij = F o ( v i ( t ij - ), v j ( t ij - ) ) δ ( t-t ij ), where v i = d r i /dt , i.e. v i ( t ij - ) and v j ( t ij - ) are the velocities of the balls j ust be fore the collision. Since the f unction F o ( . , . ) can be f ound from energy and impulse c onservation laws, its f orm is universal for t he collisions, and let us focus only on the time-dependent factor δ ( t-t ij ) that includes the shi ft. In view of the dynamic law Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 16 1 1 ( ( ), ( )) ( - ), , ij j i ij ij d v i F dt m F v t v t t t i o i j ij m j i δ ≠ − − ∑ = = = ∀ ∑ ≠    { v i ( t )}, and thus { r i ( t )}, can be presented a s some explicit f unctions of { t-t ij }. However since { } t ij themselv es are defined by the unknow n { r i ( t )} (our x here), the above system of equations is nonlinear in the s ense of the shifting ( t* ( x )-) nonlinearity, i.e. the { } t ij are quite as some { t k } in a electronic switched circuit. We thus can see such an ensemble of the colliding particles as a t* ( x )-system. Wishing to see the reason for the nonlinearity, we just described the ba sic map { r i }  { } t ij . Of course, in order to f ind t he numerical values of { } t ij , F o ( . , . ) has to be determined, and one has to decide whether the collisions are elastic or not, etc.. We conclude that it is possible that the t* ( x )-nonlinearity is always around us and is "as old as this world", though, of course, definition of a " system" s hould be properl y extended. 5.3. The Gaussian distribution as another aspect of the tendency of a s ystem to a statistically equilibrium state through the nonlinear t* ( x )- dynamics This kind of nonlinearity also causes Gaussian dist ribution, because this statistical distribution is associated wit h the tendency of the entropy of the system towards maximum, which is seen as follows. From Boltzman's formula c onnecting entropy with probability, S = k ln P , we have P = exp{ S/ k }, and since near its maximum that ta kes place at some x = x o ( x is a parameter being observed), S ( x ) ≈ S ( x o ) - d ( x − x o ) 2 , with a constant d > 0, we obtain P ( x ) = K exp{ − ( d / k )( x - x o ) 2 }, where K = exp{ S ( x o ) 2 / k }, i.e. near the m aximum of entropy of the system in which x is observed, P ( x ) becomes the Gaussian dis tribution. Thus, the coordinates of t he movement of a Brownian particle, found inside ensemble of small pa rticles that are in thermodynamic e quilibrium, have t he Gaussian distribution. The a ssumption can be expressed that if the distribution of fish in se a were to be homogeneous in the proper scale, then the move ment of the hunting albatross in the air over the se a ([23,24] and the works quoted there ) c an a lso be e xpected t o be Gaussian. S eeing/attacking a fish defines th e instant t k of t he singularity of the trajectory. That the movement of the hunting a lbatross is defined/influenced by the statistically cha racteristics of t he movements in the e nsemble of t he fishes is obvious. Both the Brownian particle and the albatross can be interpreted as some devices or sensors mea suring t he sta tistical parameters of the a ssociated me dium through the (event-defined, i. e. "functional" i n the sense of Section 4) t* ( x )-dynamics of t heir movement. This furt her stre sses the role of the t* ( x )-nonlinearity in the natural pro cesses, -- a pedagogical/educational point, not to be missed bay a teacher. We turn now to another important class of singular systems, sampling systems. Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 17 6. Comments on nonlinear s ampling What was said about t* (.) in the cases of switched systems and system s including passive ele ments with singular characteristics, can be also said a bout any singular systems, among which sampling systems are very important. For a linear sampling procedure , in the functional f (.)  f ( t* ) , (31) t* i s fixed, independent of f , and th e relevant realization, block-scheme , [5] does not differ strongly from the scheme in Fig. 1. It is correct for (31) that ( f 1 +f 2 )( t* ) = f 1 ( t* ) + f 2 ( t* ) , (32) i.e. summi ng the sha pes of the functions and then sampling is the same as to sam pling the shapes separately and then summing the results. Nonlinear sampling appear s when the sampling instants are dependent on the input function of the sampling (sub)system, t* = t * ( f ). This the sis (considered a lso in [5]) i s developed in the following subsec tions and Section 7. The point here is that such nonlinearity avoids pre scription of the sampling rate and make s the s ampling adapt ive in a sense that helps in the analysis of the input function. 6.1. The sampling "by definition" In the following a rgument, the input f unction of the sampling subsystem will be sampled by itself at its own level-crossings, which introduces th e nonlinearity of the sampling i n the simplest possible manner. We j ust observe tha t for the nonlinear expressions, ( f 1 +f 2 )(at its t * ) (33) is, generally, not equal to f 1 (at its t* ) + f 2 (at its t* ) , (34) and the distinction can be very significant. Let us int roduce the dependence of the sampling instants { t k } = t* on the sam pling function f , t k ( f ), by using level-crossings of f ( t ) with some fixed constant level f ref ( t ) ≡ D . That is, we shall s ample f ( t ) jus t at the instants where the se level-crossings occ ur. We shall obviously obtain the sampled values as f ( t k ) = D . (35) If D is unknown , it can thus be measured, but, i n general, (35) is correc t "by definition". Such an equality, illustrated by Fig. 3, ce ases, however, to be trivial in the construction of Lebesgue's integral in Section 6.2 when t k have to be found. Since dif ferent t ypes of nonlinearity can c ause c haos, t his realizing sc heme ca n be investigated as regards the stability of its operation, and the filtration of noise in f ( t ). Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 18 Samp l e ( and hol d) trig ger inp ut Co m para tor f ( t ) D t* Fig. 3: Sa mpling " by defi nition". The instant t* is defi ned by the crossing by f ( t ) of the l evel D , i.e. as f − 1 ( D ), and D is sim ultaneously defined as f ( t * ). Mathe matically, this is the identit y D = f ( f − 1 ( D )), though not a universal one since f − 1 ( . ) can be only l ocally defi ned, but rea lizing t his scheme and studying its stability ca n be relevant t o modeling t he f inite su ms a pproximating Le besgue's integral, considered in Section 6.2. For thus respec tively sampled values, the quantity (33) is, generally, not equal to (34). For instanc e, for f 1 ( t ) = α t, a nd f 2 ( t ) = β t , where sign[ α ] = sign[ β ] = sign[ D ] and thus the level-crossings obviously exist, we obtain ( f 1 +f 2 )( at its t* ) = D ( f 1 )( at its t* ) = D , ( f 2 )( at its t* ) = D , and (33) is D , while (34) 2D . T he only exce ptional c ase when (33) equ als (34) here is that of D = 0. The situa tion regarding an osc illating or just a bounded function, when the existence of the l evel-crossing is not e nsured, is somewhat more complicated. For instance, for f 1 ( t ) = A sin ω t , f 2 ( t ) = B sin ω t , 0 < A < B , there are different cases re the equality of (33) and (34), associated with the possibilities of D ∈ ( A , B ) and D ∉ ( A , B ). For A < D < B , we have both ( 33) and (34) equal D , but for B < D < A + B (33) is D , but (34) is 0. We conclude that a t* ( f )-nonlinearity of the sam pling can make (33) strongly different from (34). 6.2. A comment on Lebesgue's integral: nonlinearity of the approximating sums That we were taking zero in the a bove c omparison of (33) and (34) (as in the case of D = 0) when the re is no le vel-crossings , h as to be j ustified. This is a decision t aken by the analogy with the measure that is used f or the creation of Lebesgue's integral Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 19 [15]. In this integral, instead of starting from some portions of the a rgument, as in Riemann's integral, we s tart (Fig. 4) from some portions (∆ f ) k of the f unction, and seek, at a level D = f k , f or associated support on the axis of the a rgument ( t ), obtaining, in the f inite in te rval of the i ntegration, a set of the associated (supporting ) values (∆ t ) k,m . Figure 4 illustrates our notations; the level D = f k contributes several such " blocks" as D ( ∆ t ) k,1 to the a pproximating sum . It is relevant for us that ( ∆ t ) k,m are de pendent on f , and thus the blocks f k ⋅ ( ∆ t ) k,m a re nonline ar by f , i.e. the whole approximating sum is nonlinear. D = f k (∆ t ) k,1 t k, 1 t k, 2 ( ∆ f) k t f (t ) t k,3 Fig. 4. Construction of the Lebesgue's integral. The inter vals ( ∆ t ) k,m a re taken not arbitraril y a s in Riemann's integral; the y depend on f . This leads to n onlinearit y of the approximating sum. In simple details, since for the function possessing the needed derivatives 2 2 2 1 ( ) ( ) ( ) ( )( ) ... 2 k k k k k df d f f t t t t dt dt ∆ = ∆ + ∆ + , if (d f /d t )( t k ) is nonzero, we have , , ( ) ( ) ( / )( ) k k m k m f t df dt t ∆ ∆ = , at the level f k = D = f ( t k , m ) , ∀ m . (36) If (d f /d t )( t k ) is zero, but the second derivative at t k is nonzero, then, taking posit ive values, , 2 2 , 2( ) ( ) ( / )( ) k k m k m f t d f d t t ∆ ∆ = , (37) etc., with the value of the derivative take n at each ti me according to t he level f k = D = f ( t k , m ), i.e. having each t ime each ( ∆ t ) k,m dependent on f . . The approximating "blocked" sum uses the measures Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 20 meas ( f k , ∆ f k ) = |( ∆ t ) k,1 | + |( ∆ t ) k,2 | + … , for each level D = f k , and it is , ( , ) ( | |) k k k k k m k k m f meas f f f t ∑ ∑ ∑ ∆ = ∆ , (38) having each term nonlinear by f , and thus being nonlinear as a whole. Contrary to that, th e finite sums used in constructing R iemann's inte gral of f ( t ) are a lways linear by f ( t ). The nonlinearity of eac h approximating s um is be cause of the m apping, via f ( t ) , a finite given / fixe d set { f k } (or { D k }) on the t -axis, which creates/c hooses the set { t k }. This nonlinearity even can be demonstrated by using only one point ( D ) on the f -axis, having the approximation sum composed of only two blocks. All of t he ∆ t are obtained as equal only if |d f /d t | = consta nt, which requires f (. ) to be a straight line or some saw-tooth wave. That Lebe sgue's int egral exists for a muc h wider class of functions than Ri emann's integral is not im m ediately import ant for this point, and while thinking about an application of the nonlin earity, one can consider only ea sily re alizable functions for which both inte grals exist. However, an attempt to conne ct the fact that phase modulation is much better t han am plitude modulation in t he sense of rejecting noise, with t he fact t hat Lebe sgue's inte gral e xists f or a m uch wider class of (not s mooth) functions might be interesting. We shall touch here (Sec tion 7) the topic of spectrum analysis, but in a much narrower scope. 6.3. Remarks Returning to the probl em of comparison of (33) and (34) i n Section 6. 1, we note that the terms i n (38) f k ⋅ ( ∆ t ) k,m are zero if D = 0, a nd/or if there are no crossings, since D > f max , geometrically means that all th e ( ∆ t ) k,m became zero. Thus it was a ccepted regarding (33) and (34) that for bounded f 1 a nd f 2 , t he case of D > max { max { f 1 ( t )}, max { f 2 ( t )}, max {( f 1 +f 2 )( t )}}, without any z ero-crossings, is e quivalent to the case of D = 0. One sees from the above why Lebesgue's integral is used more in real, and not complex analysis, -- the nonlinearity associated w ith the level-crossings is of an essentially rea l t ype; the conce pts of '>' and '<' cha racterizing the work of a comparator a s in Fig. 1, do not relat e to complex numbers. Work [ 5] connects thi s fact with the wide use of the real-valued δ -function in physics and engineering. The nonli nearity of the sum i n the case of Lebegue's integral and the abs ence of the prescribed time or frequency uni t of measurement for the Lesbegue's int egral ar e two closely connected distinctions between Lebesgue's and Riemann's schemes. In the construction of R iemann's integral, one " chops " the time-axis sufficiently finely, obviously adjusting the time uni t of measurement to the finest variations of the waveform in order f or the sum of the blocked area to be sufficiently close to the precise area/integr al. T hus , this constr uction i ncludes measurement of the upper bound of the spectrum of the i ntegrand, by a proper "c lock". For the construction of Lebesgue's i ntegral the relevant points on the time (argument) axis are chosen automatically, and no time-unit is prescribed. This a spect is interesting in v iew of the method of spectrum an alysis of Section 7, where we sample a specific transform of the input function (signal, process) at the Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 21 level- (zero-) c rossings of the input function, and thus study the f requency fea tures of the input funct ion wit hout using some a priori given f requency unit, or time me asure , which is a de ep distinction, with respect to the usual methods of spectrum a nalysis. This met hod inevitably must a void Riemann integration of the instantaneous power of the signal. One notes, however, that the deletion of t he unit of measurement would be also obtained if one could use Lebesgue's integration of the power in the hardware. 7. The ψ ψ ψ ψ -transform and the nonlinear s ampling Consider the following nonlinear and singular transform f  ψ o f th e inp ut f unction f ( t ) (< f > = 0) 2 0 ( ) ( ) [ ( )] t t f sign f d ψ λ λ λ ∫ = . (39) The point is t hat ψ ( t ) may be assumed t o be limited, at lea st for the period of the processing, and thus can be generated in an analog (quick) manner. Numerical integration of f 2 ( t ) which leads to the unlimited (and thus irrelevant f or ana log integration) "energy" function 2 0 ( ) ( ) t E t f d λ λ ∫ = (40) (which is a "competit or" of ψ ( t ) in the procedure below) is reje cted because it requires prescribing ra te of sampling of f 2 , i. e. giving a time or frequency unit that we wish to generally avoid here. Consider sampling of ψ ( t ) by a sampling s ystem at the zero-crossin gs { t k } of f ( t ). Contrary to the sampl ing done by me ans of a prescribed gene rator of sam pling pulses, sampling at ze ro-crossings of an i nput funct ion is a nonlinear procedure of the t* ( u )- type, or, rather, of t* ( x )-type, sinc e f is unknown. Thus, t he nonl inearity of the problem is not only because the ma p f  ψ is nonlinear; the nonlinearity of the t* ( x )- type is neces sary for the est imation of the s pectrum of f ( t ) to be done without a ny clock or band-pass filters. Figure 5 schem atically illustrate s the sampling. That thus s ampled v alues of ψ ( t ) are its extreme values is very easy to see ([17] or [18]). Nonlin ear T ra nsf orm Z- cs' dete ctor Sampling dev ice T rig ger input Ψ( t ) Ψ extr f( t) Fig. 5. The schematic f or obtaining { ψ ( t k )}, these parameters to be used for estimation of the average period/frequency of the proce ss f ( t ). The method t hus incl udes the nonlinearit y of t* ( x )-type, or t* ( u )- t y pe, depen ding on the outl ook on f ( t ). This sc heme is relevant/effective i n the c ase when the integration inside block "Nonl inear Transform" i s Riema nnian. If the integra tion is Lebesgue's one, then, as is seen from the basic fo rmulae of the method, one can ob tain the estimation of the average period, also not using an y prescribed time- unit, but not employing the zerocr ossings. Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 22 Using the obtained seque nce ψ ( t k ), we create the f ollowing arithmetic average, measured on line, 1 ( 1 ) ( ) N k k t N ψ ∑ − , where N is the num ber of the registered input zero-crossings. (Below N will completely replace ' t '.) We also use the online-measured "average power" 2 0 1 ( ) ( ) / t P f d E t t t λ λ ∫ = = (41) of the signal, obta ined on line by the low-pass filtering of f 2 ( t ). These two value s lead us to the pos sibility of estimating the average pe riod (frequency) of a compact spectrum of the signal, which is t he basic step (t o be then repeated in some new channels of the processing system [18]) in the in estimation of the spectrum. Assuming that on average t here should be two zero-crossings per average period T a , we define T a acc ording to t he equality 2 a t N T ≈ , (42) i.e. as ( ) 2 2 a t E t T N NP ≈ = . (43) Of course, ( 1 ) , 1 a a NT t N T N < < + >> , and intending to make N be the main parameter, we shall denote t as t N . We perform the f ollowing transformation of E ( t ) [ 17] in which t he main part /term is associated with the integral over the interval ( t 1 , t N ): Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 23 [ ] 1 1 3 2 4 1 1 2 3 3 2 4 1 2 3 2 2 2 2 0 0 2 2 2 2 2 0 2 2 2 2 1 3 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ... ( ) ( ) ( ) sign ( ) sign ( ) sign ... ( ) ( ) ( ) N N N t t t t t t t t t t t t t t t t t t t t t E t f d f t dt f t dt f d f t dt f t dt f t dt f t dt f t dt f t fdt f t f dt f t fdt t t t λ λ λ λ ψ ψ ψ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ = = + + = + + + + + = − + − + = − − − [ ] [ ] ( ) 1 2 4 3 1 2 3 4 2 2 2 1 2 2 1 1 0 ( ) ( ) ( ) ... ( ) 2 ( ) 2 ( ) 2 ( ) ... ( ) 2 ( ) ( ) ( ) ( ) . N t t N N p p t t t t t t t t t t t t f t dt f d ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ λ λ − ∑ ∫ ∫ + − − + = − + − + +   = − + − + +   (44) Since as , ( ) t E t → ∞ → ∞ , for N large the terms in (44) which a re not bracketed are relatively small compared to the value of the s um, and we c an write, for the larg e N , (44) as the sum of the (positive) bracketed values: 2 2 2 1 1 ( ) 2 ( ) ( ) N p p E t t t ψ ψ − ∑   ≈ −   . This yields according to (40) 2 2 2 1 1 2 ( ) lim ( ) 4 ( ) ( ) lim . ( ) ( ) a t N p p t E t T N P t t t N t P t ψ ψ →∞ − →∞ ∑ = =   −   = (45) In this expression we consider the average [ ] N ∑ ⋅ as a limited function of the generally unlimited variable N . We finally write (45) in terms of only the countable zero-crossings Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 24 2 2 2 1 1 4 ( ) ( ) lim ( ) N p p a N N t t T N P t ψ ψ − → ∞ ∑   −   = . (46) The central parameter finally is N and not t . It may be assumed that 2 0 1 lim ( ) l i m ( ) N t N N N N P P t f d t λ λ →∞ →∞ ∫ = = exists, but in any case, we can assume P ( t N ) in (46) be limited. For a T -periodic f ( t ), such that f ( t + T /2) = - f ( t ), this proc edure is very simply graphically i llustrated i n [16], and one sees t hat also f or a non-peri odic process it can be a ssumed that on average , sign [ ψ ( t 2p-1 )] = - sig n [ ψ ( t 2p )], i.e. (46) c an be simplified (notice the change in the upper bound of the sum) to 1 4 | ( ) | lim ( ) N k a N N t T N P t ψ → ∞ ∑ = . (47) In the simple periodic case of [16], we obtain from by taking N = 2 that 1 4 ( ) t T P ψ = ( ψ ( t 1 ) = - ψ ( t 2 ) > 0), which is also obtained from (47) rewritten as 4 | ( ) | ( ) k N a N t T P t ψ = (47a) with the averaging (<> N ) by N . No clock or unit of time or f requency is used. It is just required tha t some limitation the spectrum of f ( t ) to be roughly known, in order the integrator, the sampler, a nd other involved devices to work. The latter is however, a general requirement of any technical device, a nd this is very f ar f rom giving a ny uni t of measurement. This is the ba sic step of the procedure of spectrum a nalysis ba sed on the sampling of the ψ -transform at t he zero-crossings of f ( t ). This step will be applied not only to f ( t ), but also to the time proc esses obtained from f ( t ) by some sequential reductions of the spectrum of f ( t ), by mea ns of c hannel splitting. At the i nputs of the new cha nnels, Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 25 the cutoff frequencies of these reductions will be defined on-line by parameters of the type T a , f ound in the prev ious proces sing step. Since we use P for finding T a , a t each application of the ste p we have the pairs { P , T a } that a re just th e points on the shape of the power spectrum. See [18] for a schematic of suc h sequential e stimation of the compact spectrum, and see a lso [19] where a wide class of the some stochastic functions is shown to be relevant to the method. We rejected the pos sibility of numerically estimating E ( t ) beca use of a clock needed f or sampling f 2 ( t ) in Riemann's c onstruction of th e int egral. As was noted in Section 6 . 3, i f th e integration of f 2 ( t ) could be done a ccording to Lebesgue's scheme, then, in principle, we could us e E ( t ), not introducing ψ ( t ), for the determination of T a . It would be inte resting to consider technical realizations and the us e of Lebesgue's scheme. Regarding possible applica tions, one notes that since no clock is us ed, it can be assumed that not a lot of numerical calculations should be involv ed in the method, and thus the computation stage will be quickly done. The intention is to use as far a s possible, analog dev ices. Such a method c annot be precis e, but suitable for a qui ck preliminary estimation of spectrum, when the frequency range of the signal is not we ll known, which ca n m ean that the signal a ppears unexpectedly. The associated point is that it m ay be not known wheth er the i nformation c ontained in the signal is at all of interest for the receiver, and t hus a very rough understanding the signal can be sufficient for deciding whether or not to analyze the signal using more precise and costly met hods. These items can make the m ethod relevant for c ommunication between artificial intelligence systems; howe ver permi tting the side that initiates a contact by sending t he signal to define the rat e of the processing, or the basic time unit, is a gentlemanly behavior for any receiver. Consider that the very concept of a fun ction is introduced at a n early stage of our mathematical education ( in t he secondary s chool) in a discr ete manner, namely using a t able of v alues representing the function, i.e. by means of a set { f ( t k )}, and t his is always done with equidista nt points. Sometimes this rel ates to a known function, e.g., y = x 2 , that has to be drawn, and some points for the graph are needed, a nd somet imes this relates to a measurement of an initially unknown function like, say, t e mperature measured in shadow in a town, every three hours. It is important that even in the latter case one knows that f or the actual variation-rate of the parameter being measured, th e freque ncy of mea surement/sampling is sufficient, i.e. nothing essential can be mis sed. This is also the situation of communication established betwe en systems which know e ach other , and t hus choose, for working with each other, the same frequency range. However, if one does not sufficiently well kn ow the range of the unknown input function/process, but applies the same m ethod of the equidistant sampling, t hen one can ea sily have e ither an excessive sampling rat e and too slow processing, or an insufficient sampling rate a nd some information m issed. Thus, the use of equidistant sampling in the situation whe n the range of the signal is poorly know n see ms to be awkward, w hi le relaying on the co mputer's power by always taking the maximal sampling rate (a s permitted by t he devices involved such as the sam pler ) se ems to even be crude. " Sampling by defini tion", in the spirit of the Lebesgue's scheme, creates f ( t k ) f or a m ore f lexible set { t k } that adjusts itself to the rate of th e c hanges of the input. Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 26 8. Concl usions and final re marks Continuing the line of [3], the prese nt work s uggests tha t for some proble ms, presentation (1a) of a nonlinear system may be, at least heuristically, more usef ul than (1). This, of course, is not an obj ection to a ny known rese arch di rectly based on (1), and for a problem where F ( 0 , u , t ) is nonzero, (1) can be preferable. However one sees t hat even when int roduction of [A( x )] is a nalytically unnatural, the very argument regarding the nonlinearity of a sy stem to be seen via (1a) remains in force, and, i n general, t he poi nt of our out look is first of al l logical , and only the n a nalytical. That is, the question is, first of all, w hether or not the ma p x  t* exi sts , and only then what is it, while, in the pre sent c ontext, this "e xistence" is not just a forma l mathematical problem, but something directly seen (defined) by the designer. The similarity of the forms of (2) and (1a) leads to quite practical outlooks i n the main sections and to some ax iomatic poi nts that are considered in d etail in Appe ndix 2. The overall list of the main points, including those in the Appendices is as follows. 1. The structural presentation of nonlin earity allows one to spea k about a " nonlinear system" in terms of c onstructive def initions, associated with the s tate-equations. W e thus speak about " t -systems", " x -systems", and " u -systems", of which the latter t wo relate to nonlinear systems. The (similar to the l inear c ase) matrix-presentation of nonlinearity (1a) is strongly supported by the theory of switche d systems which all are, first of all, some t* (.)-systems. Form (1a) also seems t o be more relevant than (1) as regards possible generalizations of nonlinear structures. 2. For the switched systems, t* ( .), i.e. e ither f ( t ) → t* , or x ( t ) → t* , is , gene rally, a map which is given in the operational terms, not a s an a nalytical dependence or an explicit oper a tor. In the case of t* ( x ), some components of x ( t ) a re the in puts of the triggers, i.e. define the insta nts of s ingularity, a s is illustrated by Fig. 1 (let there f 1 ∈ x ). T his outlook should be clea r for designer. However, e.g. [8], gives a n example of another option to create t* ( x ). In either case, t* are some numbers, i.e. t* ( x ) i s a functional. The prese nt method of using level-crossings (i.e. the inequalities, '<', '>' between the l evels around a t k ) better stresses that t* has to be a real valued vector; see [5] for a special discussion of this point. 3. Many switche d systems can not be described at the instants of switching (s ee Appendix 1) as lumped systems, and t he radiation at these insta nces has to be considered. T he impossibility of incl uding the radiation problem int o the general theory of switched c ircuits is a serious lack of this theory. In the sense of positivistic philosophy (Appendix 2), these circuits a re not sufficie ntly well theoret ically described. 4. The numerous possibilities to introduce connections and, in particular, inputs, appearing in the field of modern electronic systems, suggest that we rec onsider our general understanding of the differential e quations. Namely, it is argued tha t " a given input-function " is not at all the sam e as " a given funct ion " appearing in a des cription of the internal system operation. The inpu t-function must be seen as belo nging to a set of functions, a nd it should be c hanged in order to make cl ear the input-output map Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 27 of the sy s tem. For instanc e, as the te st of linearity shows, if a ( t ) is the i nput of the associated system, then ( ) ( ) 0 dx a t x t dt + = is a nonlinear equation. An operator related to system description can be linear only if the functions included in it are properly fixed. A simplified formulation, r elevant to both u - and u -sy stems, mig ht be here as follows. Checking the linearity of a sys tem should not change this system; if the structure i s changed, the system is nonlinear. 5. Hydrodynamics gives a remarkable example of nonli nearity in the sense of (1a) , i.e. [A( x )]. The velocity field is both the unknown vector field to be found a nd the structure of the liquid system; thus, the [A( x )]-nonlinearity is inherent here. The nonlinear term in the Navier-Stokes equation, ( ) v v ∇   , which is ana logous to [A( x ) ] x in (1a), has the very basi c origination from the "substation derivative" ( ) d v dt t ∂ = + ∇ ∂  , in whic h the nonlinear part cannot be eliminated by movement of the observer with the flow because v  is not the same everywhere. Since thus, t he stronger the vorticity, the stronger the nonl inearity is expressed, one see s that turbulence is caused by the nonlinear term, and that the turbulence and the degree of the nonlinearity enha nce each other. Of course, linear t erms of the e quation should not ca use turbulence, but we seek direct arguments for nonlinear effects. One of the i nteresting ques tions here is whether or not the visible properties of a ( ) v ∇  -system can be helpful in understanding an [A( x )] - system. 6. Regarding the linearization of the Navier-Stokes equation w hich was us ed in Section 5.1, a nd the more general interpretation of (2) as a linearized (1a), it ne eds to be noted that in tec hnical literature diffe rent ways of linea rization ("in average", in the sense of minim al r.m.s. error , etc.) o f nonlinear characteristics and pr oblems are found. Though after Lyapunov a nd Poincare, " linearization at a poi nt" associated with stability theory (and al so closest to elementary calculus concepts), is perhaps most well known, no general definition of l ineariza tion, as a general method of coming from a nonlinear to a linear problem exists. 7. If it can be legitimate to see an ensemble of c olliding particles as a s ystem in the circuit-theory sense, then, as we argue in Section 5.2, the chaos of the mov ement of the particles results from t he system's nonl inearity which i s of the t* ( x )-ty pe. This observation touches different natural phenomena, and one can find a dditional examples of such a role of the t* ( x )-nonlinearity. Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 28 8. " Sampling by definition" is the simplest example of nonlinear sampling, and the nonlinear sampling is furthe r conside red using ψ -transform. Namely, we sam ple the nonlinear transform 2 0 ( ) ( ) [ ( )] t t f sign f d ψ λ λ λ ∫ = at the zero-crossings of the input process f ( t ) allows one to de velop a procedure for analyzing the spectrum of f ( t ) without using any clock or ti me or f requency unit of measurement. The nonlinearity by f o f the set { ψ ( t k )} used in Section 7 is a double one, -- tha t of the nonlinearity of ψ ( t ) and that of the sam pling of ψ ( t ) at the zerocrossings of f ( t ). Sinc e, as one note s, taking higher powers in the integrands of both P and ψ ( t ) (or the associated " E ") also lead s to determination of some average period, t he a nalytical component of the nonlinearity is not unique for the m ethod, but the sampling at { t k } is such. We find it relevant he re to consider the nonlinearity of the a pproximating sums used in Lebesgues' scheme of construction of t he integral, and a lso connecting this point with the elimination of a prescribed time-unit. Ca n one use t hese sums to obtain some independent nonlinear effects? Among the physical problems that might, perhaps, be relevant, the nonli nearity of the time -space metric in large-scale gravitational problems where the time axi s/scale is c onnected wit h the s patial axis/scales and integrals can appear in calculations of total masses, can be noted. 9. From the pos ition of engineering (i.e. r eliable in a pplications) system theory, "nonlinear system" should be defined (Section A2.2) through structura lly sta ble features of a system, which can be clearly observed. This rejects the definition of "nonlinear" as "not a li near one", because not pa ssing a test of linearity is not yet evidence of clear and reliable nonlinear features, a nd simply because in order to become a rea lly usef ul instrument, eac h concept has to be i ndependently de fined in physics terms. For instance, t he two states of a trigger, which denote '1' and/or '0' have to be independently physically de finable and reliably rea lizable. No enginee r would accept, -- without indepe ndently checking both these modes, -- that 'not 1' is '0', and in a more general outlook, no reason e xists for the particular case of binary logic with its however grammatically attrac tive use of the prefix " non" to be exception in the com mon rule of using in the t ables o f l ogical connect ions only signs/nota tions o f some clearly defined objects or processes. The requirement of st ructural stability is seen to be a xiomatically important and, in this spirit, the effect chosen in [4] for observation of nonlinearity (the " Small Theorem") is a surely detectable effect. 10. In close connection with item 9, our position is that { t -systems} ∩ { x -systems} = ∅ , (i.e. {LTV-systems} ∩ {NL s ystems} = ∅ ) and, in particular, f or the respective subsets, { t* ( t )-systems} ∩ { t* ( x )-systems} = ∅ , Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 29 which is because of the requirement of the struc tural stability of the nonlinear effects. Figuratively spea king, the a reas of linear and nonlinear systems do not touch eac h other. It follows, in parti cular, that L TI systems should not be interpreted as both a limiting case of { t -systems} and a limiting case of { x -systems}, when one might write { t -systems} ∩ { x -systems} = {LTI -systems} . 11. The princi ple of c onstructiveness agrees with the positions of positivistic philosophy a nd intuinitionistic logic, and with practical and pedagogical rea soning. System theory is not just a branch of applied mathematics. However even from the mathematical point of view it has the right to its own axiomatization, as the consideration of u -systems as nonlinear s ystems, the argument of structural stability and some other arguments show. Appendix 1: Some basic phys ical aspects of sw itched systems The pres ent app endix discusse s some physical specificity of switched circuits, which is importa nt for unde rstanding these circuits a nd their a pplications. W e point out one advantage and one disadvantage. A1.1. The advantage of good reliability An important advantage of electronic swit ched systems is the pos sibility to realize a strong nonlinearity w hile ha ving a highly reliable circuit. We sw itch LTI elements, and the operational range of the se elements, i.e. t he range where they are very reliable, can be m uch wider tha n the range in which the se elements are used for realizing the strongly nonlinear switching units. This is distinct f rom the use of saturated inductors, or ferroelectric ca pacitors. Though us e of analytical-nonlinearity will remai n in basic micro and nano technology, such lumped elements hardly have a good f eature in the industrial scheme-technique, and switched units are preferable there. One has to see tha t the reliability problem regarding the " analytical nonlinearity" can prevent one from obtaining a n acceptable " nonlinear circ uit", because the official industrial acceptance of the " reliability " of an element can be a problem. The following example that the aut hor met i n his pra ctice ca n be formulated a s an "industrial theorem". It is impossible to pr oduce, -- in the sense of a mass produc tion that has t o be done in a factory (not j ust in a laboratory), -- any c ircuit in w hich nonlinearity of ferroelectric capacitors is used. Indeed, though as a rule, the given (specified) voltage range of ferroelectric ceramic capacitors ([25] a nd refe rences there) i s significantly lower than the ir breakdown voltage, the specification range is defi ned, by t he producers, by the re quirement of a f airly good linearity of the capacitors. The latter is natural s i nce the m ain consumers are the electronics factories w i shing to bu y capacitors of a certain capacitance , i.e. linear capacitors. Since the producers of any element al ways give data about the el ement only in the specified range, but electronics f irm producing devices use elements only in the range for which several big producers of the elements give a specification with an associated guarantee of reliability , one cannot strongly exc eed the linearity range in Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 30 order to use the nonlinea rity of the se capacit ors in a de vice intende d for mas s production. One sees t hat the fact that a physical system does not pass a te st of linearity in laboratory does not mean that we ha ve a nonlinear " system" in the se nse of engineering, applicative system theory. The industrial reality inf luences the theoretical concepts and definitions, and students too have to be informed about that. A1.2. The radiation problem Any switching instant is a point of singularity, at which some of the components of the ve ctor d x /d t do not e xist. However, the solutions of differential equa tions are defined on ope n sets where a ll c omponents of d x /d t should exi st. Thus, at the point s of switc hing we have to "sew" the local solutions of the equations, which are obtained in the intervals of a nalyticity, using some physical conditions. The m athematical fa ct that w e cannot spe ak a bout any di fferential equations a t the i nstants of s witching ha s an interesting physical meaning; we have not , generall y, any l umped system described by Kirchhoff's equations, at these instants. Assume f irst t hat we wish to increase a capacitance by connecti ng in paral lel to a charged capacitor a capacitor having a differe nt voltage, the n the condition of conservation of electrical c harge requires part of the electrost atic energy to be "lost" in t he transient, whic h i n the case of the really quick proce ss means r adiat ion out of the c ircuit . Though lumpe d resist ors can absorb some energy, a t l east part of the losses will be immediately radi ated. This situation contradicts the known condition [15] of suff iciently la rge wavelength needed for the lumped circuit description using Kirchhoff's equations, a nd Maxwell's field equations ha ve to be used for t he analyzing of the switching process. The same situation is for switching of inductors when (e.g., because of the duality; C  L and v  i ) conservation of q = Cv is replaced by conservation of ψ = Li , and just as Cv 2 / 2 = q 2 /( 2C ) is, generally, not conserved because C is changed, also Li 2 / 2 = ψ 2 /( 2L ) is, generally, not conserved because L is changed. As a rule, the switching of resistors doe s not ca use current pulses, but such a switching can m ake a current function disc ontinuous (e.g. when the resistor through which a capacitor i s charged from a battery, is c hanged), i.e. the derivative of the current is pulsed. This also is a singularity extending the frequen cy spectrum of th e process, which can cause radiation, e ven if not so intens ive as whe n the current by itself is pulsed. It follows that switched c ircuits are "noisy" in the sense of the ra diation and the resulted e lectromagnetic interference. Though there are screening methods for reducing the e lectromagnetic inte rference, this is a di sadvantage of t hese important circuits in the sense that their theoretical description cannot be complete in principle . A simila r remark relates to power-loss minimization proble ms for such a circuit. When trying to choose the optimal v ( q ), or i ( ψ ), or v ( i ) chara cteristic of an element, which is intende d to provide the circuit currents suc h that the power losse s in the lumped resistors in the circuit would be reduced as muc h as possible, one may well increase the electromagnetic radiation by thus-optimized s witched uni ts. Though the radiation is problematic and should be reduced, its power is usually much smaller than the power saved in the resistors because of the opti mization. Thus, when the electromagnetic interference is also the concern, there is no universal energetic criterion for circuit-optimization in switched circuits. Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 31 It is interest ing to not e, however, that the radiation ca n be a good indication of switching in a system ( see [ 4]), which presents switching as a c learly registerable phenomenon. This is a side-effect, but a positive aspect of the radiation problem. One c an c ompare the s witched c ircuit w ith a quantum system, in whic h the instants of emission or absorption of a phot on separate be tween well-describable stationary states of t he system. Another comparison can be with a m echanical sys tem, as, e. g. that in Section 5.2. Appendix 2 : On th e bas ic concep t of nonlinearity a n d t he r equirement of cons tructive definitions in sys tem theory The conce pt of linearity is defined purely mathematically, and e ven when not yet deepening, a s below, on the very important aspect of c onstructiveness , but j ust requiring, for seek of justice, the concept of nonl inearity to also be independently mathematically defined, one already com es through (1a) to the notations " x -system" (or " t* ( x )-system") and " u -system". However these not ations, characterizing different nonlinear systems can/should hardly replace the v ery concept of nonlinearity , which is a " collecting concept", useful for our i ntuition , and we only c an stress here the importance of the constructive aspect, not directly presented in this concept. The stress on c onstructiveness is seen to be more necessary whe n one not es that s ystem theory needs more constructive formalizations i n some other fields as well, e.g. in the field of infinite systems. A2.1. Some comments on the definition "nonlinear is not a linear one" The "grammatical" definition, which, in its simplest form, is " nonlinear is not a linear one " (A 1) and in [26] " The c ircuit is c alled nonlinear iff it is not linear ", is non-constructive, since an unknown is defined via something not known. In order t o see what (A1) i s about, one has to know that circuits/s ystems other than linear ones exist at all. For this, some constructive examples of nonlinear systems have to be known to one before (A1). In othe r words, for (A1) to be understood, one a lready has t o know wha t "nonlinear system" is. It is hardly de sirable to c onnect th e con cepts of linear and nonlinear systems at a ll because of the very dif ferent fea tures of these sy stems (one of the conc epts does not help one, rather the oppos ite, to t hink in te rms of the other, because of the very different features of the different systems), but if one wishe s, one can use the statement: " linearity is a particular case of nonlinearity ". (A2) Indeed, one can consider (see F ig.6) in a plane, or 3D spac e, any c urve, here representing " nonlinearity ", a nd a straight l ine, here r epre senting the l inearity, a sking the quest ion of which is more c orrect, -- t o sa y that the curve is " not a s traight l ine", or that the straight line is a particular case of the curv e? Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 32 Fig.6: Which line is initial? From the comparative geometrical point of view and also f rom the positions of the variational c alculus, a st raig ht line (li nearity) is just t he particular/special case o f the curves (nonlinearity) -- that which minimizes the c urve's length between two given points. It thus seems that the conce pt of the curve is more general and need not be defined using the concept of the straight line. Quite similarly, it is obvious that (1a) is more g eneral than (2), a nd wh en [A( t , x )] and [B( t , x )] in ( 1a) a re independent of x , or bec ome known/prescribe d by s ubstituting in them a c ertain known appr oximation/estimation of x , w e obtain a linear system, as a special case of (1a). Certainly, one should not define (1a) as "not (2)". Of c ourse, the analogy betwee n the l ines in Fig. 6 and equations (1a) and (2) not only simplifies the situation. The definition of line in topology is problematic [26,27], and for one the analogy of Fig. 6 m ay even raise the ques tion of whether or not we can w rite state-equations for a very compli cated physical system. This comment, however, does not increase the importance of linear systems here, and is not practical regarding the electrical sy ste ms that ca n c ome int o the consideration. It is cl ear t hat any, not too complicated, c urve can be bui lt (drawn) without a ny definiti onal treatment. On t he psychological re gard, one notes, however, t hat if one has in sight many drawn linear pieces and only one drawn curve, then it is natural for one to call the curve "not straight li ne". T he la tter situation is the real one since one usually knows much more about l inear systems, and the concept of linear system thus natura lly becomes the " logical reference". Since we live i n wea k fields, apply wea k forces, which all lead t o li nearizations, and als o li ke sinusoids that are associated with sim ple rotational movement, it was natural and mathematically eas ier to start the science s from linear s ystems. However, this situation is changing. Because of the wide use of electronic switched systems, nonlinear systems have become v ery common, perhaps more common than linear, and this should influence t he interests and education of future enginee rs a nd circuit specialists. Acc ording to thes e arguments and the mentioned position of [1], nonlinear system-theory should be founded independently. A2.2. The requirement of structural stability of the operation of a system. Since nonlinear science is pa rtly e mpirical, even such a s pecial case wher e a system fails a test of lin earity just be cause i t is faulty , or e xploded, ca nnot be ignored i n the criticism of (A1). Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 33 Not every phy sical system/object is a "system" in the sense of circuit theory, and the problem of having a nonlinear system may be, in a particular ca se, the problem of having a " system" at all. Thus, any de finition of "nonlinear system" must be subjected to a general definition of " system" w hich would be accepte d in engineering circuit theory. Perhaps t he m ost practical reason explaining why if a sys tem does not pass a test of linearity (i.e. is "not a linear one") t his doe s not yet me an that it can be considered as "nonlinear" and operated as such , is giv en by t he basic requirement of engineering system the ory t o dea l only with those systems whos e feature s are sure ly (structura lly stably) measured. This r equirement relat es, of course, to a s ystem taken as a w hole, i.e. to i ts overall application side. A cha os gene rator c an unsta bly undergo numerous bifurcations of its inte rnal dynamics which l eads to the cha otic attractor, but t his very attractor has to be stably observed f or the system to be re liably used as the chaos generator, i.e. the statistical pr operties of the attrac tor (or chaos) should be stable. Recall also the argument of Section A1.1 associated with the problem of reliability. Since not passing a test of li nearity doe s not ens ure structural stability of any nonlinear features, one can schematically imagine the fields of linear and nonli near systems as some geometric a reas not tou ching each other. Statement (A1) require s the systems in the "separati ng area" to be clas sified a s " nonlinear systems", but the physical s ystems belonging to the sepa rating area at all should not be considered a s "systems" in the sense of the applicable system theory. A2.3. The axiomatic side and a historical view The requirement of the constructiv eness of scientific definitions is not ne w, and the "intuitionistic" logic/mathematics " by L. Brouwer [29,30] is e ven ce ntered on such a requirement. Namely, it is assumed that no " A " ca n b e prov e d/obtained by re jection of "not A ", i.e. the very common ter tium non datu r ("third not given") principle saying that the disjunction A A ∪ is complete , is not accepted in the intuitionistic logic. However, the basic point here is not that one thinks t hat there can be some thing third , not A , and not A , but the opinion/belief that rejection of one thing cannot prove the existe nce of another thi ng, -- a direct intuitively convincing example of construction of the latter thing is needed . In terms of probabilities, the c ommonly used equality { } 1 P A A = ∪ , (A3) taken by itself , is rejected if at lea st one of the symbols, A , or A , is not clearly constructively defined. In the c ontext of the definition of nonlinearity, this mea ns that it i s insufficient to know tha t the test of linearity is not pas sed in order to ha ve a construction that might be seen as a nonlinear system . Since the de velopment of c omputers through the performance of t he arithmetical operations did not require that intuitive arguments be involved, the intuitionistic logic, developed at the beginning of the past century, "lost" the "competition" against the usual logic that operates, in particular, Boolean a lgebra. However, from t he e mpirical point of view, int uitionism agre es, in its bas ic point with t he usual logic . Inde ed, the success o f t he usual logic in the field of com puters i s ba sed on th e rel iable technical realization of 1 and 0 ("trut h", "false", both constructively well-define d). If non- realization of a certain s tate of a flip-flop d id not tec hnically mean certain rea lization of another state (if not '1', then sur ely '0', and op posite), -- then engineers too would Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 34 be against the use o f the te rtium non datur principle a nd Boolean algebra. The order of the things is such t hat one f irst fills a table with the nota tions (as '1' and '0') of some existing physical states that can be reliably used , and only then, if there are only two such states, can make such conclusions as 'not 1' is '0', etc.. However successful in the applications (and simply gramma tically formulated using t he prefix "non" ), the binary case is j ust a particular one here, and nothing can be cha nged in thi s basic order/scheme. Thus, in its rejection of non-c onstructive objects or procedures, intuitionism tried to extend the same principle of the good " technical realizability" that was a dopted by conventional l ogic, to other branches of mathematics. H ow ever, being occupie d with mathematical axioms, Brouwer and, e.g., the authors of [30] c ould not express themselves in such convincing physical terms. The role of s witched sy stems in the main te xt is not occasional a lso reg arding the intuitionistic logic, because switchings are easily registerable/measurable parameters (again, the Small Theorem of [4] i s an example.) It is interesting to observe that in the history of science, switching systems a lready played important role when Shennon used, afte r almost 100 years sinc e the Boolean a lgebra appea red, this algebra f or creating his theory of switched circuits that lead to the development of computers. More about intuitionistic logic can be found in [30], and more about Brouwer's overall c ontribution to ma thematics (including the proof of t he fixed-point theorem) and personality in [31,32]. Regarding the historical background, it should be als o noted that the requireme nt of constructivism itself also c omes f rom the much ea rlier, and more widel y known than the intuitivism, "positivistic phi losophy" [29] supported, in pa rticular, b y K i rchhoff, which must have influe nced Brouwer. According to positivist philosophy, one al ways has first to learn how to well ( completely and as simply as possible ) des cribe things and only then how to e xplain them, so t hat the primary goal of science is desc ription of natural phenomena, which obviously requires constructivism. Of c ourse, cons tructiv ism is rele vant to the purel y theore tical treatments too; Kirchhoff would not be satisfied by the existing theory of switched circuits bec ause of the lack of description of the radiation at the sw itching instants, as i s consi dered in Section A1.2. The positivistic philosophy ha d been proved i mportant ea ch time when ne w sciences started a nd when a difficult field (e.g., t hermodynamics, or qua ntum mechanics) had bee n developed. The field of nonlinear systems seems to be sufficiently dif ficult f or the posit ivistic approac h to be acceptable for it, a nd even a theoretician can accept that the phy sical existence of things is more important than their theoretical explanations . Statement (A1) is very fa r from any such construct ive thesis. Emanuel Gluskin, " On the representation .." arXiv:0807.0966v2 [nlin.SI], c o rr. posted 24 Oct., 2008 . 35 Referen ces [1] L .O. 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