On the Difficulty of Deciding Asymptotic Stability of Cubic Homogeneous Vector Fields

On the Difficulty of Deciding Asymptotic Stability of Cubic Homogeneous V ector Fields Amir Ali Ahmadi Abstract — It is well-known that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decid ed in polynomial ti me e.g. by searc hing for a quadratic L ya punov function. Since homogeneo us vecto r fields of e ven degr ee ca n never be AS, the next interesting degree to consider is equal to th ree. In this paper , we prov e that decidi ng AS of h omogeneous cubic vector fields is strongly NP-hard and pose the question of determining whether it is ev en decidable. As a b yproduct of the reduction that establishes our NP- hardness result, we obtain a Ly apunov-inspired technique fo r proving positivity of forms. W e also show that for asymptotically stable homogeneous cu bic vecto r fields in as few as two variables, th e minimum degree of a polynomial Ly apunov function can be arbitrarily larg e. Finally , we show that there is no monotonicity in the degree of polynomial Ly apunov functions that prov e AS; i.e., a h omogeneous cubic vector field with no homogeneous polynomial Ly apunov fun ction of some degree d can very well hav e a homogeneous polynomial L yapunov function of degree less than d . I . I N T R O D U C T I O N A. Backgr ound W e are con cerned in this paper with a co ntinuou s time dynamica l sy stem ˙ x = f ( x ) , (1) where f : R n → R n is a polyno mial and has an eq uilibrium at the origin, i.e., f (0) = 0 . Polynomial d ifferential eq ua- tions appear ub iquitously in engineer ing and s ciences either as tr ue mo dels of physical systems, or as ap proxima tions to other families of nonlin ear dynamics. The pro blem of deciding stability of equilibriu m p oints of such systems is o f fundam ental importance in control theory . The goal of this paper is to de monstrate some of the difficulties associated with answerin g stability q uestions abou t polyno mial vector fields in terms of b oth comp utational co mplexity and non- existence of “simple” L yapunov f unctions, even if one lim its attention to very restricted settings. The notion of stab ility of in terest in this paper is (loca l or g lobal) asympto tic stability . The or igin of (1) is said to be stab le in the sense o f Lyapunov if for every ǫ > 0 , th ere exists a δ = δ ( ǫ ) > 0 such tha t || x (0) || < δ ⇒ || x ( t ) || < ǫ, ∀ t ≥ 0 . W e say that the origin is asymptotically stable (AS) if it is stable in the sense of L yapu nov and δ can be ch osen such that || x (0) || < δ ⇒ lim t →∞ x ( t ) = 0 . Amir Ali Ahmadi is with the Laboratory for Computer Science and Artificia l Intelli gence, Department of Electri cal Enginee ring and Computer Science , Massachusetts Institute of T echnology . E -mail: a a a@ mit.edu . The origin is globally a symptotically stable (GAS) if it is stable in the sense of L y apunov and ∀ x (0) ∈ R n , lim t →∞ x ( t ) = 0 . The degree of the vector field in (1) is defined to be th e largest d egree of the compo nents of f . Our focus in this paper is on homogeneous polyn omial vector fields. A scalar valued fun ction p : R n → R is said to be hom ogeneo us (of degree d ) if it satisfies p ( λx ) = λ d p ( x ) for all x ∈ R n and all λ ∈ R . A homogeneou s p olynom ial is also called a form . All monom ials of a form share the same degree. W e say that the vector field f in (1) is ho mogen eous if all com- ponen ts of f are fo rms of th e same degree. Homog eneous systems are extensi vely studied in the literatur e on non linear control; see e.g. [1], [2], [3 ], [ 4], [5] , [6], [7]. Since our results a re negative in n ature, th eir validity f or h omogen eous polyno mial systems obviously also imp lies th eir v alidity f or all poly nomial systems. A basic fact about homo geneou s vector fields is tha t for these systems the notions of local and g lobal asymptotic stability are equiv alen t. Indeed, a homoge neous vector field of d egree d satisfies f ( λx ) = λ d f ( x ) fo r any scalar λ , and therefor e the value of f on the u nit sphere deter mines its value everywhere. I t is also well-known that an asymptot- ically stable hom ogeneo us system admits a hom ogeneo us L yapunov function [8, S ec. 57], [ 6]. B. An open question o f Arnold It is natural to ask wh ether stability of equilibrium points of polyno mial vector fields can be decided in finite time. In fact, this is a well-kn own question of Arno ld that appears in [9 ]: “Is the stability prob lem fo r stationary poin ts alg orith- mically decid able? The well-k nown L yapunov theorem 1 solves the problem in th e absence of eigenv alues with zero r eal parts. In more complicated c ases, where the stability d epends on h igher o rder terms in the T ay lor series, there e x ists no algeb raic criterion. Let a vector field be given by polynomials of a fixed degree, with ratio nal coefficients. Does an algorithm exist, allowing to decide, whether the station ary poin t is stable?” T o our k nowledge, there h as b een no for mal resolutio n to this question, ne ither for the case o f stability in the sen se of L yapun ov , nor fo r the case of asymp totic stability (in its local or global version). In [10], da Costa an d Doria 1 The theorem that Arnold is refe rring to here is the indir ect method of Lya punov relat ed to lineariza tion. This is not to be confused with L yapunov’ s direct method (or the second method), which is what we are concern ed with in secti ons that follow . show th at if th e right hand side of the differential equatio n contains elemen tary functions (sine s, cosines, expon entials, absolute value fu nction, etc.), then there is no algo rithm for deciding wh ether the origin is stab le o r un stable. They also present a dynamical system in [11] wh ere on e c annot decide whether a Hopf bifu rcation will occur or whether there will be parameter v alues such that a s table fixed point becomes unstable. A relativ e ly larger nu mber of u ndecidab ility re- sults are av ailable for q uestions related to other proper ties of polyn omial vector fields, su ch as reachability [12 ] or bound edness of domain of definition [13], or f or question s about stability of hy brid systems [14], [15] , [16], [17]. W e refer the interested reader to the sur vey pap ers in [18], [1 2], [19], [20 ], [21]. W e ar e also interested to know whether the pr oblem of deciding asym ptotic stability of homogeneo us po lynom ial vector fields is un decidable for some fixed degree, say , equal to 3 . Th e answer to such d ecidability q uestions, or at least the le vel of difficulty associa ted with proving such results, can depend in a subtle w a y on the exact c riteria in question. For example, it h as b een k nown for a while th at the question of determining boundedn ess of trajectories for arbitrarily switch ed linear systems is und ecidable [1 5] even when on e restricts a ttention to switched systems defined by nonnegative matrices. On the other hand, the complexity o f testing asymptotic stability for the same class of systems remains o pen a nd in fact is conjectu red to be decidab le [22 ]. C. Existence of po lynomial Lyapunov fun ctions For stability analysis of poly nomial vecto r fields, it is most commo n ( and qu ite natural) to search for L yapunov function s that are polynomials themselves. This approac h h as become f urther prevalent over the pa st d ecade du e to the fact that techniqu es from sum of sq uares optimization [23 ] hav e provided f or algor ithms tha t giv en a polynom ial system can efficiently search for a polyno mial L yapun ov function [23], [24]. T he question is therefore naturally motiv ated to de- termine whether stable po lynomial systems always a dmit polyno mial L yapunov function s, and wh ether o ne can giv e upper bounds o n the d egree of such L yapunov fu nctions in cases wh en th ey do exist. A study o f qu estions of this type for different notion s of stability h as recen tly been car ried out in [2 5], [26], [27 ], [ 28], [2 9, Chap. 4]. In this paper , we continue this lin e o f research b y studying the case wher e th e vector field is ho mogen eous. Throu ghout this paper, by a (polyn omial) L yapunov func- tion for ( 1), we mean a positiv e definite poly nomial fu nction V whose deriv ativ es ˙ V along trajectories of (1) is negati ve definite; i.e., a function V satisfy ing V ( x ) > 0 ∀ x 6 = 0 (2) ˙ V ( x ) = h∇ V ( x ) , f ( x ) i < 0 ∀ x 6 = 0 . (3) Here, ∇ V ( x ) d enotes the gradient vector o f V , an d h ., . i is the stand ard inner pro duct in R n . If such a V is also radially u nboun ded, then the ineq ualities in (2) and (3 ) imply that the or igin of (1) is GAS. When the dynamics f is homog eneous, we can restrict our search to ho mogene ous polyno mials. Such a L yapu nov functio n is automatically radially unbo unded a nd proves (lo cal o r equivalently glo bal) asymptotic stability o f the h omog eneous vector field. Naturally , que stions regardin g comp lexity of d eciding asymptotic stab ility and question s about existence of L ya- punov functions are related. For instance, if one proves that for a class of poly nomial v ector fields, asymptotic stability implies existence of a polynomial L yapun ov function to- gether with a computable upper bound on its degree , then the question of asymptotic stability f or that class becomes decidable. Th is is due to the fact that given a po lynomial system and an integer d , the question of de ciding whether the system admits a po lynom ial L yap unov function of degree d can be answered in finite time using quantifier elimina- tion [30 ], [31]. For the c ase of linear systems ( i.e., homog eneous systems of degree 1 ), the situation is par ticularly nice. If such a system is asymptotically stable, then there always exists a qua dratic L yapun ov fun ction. Asymptotic stability of a linear system ˙ x = Ax is equiv alen t to the easily checkable algebraic criterion that the eigenv alues of A be in the open left h alf complex plane. Decidin g this prop erty of the matrix A can formally be done in p olyno mial time, e.g. by solving a L yapu nov eq uation [21] . Moving up in the degree, it is not dif ficu lt to sho w that if a homogene ous polyn omial vecto r field has even d egree, then it can never be asym ptotically stable; see e.g. [8 , p. 283]. So the next interesting ca se occur s for ho mogen eous vector fields of degree 3 . W e will prove three r esults in this paper which demonstrate that already for cubic homogen eous systems, the situation is significantly mor e complex th an it is for line ar systems. W e outline our co ntributions next. D. Contributions and organization of this paper In Section II, we p rove that determin ing asymptotic stabil- ity for ho mogen eous cubic vector fields is strongly NP-hard (Theor em 2.1). Altho ugh th is o f cou rse d oes no t resolve the question of Arnold, the result gives a lower boun d on the complexity of this pro blem. It is an interesting op en que stion to investigate whe ther in this specific setting, the problem is also un decidable. The implication of the NP-hardn ess of this problem is that unless P=NP , it is im possible to design an algor ithm that can take as in put the (ration al) coefficients of a h omogen eous cu- bic vector field, hav e r unning time b ounded by a polyn omial in the num ber of b its needed to repr esent these coefficients, and alw ay s output the correct yes/no a nswer on asym ptotic stability . M oreover , the fact that our NP-ha rdness resu lt is in the strong sense (as o pposed to weakly NP-hard problems such as KNAPSA CK, SUBSET SUM, etc.) implies that the problem rem ains NP-hard e ven if the size (bit len gth) of the coefficients is O (log n ) , where n is the dime nsion. For a strongly NP-hard p roblem, e ven a pseudo-poly nomial time algorithm cannot exist u nless P=NP . See [ 32] for precise definitions and more d etails. In Section II, w e also p resent a L yapun ov-inspired t ech- nique for p roving positi v ity of forms that comes direc tly out of the r eduction in the proof of our NP-hardn ess re sult (Corollary 2.1). W e show the potential ad vantages of th is technique ov er stand ard sum of squar es techniq ues on an example (Example 2.1). In Section III, we prove that unlike AS linear systems that always admit quadratic L yapunov functions, AS cubic ho mo- geneou s systems may need polynomial L yapu nov func tions of arbitr arily large d egree, e ven wh en the dim ension is fixed to 2 (Theo rem 3.1). Finally , in S ection IV, we show th at there is no monotonicity in the degree of homogen eous polynomia l L yapunov functions for ho mogen eous cu bic vector fields. W e gi ve an example of such a vector field which ad mits a homog eneous po lynomia l L ya punov function of de gree 4 but n ot o ne of degree 6 (Th eorem 4.1). I I . N P - H A R D N E S S O F D E C I D I N G A S Y M P T OT I C S TA B I L I T Y O F H O M O G E N E O U S C U B I C V E C T O R FI E L D S The main r esult of th is section is the following theo rem. Theor em 2.1: Deciding asymptotic stability of homog e- neous cub ic polyno mial vector fields is strongly NP-hard . The key idea behind the proof of this theor em is the following: W e will relate the solution of a com binatorial problem no t to th e b ehavior of the trajector ies of a cubic vector field that are h ard to get a handle on , but instead to properties of a L yapunov function that proves asympto tic stability of this vector field . As we will see shortly , in sights from L yapunov theory make the proof of this theorem quite simple. The re duction is br oken into two steps: ONE-IN-THREE 3SA T ↓ positivity of quartic forms ↓ asymptotic stability o f cubic vector fields A. Reduction fr om ONE-IN-THREE 3SAT to positivity of quartic forms A f orm q is said to be nonnegative or positive semidefinite if q ( x ) ≥ 0 for all x in R n . W e say that a form q is positive definite if q ( x ) > 0 for all x 6 = 0 in R n . (Note that forms necessarily vanish at the origin.) It is well-known that decid- ing non negativity of quar tic forms is NP-hard; see e.g . [33] and [34]. For r easons that will become clear shortly , we are interested instead in showing hardness of decidin g positi ve definiteness o f qu artic form s. This is in so me sense ev en easier to ac complish. A very straightforward reduc tion from 3SA T proves NP-hardness of decidin g positive definiteness of polynomials of degree 6 . By using ONE-IN-THREE 3SA T instead, we will red uce the degre e of the polynom ial f rom 6 to 4 . Pr oposition 1: It is strongly 2 NP-hard to decide wh ether a hom ogeneo us polyn omial of degree 4 is positi ve definite. Pr oof: W e give a reductio n from ONE-IN-THREE 3SA T which is known to be NP-complete [32, p. 259]. Recall that in ONE-IN-T HREE 3SA T , we ar e giv e n a 3SA T instance 2 The NP-hardness results of this section will all be in the s trong sense. From here on, we drop the prefix “strong” for bre vity . (i.e., a collection of clauses, where each clau se consists of exactly three literals, and each literal is either a v ar iable or its negation) an d we are asked to decide whether there exists a { 0 , 1 } a ssignment to th e variables that makes the expression true with the ad ditional pro perty that each clause has exactly one true liter al. T o av o id introducin g unn ecessary n otation, we presen t th e reduction on a specific in stance. The pattern will make it obvious that the g eneral construction is no dif fe rent. G i ven an instanc e of ONE-IN- THREE 3 SA T , such as the fo llowing ( x 1 ∨ ¯ x 2 ∨ x 4 ) ∧ ( ¯ x 2 ∨ ¯ x 3 ∨ x 5 ) ∧ ( ¯ x 1 ∨ x 3 ∨ ¯ x 5 ) ∧ ( x 1 ∨ x 3 ∨ x 4 ) , (4) we defin e the quartic p olynom ial p as follows: p ( x ) = P 5 i =1 x 2 i (1 − x i ) 2 +( x 1 + (1 − x 2 ) + x 4 − 1) 2 + ((1 − x 2 ) +(1 − x 3 ) + x 5 − 1) 2 +((1 − x 1 ) + x 3 + (1 − x 5 ) − 1) 2 +( x 1 + x 3 + x 4 − 1) 2 . (5) Having do ne so, our claim is that p ( x ) > 0 for all x ∈ R 5 (or gener ally for all x ∈ R n ) if and o nly if the ONE- IN- THREE 3 SA T instance is not satisfiable. Note that p is a sum of squa res and therefor e no nnegative. The only possible locations for ze ros o f p a re b y con struction amon g the points in { 0 , 1 } 5 . If th ere is a satisfy ing Boolean assignme nt x to (4) with exactly one true litera l per clause, then p will vanish a t poin t x . Conversely , if ther e are n o such satisfyin g assignments, then for any point in { 0 , 1 } 5 , at least one of the term s in (5) will b e positive and hence p will have no zeros. It remains to make p ho mogen eous. Th is c an b e done v ia introdu cing a ne w scalar variable y . I f we let p h ( x, y ) = y 4 p ( x y ) , (6) then we claim that p h (which is a quartic form) is po siti ve definite if and only if p constructed as in (5) has no zeros. 3 Indeed , if p has a zer o at a p oint x , th en that zero is inher ited by p h at th e point ( x, 1 ) . If p has n o zer os, then (6) shows that p h can only possibly hav e zeros at points with y = 0 . Howe ver, from the stru cture of p in (5) we see that p h ( x, 0) = x 4 1 + · · · + x 4 5 , which cann ot be z ero (except at the origin). This con cludes the pr oof. B. Reduction fr om positivity of quartic forms to asymp totic stability of cu bic vector fields W e n ow present the secon d step of the red uction and finish the pr oof of The orem 2.1. 3 In gene ral, the homogenizat ion operati on in (6) doe s not preserve positi vity . For example, as shown in [35 ], the polynomial x 2 1 + (1 − x 1 x 2 ) 2 has no zeros, but its homogenizat ion x 2 1 y 2 + ( y 2 − x 1 x 2 ) 2 has zeros at the points ( 1 , 0 , 0) T and (0 , 1 , 0) T . Ne vert heless, posit i vity is preserv ed under homogeniz ation for the special class of polynomials constructe d in this redu ction, essential ly because polynomia ls of type (5 ) ha ve no zeros at infinity . Pr oof: [Proof o f T heorem 2. 1] W e give a r eduction from the pro blem of decidin g p ositiv e definiten ess o f qua rtic forms, whose NP-hardness was established in Proposition 1. Giv e n a quartic f orm V : = V ( x ) , we d efine the polynomial vector field ˙ x = −∇ V ( x ) . (7) Note that the vector field is homogeneo us of degree 3 . W e claim that the ab ove v ector field is (locally or equi valently globally) a symptotically stable if and only if V is positi ve definite. First, w e obser ve th at by constru ction ˙ V ( x ) = h∇ V ( x ) , ˙ x i = −||∇ V ( x ) || 2 ≤ 0 . (8) Suppose V is positi ve defin ite. By Euler’ s identity for ho - mogene ous function s, 4 we have V ( x ) = 1 4 x T ∇ V ( x ) . There - fore, positive defin iteness of V implies that ∇ V ( x ) can not vanish anywhere except at the origin. Hence, ˙ V ( x ) < 0 for all x 6 = 0 . In view of L y apunov’ s t heorem (see e.g. [36, p. 124]) , and the alr eady mention ed fact that a positiv e d efinite homog eneous function is radially un bound ed, it follows that the system in (7) is glob ally asymptotically stable. For the converse d irection, su ppose ( 7) is GAS. Our first claim is that globa l asymptotic stability together with ˙ V ( x ) ≤ 0 imp lies that V must be po siti ve semidefin ite. This fo llows fro m th e following simple argum ent, which we have also previously presented i n [37] for a different purpose. Suppose for the sake of con tradiction that for some ˆ x ∈ R n and some ǫ > 0 , we had V ( ˆ x ) = − ǫ < 0 . Consider a trajectory x ( t ; ˆ x ) of system (7) that starts at initial cond ition ˆ x , and let us ev alu ate the function V on th is trajec tory . Since V ( ˆ x ) = − ǫ and ˙ V ( x ) ≤ 0 , we hav e V ( x ( t ; ˆ x )) ≤ − ǫ fo r all t > 0 . Howe ver, th is contrad icts the fact that by glo bal asymptotic stability , th e traje ctory must g o to the origin, where V , being a form, vanishes. T o prove that V is positive d efinite, suppose by con- tradiction that for some nonzer o po int x ∗ ∈ R n we had V ( x ∗ ) = 0 . Since we just proved that V h as to be positi ve semidefinite, the p oint x ∗ must be a global minimum of V . Therefo re, as a necessary condition of optimality , we should have ∇ V ( x ∗ ) = 0 . But this con tradicts the sy stem in (7) being GAS, since the trajecto ry star ting at x ∗ stays there forever and can ne ver g o to th e origin. Perhaps of ind ependen t in terest, the red uction we just gave suggests a method for proving positive definiteness of for ms. Given a form V , we can c onstruct a dynamica l system as in (7), an d then any method that we may have for provin g stability of vector fields (e.g . the use of various kinds o f L yapun ov functions) can s erve as an algorithm fo r proving p ositivity of V . In p articular, if we use a poly nomial L yapunov func tion W to prove stability of the system in ( 7), we get the following cor ollary . Cor ollary 2 .1: Let V and W be two f orms of possibly different degree. If W is po siti ve defin ite, a nd h∇ W, ∇ V i is positive defin ite, then V is p ositiv e definite. 4 Euler’ s identity is easil y deriv ed by dif ferenti ating both sides of the equati on V ( λx ) = λ d V ( x ) with respec t to λ and setting λ = 1 . A p olyno mial p is said to be a sum of squar es (sos) if it can be written as p = P m i =1 q 2 i for some polynomials q i . An sos poly nomial is clearly non negativ e. Moreover, un like the proper ty of nonnegativity that is NP-hard to check, existence of a n sos decomp osition can be cast a s a semidefinite progr am [38], wh ich can be solved efficiently . Howe ver , n ot ev ery nonnegative p olynom ial is a sum of squar es. An inter esting fact a bout Corollar y 2 .1 is that its algebraic version with sum of squares replaced for positivity is not true. In other word s, w e can have W sos ( and positi ve definite), h∇ W , ∇ V i sos (an d positive defin ite), but V not sos. This giv es us a way of proving po siti vity of s ome polyn omials th at are not so s, using only sos certificates. Given a form V , since the expression h∇ W, ∇ V i is linear in th e coefficients of W , we can use semidefinite programming to search for a form W that satisfies W sos an d h∇ W , ∇ V i sos, and this would prove positivity of V . The fo llowing example demonstrates the po tential usefulness of this approach. Example 2. 1: Consider the following form of degree 6 : V ( x ) = x 4 1 x 2 2 + x 2 1 x 4 2 − 3 x 2 1 x 2 2 x 2 3 + x 6 3 + 1 250 ( x 2 1 + x 2 2 + x 2 3 ) 3 . (9) One can check th at this po lynomial is not a sum of squares. (In fact, this is the celeb rated Motzkin form [3 9] slightly per- turbed. ) On the other h and, we can use the software p ackage Y ALMIP [40 ] togeth er with the SDP solver SeDuMi [41] to search for a form W satisfying W sos h∇ W , ∇ V i sos. (10) If we param eterize W as a quadr atic form , no fe asible solution will be re turned form the so lver . However , when we increase the degree o f W from 2 to 4 , the solver returns the fo llowing polyn omial W ( x ) = 9 x 4 2 + 9 x 4 1 − 6 x 2 1 x 2 2 + 6 x 2 1 x 2 3 + 6 x 2 2 x 2 3 + 3 x 4 3 − x 3 1 x 2 − x 1 x 3 2 − x 3 1 x 3 − 3 x 2 1 x 2 x 3 − 3 x 1 x 2 2 x 3 − x 3 2 x 3 − 4 x 1 x 2 x 2 3 − x 1 x 3 3 − x 2 x 3 3 that satisfies b oth sos constraints in (10). One can easily infer from the sos d ecompo sitions (e.g. by ch ecking positiv e definiteness of th e associated “Gram matrices”) that th e forms W and h∇ W, ∇ V i are positive definite. Hence, by Corollary 2.1, we h av e a proof that V in ( 9) is positive definite. △ Interestingly , ap proach es of this typ e tha t use gr adient informa tion for proving po siti vity of po lynomials with sum of squares techniqu es have b een studied by Nie, Demmel, and Sturmfe ls in [42 ], though the derivation ther e is not inspired by L yap unov theory . I I I . N O N - E X I S T E N C E O F A U N I F O R M B O U N D O N T H E D E G R E E O F P O LY N O M I A L L Y A P U N OV F U N C T I O N S I N FI X E D D I M E N S I O N A N D D E G R E E For polynomial vector field s in gen eral, existence of a polyno mial L yapunov fun ction is not necessary for global asymptotic stability . In jo int work with M. Krstic and P .A. Parrilo [27], we recen tly g av e a rema rkably simp le example of a (n on-ho mogen eous) qu adratic p olynom ial vector field in two variables tha t is GAS b u t does not admit a polyn omial L yapunov function (of any degree). An independen t e arlier example that app ears in a book by Bacciotti an d Rosier [43, Prop. 5.2] was brought to ou r attention after o ur work w as submitted. W e refer the reader to [2 7] for a d iscussion on the d ifferences between the two examp les, the main on e being that the example in [43] doe s n ot admit a po lynomial L yapunov function ev en locally but unlike the example in [2 7] relies on u sing irration al coefficients. The situation fo r homo geneou s po lynom ial vector fields, howe ver, seems to be different. W e conjecture that for such systems, existence of a ho mogeneo us polyn omial L yapunov function is necessary and su fficient for (global) asym ptotic stability . Th e reaso n f or th is co njecture is that we expect that one should be able to approximate a continu ously differen- tiable L yapunov fu nction with a polynomial one o n the unit sphere, which b y homog eneity should be enough to imply the L y apunov inequalities ev erywhere. A form al treatment of this idea is left for futur e work . He re, we build o n the result in [4 3, Prop. 5.2] to prove th at the minim um degree of a po lynomial L yapunov function for an AS homog eneous vector field can b e arbitrar ily large ev e n when the degree and dimension are fixed resp ectiv ely to 3 and 2 . Pr oposition 2 ( [43, Pr op . 5.2–a ]): Con sider the vector field ˙ x = − 2 λy ( x 2 + y 2 ) − 2 y (2 x 2 + y 2 ) ˙ y = 4 λx ( x 2 + y 2 ) + 2 x (2 x 2 + y 2 ) (11) parameteriz ed by the scalar λ > 0 . For all v alu es of λ th e origin is a center for ( 11), b u t f or any irration al value of λ there exists no polyn omial function V satisfying ˙ V ( x, y ) = ∂ V ∂ x ˙ x + ∂ V ∂ y ˙ y = 0 . Theor em 3.1: Let λ be a po siti ve irra tional real nu mber and consider the follo wing h omog eneous cub ic v ector field parameteriz ed b y the scalar θ :  ˙ x ˙ y  =  cos( θ ) − si n( θ ) sin( θ ) cos( θ )   − 2 λy ( x 2 + y 2 ) − 2 y (2 x 2 + y 2 ) 4 λx ( x 2 + y 2 ) + 2 x (2 x 2 + y 2 )  . (12) Then for any even degree d o f a candid ate polyno mial L yapunov function, there exits a θ > 0 small enough such that the vector field in (12) is asymptotically stable but do es not admit a polynomial L yapun ov function of de gree ≤ d . Pr oof: Consider the (non -polyn omial) positive definite L yapunov function V ( x, y ) = (2 x 2 + y 2 ) λ ( x 2 + y 2 ) whose derivati ve along the trajecto ries of (12) is equal to ˙ V ( x, y ) = − sin( θ )(2 x 2 + y 2 ) λ − 1 ( ˙ x 2 + ˙ y 2 ) . Since ˙ V is negative definite fo r 0 < θ < π , it follows that for θ in th is rang e, the orig in of ( 12) is asympto tically stable. T o establish the claim in the theorem, su ppose for the sake o f con tradiction th at there exists an upper bo und ¯ d such that for all 0 < θ < π the system admits a (homogeneou s) polyno mial L yapunov function of degree d ( θ ) with d ( θ ) ≤ ¯ d . Let ˆ d be the lea st comm on multiplier of the degrees d ( θ ) for 0 < θ < π . (Note that d ( θ ) can at most range over all even p ositiv e integers less than or equ al to ¯ d .) Sin ce positive po wers of L ya punov function s are valid L yapu nov function s, it fo llows that for e very 0 < θ < π , the system admits a ho mogen eous po lynomia l L yap unov function W θ of degree ˆ d . By rescaling, we can assume withou t loss of generality that all L yapunov functions W θ have unit area o n the un it sphere. Let u s now consider the seq uence { W θ } as θ → 0 . W e th ink o f this sequ ence as residing in a comp act subset of R ( ˆ d +1 ˆ d ) associated with the set P 2 , ˆ d of (coefficients of) all no nnegative biv ariate h omog eneous polyn omials of degree ˆ d with un it area o n the u nit sphere. Sin ce e very bound ed seq uence has a co n verging su bsequenc e, it follo ws that there must exist a subsequence of { W θ } that conv e rges (in the coefficient sense) to som e polynomial W 0 belongin g to P 2 , ˆ d . Since con vergence of this subsequence also implies conv ergence of the as sociated g radient vectors, we get tha t ˙ W 0 ( x, y ) = ∂ W 0 ∂ x ˙ x + ∂ W 0 ∂ y ˙ y ≤ 0 . On the other hand , wh en θ = 0 , the vector field in (12) is the same as the one in (11) an d hen ce the trajectories starting fro m any no nzero initial condition go on periodic orbits. This howe ver implies th at ˙ W = 0 e verywhere an d in view o f Prop osition 2 we h av e a con tradiction. Remark 3.1: Unlike the result in [43, Pro p. 5.2] , it is easy to establish th e result of Theor em 3.1 without ha ving to use irrational co efficients in the vector field. One appr oach is to take an irr ational number, e.g . π , and th en think of a sequence of vector fields given by (12) th at is param eterized by both θ and λ . W e let th e k -th vector field in th e sequence ha ve θ k = 1 k and λ k equal to a rational num ber representin g π up to k decimal digits. Since in the limit as k → ∞ we have θ k → 0 and λ k → π , it sh ould be clear fr om th e pr oof of Theorem 3.1 that for any integer d , there exists an AS biv ar iate homo geneou s cu bic vector field with rational coefficients that does n ot have a poly nomial L yapunov function of d egree less than d . I V . L A C K O F M O N OT O N I C I T Y I N T H E D E G R E E O F P O LY N O M I A L L Y A P U N OV F U N C T I O N S If a dyn amical system admits a quadra tic L yapunov fun c- tion V , then it clearly a lso ad mits a polyn omial L yapunov function of any high er even degree (e.g. simply gi ven by V k for k = 2 , 3 , . . . ). Howe ver , our next theorem shows that for h omogen eous systems that d o n ot ad mit a qu adratic L yapunov func tion, such a mon otonicity pr operty in the degree of polyn omial L yapunov f unctions m ay not ho ld. Theor em 4.1: Consider the following hom ogeneo us cubic vector field param eterized by the scalar θ :  ˙ x ˙ y  =  − sin( θ ) cos( θ ) − co s( θ ) − sin( θ )   x 3 y 3  . ( 13) There exists a r ange of values fo r the p arameter θ > 0 for whic h the vector field is asym ptotically stable, h as no homog eneous p olynom ial L yapun ov fun ction of degree 6 , but adm its a h omog eneous polynom ial L yapun ov function of degree 4 . Pr oof: Consider the positiv e definite L yapun ov fu nction V ( x, y ) = x 4 + y 4 . (14) The derivati ve of this L yap unov f unction is given by ˙ V ( x, y ) = − 4 sin( θ )( x 6 + y 6 ) , which is negative definite fo r 0 < θ < π . T herefor e, when θ belongs to this ra nge, the origin of (12) is a symptotically stable an d th e system adm its the d egree 4 L yapu nov function giv en in ( 14). On the other h and, we claim that for θ small enoug h, the system can not adm it a degree 6 (h omogen eous) polyno mial L y apunov function . T o argue by con tradiction, we su ppose th at f or ar bitrarily small and positiv e values o f θ the system ad mits sextic L yapun ov fun ctions W θ . Since the vector field satisfies th e symmetry  ˙ x ( y , − x ) ˙ y ( y , − x )  =  0 1 − 1 0   ˙ x ˙ y  , we can assume th at the L yap unov fun ctions W θ satisfy the symmetry W θ ( y , − x ) = W θ ( x, y ) . 5 This means that W θ can be param eterized with no od d mono mials, i.e., in the form W θ ( x, y ) = c 1 x 6 + c 2 x 2 y 4 + c 3 x 4 y 2 + c 4 y 6 , where it is un derstood that the coe fficients c 1 , . . . , c 4 are a fun ction of θ . Since by our assumption ˙ W θ is n egati ve definite for θ arbitrarily s mall, an argumen t identical to the one used in th e p roof of Theo rem 3. 1 implies that as θ → 0 , W θ conv erges to a nonzero sextic hom ogeneo us p olynom ial W 0 whose derivati ve ˙ W 0 along the trajectories of (1 3) (with θ = 0 ) is non -positive. Howe ver , note that when θ = 0 , th e trajectories of (13) go on perio dic orb its tracing the level sets of the function x 4 + y 4 . This implies that ˙ W 0 = ∂ W 0 ∂ x y 3 + ∂ W 0 ∂ y ( − x 3 ) = 0 . If we write out this e quation, we ob tain ˙ W 0 = (6 c 1 − 4 c 2 ) x 5 y 3 + 2 c 2 xy 7 − 2 c 3 x 7 y + (4 c 3 − 6 c 4 ) x 3 y 5 = 0 , which implies that c 1 = c 2 = c 3 = c 4 = 0 , hence a contradictio n. Remark 4.1: W e h ave numerically computed the range 0 < θ < 0 . 0267 , for which the conclusion of Theorem 4.1 holds. This bound has been comp uted via sum o f squar es relaxation an d semidefinite program ming (SDP) by using the SDP solver SeDuMi [41]. What allo ws th e sear ch fo r a L yapunov function for the vector field in (13) to be exactly cast as a sem idefinite prog ram is the fact that all nonnegative biv ariate for ms are sum s of squ ares. V . 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