On a Hamiltonian version of a 3D Lotka-Volterra system

On a Hamiltonian v ersion of a 3D Lotk a-V olterra system R˘ azv an M. T udor an and Anania G ˆ ırban Abstract In this paper we present some r elev an t dynamical prop erties of a 3D Lotk a- V olterra system from the P oisson dynamics p oin t of vie w. AMS 2000 : 70H05; 3 7 J25; 37J35. Keyw ords : Hamiltonian dynamics, Lotk a-V olterra syste m, stabilit y of equilibria, P o incar ´ e compactification, energy-Casimir mapping. 1 In tr o duction The Lotk a-V o lt erra system has b ee n widely in v estigated in the la st y ears. This system, studied b y Ma y a nd Leonard [ 8 ], mo dels the evolution o f comp etition b et w een three sp ecies. Among the studied to pics related with the Lotk a- V olterra system, w e recall a few of them together with a partial list of references, namely: integrals and in v ar ia n t manifolds ([ 2 ], [ 4 ]), stabilit y ([ 7 ], [ 2 ]), analytic b ehavior [ 3 ], nonlinear a nalysis [ 8 ], and man y o thers. In this pap er w e consider a sp ecial case of the Lo tk a-V olterra system, recen tly in- tro duced in [ 4 ]. W e write the system as a Hamiltonian system of P oisson t ype in order to analyze the system from the P oisson dynamics p oint of view. More exactly , in the second section of this paper, w e prepare the framew ork of our study b y writing the Lotk a- V olterra system as a Hamilton- P oisson system, and also find a S L (2 , R ) parameterized family of Ha milton-P oisson realizations. As consequence of the Hamiltonian setting w e obtain t w o new first in tegrals of the Lotk a-V o lterra system t ha t generates the first in- tegrals o f this system found in [ 4 ]. In the t hird section of the pap er w e determine the equilibria o f the Lotk a-V olterra system a nd then a na lyze their Ly apuno v stability . The fourth section is dedicated to the study of the Poincar ´ e compactification of the Lotk a- V olterra system. Mor e exactly , we in tegrate explicitly the P oincar ´ e compactification o f the Lotk a- V olterra system. In the fifth section of the article w e presen t some conv exit y prop erties of the image of the energy-Casimir ma pping and define some naturally as- so ciated semialgebraic splittings of the ima g e. More precisely , we discuss the relation b et w een the imag e thro ugh t he energy-Casimir mapping of the families of equilibria of the Lo t k a-V olterra system and the canonical Whitney stratifications of the semialgebraic splittings of the image of the energy-Casimir mapping. In the sixth part of the pap er we 1 giv e a top ological c lassification o f the fib ers of the energy-Casimir mapping, classific ation that follow s naturally from the stratifications in tro duced in the ab ov e section. Note that in our approac h we consider fib ers o v er the regular and a lso ov er the singular v alues of the energy-Casimir mapping. In the last part of the article w e giv e tw o Lax form ulations of the system. F o r details o n P o isson geometry and Ha milto nian dynamics see e.g. [ 6 ], [ 9 ], [ 1 ], [ 11 ]. 2 Hamilton-P o isson realizati o ns o f a 3D Lotk a-V olterra system The Lo tk a-V olterra system w e consider for our study , is gov erned by the equations:    ˙ x = − x ( x − y − z ) ˙ y = − y ( − x + y − z ) ˙ z = − z ( − x − y + z ) . (2.1) Note that t he ab o v e system is the Lotk a-V olterra system studied in [ 4 ] in the case a = b = − 1. In [ 4 ] it is show n tha t this system admits the follo wing p olynomial conserv at ion la ws: f ( x, y , z ) = xy z ( x − y )( x − z )( y − z ) , g ( x, y , z ) = x 2 y 2 − x 2 y z − xy 2 z + x 2 z 2 − xy z 2 + y 2 z 2 . Using Ha miltonian setting of the problem, w e provide tw o degree-t w o p olynomial conserv a tion laws o f t he system ( 2.1 ) whic h generates f and g . Thes e conserv ation law s will b e repres en ted b y the Hamiltonian and resp ectiv ely a Casimir function of the Pois son configuration manifold of the system ( 2.1 ). As the purp ose of this pap er is to study the ab o v e system from the P oisson dynamics p oin t of view, the first step in this approach is to g ive a Ha milton-P oisson realization of the system. Theorem 2.1 The dynamic s ( 2.1 ) has the fol lowing Hamilton-Poisso n r e alization: ( R 3 , Π C , H ) (2.2) wher e, Π C ( x, y , z ) =   0 y x − z − y 0 − y − x + z y 0   is the Poisson structur e gene r ate d by the smo oth function C ( x, y , z ) := − xy + y z , and the Hamiltonian H ∈ C ∞ ( R 3 , R ) is given by H ( x, y , z ) := xy − xz . Note that, by Poisson structur e gener a te d by the smo oth function C , we me an the Poisson structur e gener ate d by the Poisson br ac k et { f , g } := ∇ C · ( ∇ f × ∇ g ) , fo r any smo oth functions f , g ∈ C ∞ ( R 3 , R ) . 2 Pro of. Indeed, we ha v e successiv ely: Π C ( x, y , z ) ·∇ H ( x, y , z ) =   0 y x − z − y 0 − y − x + z y 0   ·   y − z x − x   =   − x ( x − y − z ) − y ( − x + y − z ) − z ( − x − y + z )   =   ˙ x ˙ y ˙ z   , as required. Remark 2.2 Sinc e the s ignatur e of the quadr atic fo rm g ener ate d by C ( x, y , z ) = − xy + y z is ( − 1 , 0 , +1) , the triple ( R 3 , Π C , H ) it is isomorphic with a Lie-Poisson r e alization of the L otka-V ol terr a system ( 2.1 ) on the dual o f the semi d ir e ct pr o duct b etwe en the Lie algebr a so (1 , 1) and R 2 . Remark 2.3 By definition we ha v e that the c enter of the Poisson a lgebr a C ∞ ( R 3 , R ) is gener ate d by the Casimir invariant C ( x, y , z ) = − xy + y z . Remark 2.4 The c o nservation laws f and g found in [ 4 ], c an b e written in terms of the Casimir C and r esp e ctively the Hamil toni a n H as fol lows: f = C H ( C + H ) , g = 1 2  C 2 + H 2 + ( C + H ) 2  . Next prop osition giv es others Hamilton-P oisson realizations of the Lo tk a-V olterra system ( 2.1 ). Prop osition 2.5 The dynamics ( 2.1 ) admits a famil y of Hamilton-Pois s o n r e alizations p ar ameterize d b y the gr oup S L (2 , R ) . Mor e exactly, ( R 3 , {· , ·} a,b , H c,d ) is a Hamilton- Poisson r e alization of the dynamics ( 2.1 ) wher e the br acket {· , ·} a,b is defi n e d by { f , g } a,b := ∇ C a,b · ( ∇ f × ∇ g ) , for any f , g ∈ C ∞ ( R 3 , R ) , and the functions C a,b and H c,d ar e given by: C a,b ( x, y , z ) = ( − a + b ) xy + ay z − bxz , H c,d ( x, y , z ) = ( − c + d ) xy + cy z − dxz , r esp e ctivel y, the m atrix of c o efficients a, b, c, d is  a b c d  ∈ S L (2 , R ) . Pro of. The conclusion follows directly taking into accoun t that the matrix form ulation of the Pois son brac k et { · , · } a,b is give n in co ordinates by: Π a,b ( x, y , z ) =   0 − bx + ay ( a − b ) x − az bx − ay 0 ( − a + b ) y − bz ( − a + b ) x + az ( a − b ) y + bz 0   . 3 3 Stabilit y of equilibri a In this short section we analyze the stabilit y prop erties of the equilibrium states o f the Lotk a-V olterra system ( 2.1 ). Remark 3.1 The e quilibrium states of the system ( 2.1 ) ar e given as the union of the fol lowing thr e e families: E 1 := { (0 , M , M ) : M ∈ R } , E 2 := { ( M , 0 , M ) : M ∈ R } , E 3 := { ( M , M , 0) : M ∈ R } . Figure 1 presen ts the ab o v e defined families of equilibrium states of the Lotk a- V olterra system. Figure 1: Equilibria of the L otka-V olterr a system ( 2.1 ) . In the following theorem w e describe the stabilit y prop erties of the equilibrium states of the system ( 2.1 ). Theorem 3.2 Al l the e quilibrium states of the L otka-V olterr a system ( 2.1 ) ar e unstable. Pro of. The conclusion follows from the fa ct tha t the c haracteristic p olynomial asso ci- ated with the linear part of the system ev a luated at an arbitrary equilibrium state, is the same for an y of the fa milies E 1 , E 2 , E 3 , a nd is giv en b y: p ( λ ) = (2 M − λ ) λ (2 M + λ ) . F or M = 0, w e get the origin (0 , 0 , 0) whic h is a lso unstable since in a ny arbitrary small op en neighborho od around, there exists unstable equilibrium states. 4 4 The b eha vi o r on the sphere at infi n it y In this section we in tegrate explicitly the P oincar ´ e compactification of the Lotk a-V olterra system, and conseq uen t ly the Lotk a-V olterra system ( 2.1 ) (on t he sphere) at infinity . Recall that using the P oincar ´ e compactification of R 3 , the infinity of R 3 is represen ted b y the sphere S 2 - the equator of the unit sphere S 3 in R 4 . F or details regarding the P o incar ´ e compactification of p o lynomial v ector fields in R 3 see [ 5 ]. Fixing the notations in a ccordance with t he results stated in [ 5 ] w e write the Lotk a - V olterra system ( 2.1 ) as    ˙ x = P 1 ( x, y , z ) ˙ y = P 2 ( x, y , z ) ˙ z = P 3 ( x, y , z ) , with P 1 ( x, y , z ) = − x ( x − y − z ), P 2 ( x, y , z ) = − y ( − x + y − z ), P 3 ( x, y , z ) = − z ( − x − y + z ). Let us now study the P oincar ´ e compactification of the Lotk a-V olterra system in the lo cal c ha r ts U i and V i , i ∈ { 1 , 2 , 3 } , of the manifold S 3 . The P o incar´ e compactification ( p ( X ) in the notations from [ 5 ]) of the Lotk a-V olterra system ( 2.1 ) is the same for each of the lo cal c hart s U 1 , U 2 and resp ectiv ely U 3 and is giv en in the corresp onding lo cal co ordinates b y    ˙ z 1 = 2 z 1 (1 − z 1 ) ˙ z 2 = 2 z 2 (1 − z 2 ) ˙ z 3 = − z 3 ( z 1 + z 2 + 1) . (4.1) Regarding the Poincar ´ e compactification o f the Lotk a-V olterra system in the lo cal c ha rts V 1 , V 2 and resp ective ly V 3 , a s a prop ert y of the compactification pro cedure, the compactified v ector field p ( X ) in the lo cal c hart V i coincides with the v ector field p ( X ) in U i m ultiplied b y the factor − 1, for eac h i ∈ { 1 , 2 , 3 } . Hence, for eac h i ∈ { 1 , 2 , 3 } , the flo w of the system ( 4.1 ) on the lo cal chart V i is the same as the flo w on the lo cal chart U i rev ersing the time. The system ( 4.1 ) is in tegrable with the solution giv en b y              z 1 ( t ) = e 2 t e 2 t + e k 1 z 2 ( t ) = e 2 t e 2 t + e k 2 z 3 ( t ) = e t k 3 √ e 2 t + e k 1 √ e 2 t + e k 2 , where k 1 , k 2 , k 3 are arbitrary real constan ts. T o analyze the Lotk a- V olterra system on the sphere S 2 at infinit y , note that the p oints on the sphere at infinity are c ha r a cterized b y z 3 = 0. As the plane z 1 z 2 is inv ariant under the flow o f the system ( 4.1 ), the compactified Lotk a-V o lterra system on t he lo cal ch arts U i ( i ∈ { 1 , 2 , 3 } ) on the infinity sphere reduces to  ˙ z 1 = 2 z 1 (1 − z 1 ) ˙ z 2 = 2 z 2 (1 − z 2 ) . (4.2) 5 The system ( 4.2 ) is inte grable and the solution is given b y      z 1 ( t ) = e 2 t e 2 t + e k 1 z 2 ( t ) = e 2 t e 2 t + e k 2 , where k 1 , k 2 are arbitrary real constan ts. Consequen tly , the phase p ortrait on t he lo cal c harts U i ( i ∈ { 1 , 2 , 3 } ) o n the infinity sphere is given in Fig ure 2 . Figure 2: Phase p ortr ait of the system ( 4.2 ) . 5 The image of the energy-Cas i mir mapping The aim of this section is to study the image of the energy-Casimir mapping I m ( E C ), asso ciated with the Hamilton- Poisson realization ( 2.2 ) of the Lotk a-V olterra syste m ( 2.1 ). W e consider con v exit y prop erties of the image of E C , as w ell as a semialgebraic split- ting of the image that agree with the top olog y of the symplectic leav es of the P oisson manifold ( R 3 , Π C ). Recall that b y a semialgebraic splitting, we mean a splitting consist- ing of semialgebraic manifolds, namely manifolds that ar e describ ed in co ordinates by a set o f p olynomial inequalities and equalities. F or details on semialgebraic manifolds and their geometry see e.g. [ 10 ]. All these will b e used later on, in order to obtain a top ological classification o f the orbits o f ( 2.1 ). Recall first that t he energy-Casimir mapping, E C ∈ C ∞ ( R 3 , R 2 ) is giv en by: E C ( x, y , z ) = ( H ( x, y , z ) , C ( x, y , z )) , ( x, y , z ) ∈ R 3 6 where H , C ∈ C ∞ ( R 3 , R ) are the Hamiltonian of the system ( 2.1 ), and resp ectiv ely the Casimir of the P o isson manifo ld ( R 3 , Π C ), b oth of them as considered in Theorem 2.1 . Next prop osition explicitly giv es the semialgebraic splitting of the image of the energy-Casimir map E C . Prop osition 5.1 The image of the ener gy-Casim i r map - R 2 - admits the fol lowing splitting: I m ( E C ) = S c> 0 ∪ S c =0 ∪ S c< 0 , wher e the subsets S c> 0 , S c =0 , S c< 0 ⊂ R 2 ar e splitting further o n a union of sem i a lgebr aic manifolds, as fol lows: S c> 0 = { ( h, c ) ∈ R 2 : h < 0; c > 0 } ∪ { ( h, c ) ∈ R 2 : h = 0; c > 0 } ∪ { ( h, c ) ∈ R 2 : h > 0; c > 0 } , S c =0 = { ( h, c ) ∈ R 2 : h < 0; c = 0 } ∪ { ( h, c ) ∈ R 2 : h = c = 0 } ∪ { ( h, c ) ∈ R 2 : h > 0; c = 0 } , S c< 0 = { ( h, c ) ∈ R 2 : c < min {− h, 0 }} ∪ { ( h, c ) ∈ R 2 : c < 0; c = − h } ∪ { ( h, c ) ∈ R 2 : − h < c < 0 } . Pro of. The conclusion follows directly by simple algebraic computation using t he defi- nition o f the energy-Casimir mapping. Remark 5.2 The sup erscripts use d to deno te the sets S , ar e in agr e ement with the top olo gy of the symple ctic le aves of the Poisson manifo ld ( R 3 , Π C ) , nam ely: (i) F or c 6 = 0 , Γ c = { ( x, y , z ) ∈ R 3 : y ( z − x ) = c } is a hyp erb oli c cylinder. (ii) F o r c = 0 , Γ c = { ( x, y , z ) ∈ R 3 : y ( z − x ) = c } is a union of two in terse cting planes. The connection b etw een the semialgebraic splittings of the image I m ( E C ) giv en by Prop osition 5.1 , and the equilibrium states of the Lotk a- V olterra syste m, is giv en in the follo wing remark. Remark 5.3 The sem i a lgebr aic splitting of the sets S is describ e d in terms of the ima ge of e quilibria of the L otka-V olterr a system thr ough the map E C as fol lows: (i) S c> 0 = Σ ← 1 ∪ I m ( E C | E ⋆ 1 ) ∪ Σ → 1 , w her e E ⋆ 1 = E 1 \ { (0 , 0 , 0) } , Σ ← 1 = { ( h, c ) ∈ R 2 : h < 0; c > 0 } , Σ → 1 = { ( h, c ) ∈ R 2 : h > 0; c > 0 } . 7 (ii) S c =0 = I m ( E C | E ⋆ 2 ) ∪ Σ 0 ∩ Σ → 0 , w her e E ⋆ 2 = E 2 \ { (0 , 0 , 0) } , Σ 0 = { (0 , 0 , 0) } , Σ → 0 = { ( h, c ) ∈ R 2 : h > 0; c = 0 } . (iii) S c< 0 = Σ ← 3 ∪ I m ( E C | E ⋆ 3 ) ∪ Σ → 3 , w her e E ⋆ 3 = E 3 \ { (0 , 0 , 0) } , Σ ← 3 = { ( h, c ) ∈ R 2 : c < min {− h, 0 }} , Σ → 3 = { ( h, c ) ∈ R 2 : − h < c < 0 } . All t he stra t ification results can b e gathered as sho wn in Figure 3 . Note that f or simplicit y w e ado pted the notation I m ( E C | E ) not = : Σ. 8 I m ( E C ) S c> 0 S c =0 0 h c S 1 ¬ S 1 * S 1 ®  0 h c S 2 S 0 S 0 ® S c< 0 0 0 h c S 3 ¬ S 3 * S 3 ®  S c> 0 0 ∪ S c =0 0 ∪ S c< 0 0 0 h c Figure 3: Semi a lgebr aic splitting of I m ( E C ) . Remark 5.4 As a c o n vex set, the image of the ener gy-Cas i m ir map is c onvexly gener- ate d by the im ages of the e quilibrium states of the L otka-V o l terr a system ( 2.1 ) , namel y: I m ( E C ) = co { I m ( E C | E 1 ) , I m ( E C | E 2 ) , I m ( E C | E 3 ) } . 9 6 The top ology of the fib ers of the energy-C asimir mapping In this section we describ e the top ology of the fib ers of E C , considering for o ur study fib ers o v er regular v alues of E C as w ell as fib ers ov er the singular v alues. It will remain an op en question how these fib ers fit all tog ether in a more abstract fashion, suc h a s bundle structures in the symplectic Arnold-Liouville integrable regular case. Prop osition 6.1 A c c or ding to the str atific ations fr o m the pr evious se c tion , the top olo gy of the fib ers of E C c an b e d escrib e d as in T ables 1 , 2 , 3 : S c> 0 A ⊆ S c> 0 Σ ← 1 Σ ∗ 1 Σ → 1 F ( h,c ) ⊆ R 3 ( h, c ) ∈ A 4 ` i =1 ( R × { i } ) 8 ` i =1 ( R × { i } ) ` { pt } ` { pt ′ } 4 ` i =1 ( R × { i } ) Dynamical union o f union o f 8 union of description 4 orbits orbits a nd t w o 4 orbits equilibrium p o in t s T able 1: Fib ers cla s sific ation c orr esp onding to S c> 0 . S c =0 A ⊆ S c =0 Σ 2 Σ 0 Σ → 0 F ( h,c ) ⊆ R 3 ( h, c ) ∈ A 8 ` i =1 ( R × { i } ) ` { pt } ` { pt ′ } 8 ` i =1 ( R × { i } ) ` { pt } 4 ` i =1 ( R × { i } ) Dynamical union o f 8 union o f 8 union o f description orbits a nd tw o orbits and one 4 orbits equilibrium p oin ts equilibrium p oin t T able 2: Fib ers cla s sific ation c orr esp onding to S c =0 . 10 S c< 0 A ⊆ S c< 0 Σ ← 3 Σ ∗ 3 Σ → 3 F ( h,c ) ⊆ R 3 ( h, c ) ∈ A 4 ` i =1 ( R × { i } ) 8 ` i =1 ( R × { i } ) ` { pt } ` { pt ′ } 4 ` i =1 ( R × { i } ) Dynamical union o f union o f 8 union of description 4 orbits orbits a nd t w o 4 orbits equilibrium p o in t s T able 3: Fib ers cla s sific ation c orr esp onding to S c< 0 . Pro of. The conclusion follows b y simple computations according to the top o logy of the solution set of the system:  H ( x, y , z ) = h C ( x, y , z ) = c where ( h, c ) b elongs to the semialgebraic manifolds introduced in the ab ov e section. A prese n tation that puts tog ether the top ological class ification of the fib ers of E C and the top ological classification of the symplec tic leav es o f the P oisson manifold ( R 3 , Π C ), is giv en in Figure 4 . 11 E C − 1 ( R 2 ) = E C − 1 ( I m ( E C )) E C − 1 ( S c< 0 ) E C − 1 ( S c =0 ) E C − 1 ( S c> 0 ) E C − 1 ( R 2 ) = E C − 1 ( S c< 0 ) ∪ E C − 1 ( S c =0 ) ∪ E C − 1 ( S c> 0 ) Figure 4: Phase p ortr ait splitting. 12 7 Lax F orm ulation In this section w e presen t a La x formulation of the Lotk a- V olterra system ( 2.1 ). Let us first note that as the system ( 2.1 ) restricted to a regular symplectic leaf, giv e rise to a sypmle ctic Hamiltonian system that is completely integrable in the sense of Liouville and consequen tly it has a Lax f orm ulation. Is a natural question to ask if the unrestricted system admit a Lax form ulation. The answ er is p o sitiv e and is give n b y the follow ing prop osition: Prop osition 7.1 The L otka-V olterr a system ( 2.1 ) c an b e wri tten in the L ax form ˙ L = [ L, B ] , wher e the matric es L and r esp e ctively B ar e given by: L =   0 x − y z − x + y 0 i ( x + y − z ) − z − i ( x + y − z ) 0   , B =   0 iz i ( x − y ) − iz 0 0 − i ( x − y ) 0 0   . References [1] R.H. Cushman and L. Ba tes , Glob al asp e cts of classic al int e gr able systems (1977), Ba sel: Birkhauser . [2] P. G .L. Leach and J. Miritzis , Competing sp ecies: integrabilit y a nd s tabilit y , J . Nonline ar Math. Phys. , 11 (2004), 123– 133. [3] P. G .L. Leach and J. Miritzis , Analytic behavior of co mpetition among thre e sp ecies, J. Non- line ar Math. Phys. , 13 (2006 ), 535– 548. [4] J. L libre and C. V alls , Polynomial, rational and a nalytic fir st int egrals for a family o f 3- dimensional Lo tk a-V o lterra systems, Z. Angew. Math. Phys. , (20 11), DOI 10.100 7/s00 033-011- 0119- 2. [5] C.A. Buzzi, J. Llibre and J. C. Medrado , Perio dic o r bits for a class of reversible quadratic vector field on R 3 , J . Math. Anal . Appl. , 335 (2007 ), 1 3 35–13 46. [6] J.E . Marsden , L e ctu r es on me chanics , Londo n Mathematical So ciety Lecture Notes Series, v ol. 174, Cam bridge Univ ersity Press. [7] R.M. Ma y , Stability and Complexity in Mo del Ec osystems (197 4 ), Second E dition, Princeton Uni- versit y Pres s, Pr inc e to n. [8] R.M. Ma y an d W .J. Leonard , Nonlinear aspects o f comp etition b etw een three sp ecies, SIAM J. Appl. Math. , 29 (1975), 2 43–256 . [9] J.E . Marsden and T.S. Ra tiu , Intro duction to me chanics and symmetry , T exts in Applied Mathematics, v ol. 17, second edition, s econd prin ting, Springer, Berlin. [10] M.J. Pflaum , Analytic and ge ometric study of str atifie d sp ac es (2001), Lecture Notes in Mathe- matics, v ol. 510, Springer, B e r lin. 13 [11] T.S. Ra tiu, R.M. Tudo r a n , L. Sbano, E. Sousa Dias and G. Terra , Ge ometric Me chanics and Symmetry: the Peyr esq L e ctur es; Chapter II: A Cr ash Course in Ge ometric Me chanics , pp. 23–15 6, London Mathematical So ciety Lecture Notes Ser ie s, vol. 3 06 (2005), Ca m bridge Univ ersity Press. R.M. Tudoran The W est Univ ersit y of Timi¸ soara F acult y of Mathematics and C.S., Depart ment of Mathematics, B-dl. V asile P arv an, No. 4, 300223- Timi ¸ soara , Romania. E-mail: tudo ra n@math.uvt.ro Supp orted by CNCSIS -UEFISCDI, pro ject n umber PN I I-ID EI code 1081/2008 No. 550/2009 . A. G ˆ ırban ”P olitehnica” Univ ersity of Timi ¸ soara Departmen t of Mathematics, Piat ¸a Victoriei nr. 2, 300006- Timi ¸ soara , Romˆ ania. E-mail: anania.girban@gmail .com 14