A polynomial class of Markus-Yamabe counterexamples

Publicacions Matem`atiques, Vol 41 (1997), 85–100. A POLYNOMIAL CLASS OF MARKUS-YAMABE COUNTEREXAMPLES Anna Cima, Armengol Gasull and Francesc Ma˜ nosas Abstract In the paper [CEGHM] a polynomial counterexample to the Markus-Yamabe Conjecture and to the discrete Markus-Yamabe Question in dimension n≥3 are given. In the present paper we explain a way for obtaining a family of polynomial counterexamples containing the above ones. Finally we study the global dynamics of the examples given in [CEGHM]. 1. Introduction Let F:Rn−→ Rnbe a C1map and consider the differential system (1) ˙x=F(x). Assume that pis a critical point of (1), i.e., F(p) = 0. We say that p is a global attractor of the continuous dynamical system (1) if φ(t, x)is defined for all t>0 and tends to pas ttends to infinity for each x∈Rn, where φ(t, x) is the solution of (1) with initial condition φ(0,x)=x. The next conjecture was explicitly stated by Markus and Yamabe (see [MY]) in 1960. MYC(n) (Markus-Yamabe Conjecture). Let FbeaC1vector field defined on Rnsuch that for any x∈Rn, the jacobian of Fat xhas all its eigenvalues with negative real part. If F(p) = 0, then pis a global attractor of ˙x=F(x). This conjecture was proved for planar polynomial maps in 1988 (see [MO]) and for planar C1maps in 1993 (see [F] and [Gu]) and in 1994 (see [G1]). In [B] and [BL] there are examples of smooth vector fields defined in Rn,n≥4 satisfying the hypothesis of the Conjecture and having a periodic orbit. On the other hand in [CEGHM], the authors 86 A. Cima, A. Gasull, F. Ma˜ nosas give an example of a polynomial vector field in Rnfor n≥3 satisfying the hypothesis of the MYC such that it has orbits which scape at infinity. A non polynomial counterexample for n≥R3is presented in [G2]. The goal of this paper is to give a more general family of polynomial counterexamples and explain how they are obtained. The construction of the counterexamples is based on two points: The first one concerns on the characterizacion of vector fields which have solutions of exponential type, that lead us with the notion of linear quasi-homogeneous vector fields (see Section 2). The second one deals with the construction of nilpotent maps (see Section 3). Using the same tools we can also give an answer to the Discrete Markus-Yamabe Question, which can be established as follows. Let F:Rn−→ Rnbe a C1map and consider the sequence: (2) x(m+1) =F(x(m)),x (0) ∈Rn. Now consider the dynamics of the iterations of F. Let pbe a fixed point of F, i.e., F(p)=p. We say that pis a global attractor of the discrete dynamical system (2) if the sequence x(m)tends to pwhen mtends to infinity for any x(0) ∈Rn. The question is the following: DMYQ(n) (Discrete Markus-Yamabe Question). Let Fbe a C1map from Rninto itself such that F(0) = 0 and for any x∈Rn, JF(x) has all its eigenvalues with modulus less than one. Is it true that 0 is a global attractor for the discrete dynamical system generated by F? The above problem is introduced in [S] and [CGM]. In this last paper the authors prove that the answer is negative even in the planar case and that the conclusion is affirmative if we take into account polynomial maps defined in R2. The planar example which gives a negative answer to this question has a periodic orbit of period 4 and is given by a rational function. In [EH] the authors give a negative answer to this question for polynomial maps in dimension greather than three. A polynomial counterexample for dimension greather than two is presented in [CEGHM]. In this paper we also give a family of polynomial maps in Rnfor n≥3 satisfying the hypothesis of DMYQ(n) and such that there are some unbounded orbits. The paper is organized as follows. In the next section we give some results on quasi-homogeneous vector fields of degree one. In Section 3 we Polynomial Markus-Yamabe counterexamples 87 construct the mentioned families of counterexamples. The last section is devoted to study the global dynamics of the simplest counterexamples inside the family (the ones given in [CEGHM]). This paper is partially supported by the DGICYT grant number PB93-0860. 2. On quasi-homogeneous vector fields of degree one This section is a development of some results of [A, Chap. 1]. We say that f:Rn→Ris a quasi-homogeneous function with weigths α1,α 2,... ,α nand quasi-degree dif (3) f(λα1x1,λ α2x2,... ,λ αnxn)=λdf(x1,x 2,... ,x n), for all λ>0 and for all x∈Rn. We stress that the weights can be taken as non zero real numbers. Notice that if fis a quasi-homogeneous function with weights α1,α 2,... ,α nand quasi-degree dthen, for all constant c∈R,c=0,f is also a quasi-homogeneous function with weights cα1,cα 2,... ,cα nand quasi-degree cd. On the other hand let fbe a quasi-homogeneous analytic function with weights α1,α 2,... ,α nand write fas a sum of monomials: f(x)=  r1,r2,... ,rn Ar1r2...rnxr1 1xr2 2···xrn n. Since for all λ>0 (3) is satisfied, we have that  r1,r2,... ,rn Ar1r2...rnxr1 1xr2 2···xrn nλ(r1α1+r2α2+···+rnαn) =λd r1,r2,... ,rn Ar1r2...rnxr1 1xr2 2···xrn n. Hence, the only monomials which appear in the decomposition of fare of the form xr1 1xr2 2···xrn nwith r1α1+r2α2+···+rnαn=d. We say that F=(F1,F 2,... ,F n):Rn→Rnis a quasi-homogeneous vector field with weights α1,α 2,... ,α nand quasi-degree dif each Fi is a quasi-homogeneous function with weights α1,α 2,... ,α nand quasidegree αi+d−1, see [BDST, Chap. 7]. We say that Fis a linear quasi-homogeneous vector field if it is a quasi-homogeneous vector field of degree one. From now on we deal with linear quasi-homogeneous vector fields. 88 A. Cima, A. Gasull, F. Ma˜ nosas Proposition 2.1. Let Fbe a linear quasi-homogeneous vector field with weights α1,α 2,... ,α nand consider the differential system ˙x= F(x). Then Fis invariant by the change ¯xi=λαixi. Proof: We have to show that if x(t)=(x1(t),x 2(t),... ,x n(t)) is a solution of ˙x=F(x) then ¯x(t)=(λα1x1(t),λ α2x2(t),... ,λ αnxn(t)) is also a solution. This is done in the sequel. d dt ¯xi(t)=λαid dtxi(t)=λαiFi(x1(t),x 2(t),... ,x n(t)) =Fi(λα1x1(t),λ α2x2(t),... ,λ αnxn(t)) = Fi(¯x(t)). Given the weights α1,α 2,... ,α nwe define the “semi straight line” which passes through a point x∈Rn,as Lx={(λα1x1,λ α2x2,... ,λ αnxn):λ∈R+}. Proposition 2.2. Let Fbe a linear quasi-homogeneous vector field with weights α1,α 2,... ,α nand consider the differential system ˙x= F(x).Ifαiαj>0for all i, j =1,2,... ,n, then the knowledge of the solutions near the origin determines the global phase portrait of F. Furthermore, if 0is locally asymptotically stable, then 0is a global attractor. Proof: Clearly we can assume that αi>0 for all i=1,2,... ,n.So the origin is in the closure of Lx. Assume that we know the solutions of ˙x=F(x) in a neighbourhood B of the origin and let xbeapointinRn. Then for λsmall enough the point y=(λα1x1,λ α2x2,... ,λ αnxn) belongs to B. Let y(t) be the solution of ˙x=F(x) which passes through y. Then, from Proposition 2.1, we know that x(t)=1 λα1 y1(t),1 λα2 y2(t),... ,1 λαn yn(t) is also a solution. And it is clear that this solution passes through x.On the other hand it is clear that if the omega limit of yis the origin, the same is true for x. More interesting dynamical behaviours appear for quasi-homogeneous vector fields with weights of different sign (see Sections 3 and 4). Example. Consider the family of systems      ˙x=−x ˙y=−y+ax2z+bx4 ˙z=−z+cx2. Polynomial Markus-Yamabe counterexamples 89 All these systems are linear quasi-homogeneous with weights (1,4,2). On the other hand, the characteristic polynomial of the linear part is independent of the point and it is equal to P(λ)=(λ+1) 3. So we are in the hypothesis of the Markus Yamabe conjecture. Since 0 is locally asymptotically stable from the above result we deduce that 0 is a global attractor. Let ˙x=F(x) and let x∈Rn. We say that the solution which passes through xis of exponential type if x(t)=(x1em1t,x 2em2t,... ,x nemnt) for some m1,m 2,... ,m n∈R. Proposition 2.3. Let Fbe a linear quasi-homogeneous vector field with weights α1,α 2,... ,α nand let Lxbe the “semi straight line” which passes through x. Then Lxis invariant by the flow of ˙x=F(x)if and only if the solution which passes through xis of the form xi(t)=xiemit where mi=cαifor some c∈R. Proof: Let Lx={(λα1x1,λ α2x2,... ,λ αnxn):λ∈R+}and consider the parametrization λ=et, i.e., Lx={(eα1tx1,e α2tx2,... ,e αntxn):t∈R+}. If Lxis invariant by the flow, then it exists some µ(t)∈Rsuch that αieαitxi=µ(t)Fi(eα1tx1,e α2tx2,... ,e αntxn). Due to the homogeneity of Fthis last condition can be written as αixi= µ(t)Fi(x). Hence µ(t)≡µis independent on t. Hence (4) αixi=µFi(x). If µ= 0, then x= 0. Since 0 is a critical point of ˙x=F(x) the solution can also be considered as a solution of exponential type. If µ= 0, consider xi(t)=xieαi µt. Then x(t) is the solution which passes through x: x i(t)=αi µxieαi µt=Fi(x)eαi µt=Fi(x1eα1 µt,x 2eα2 µt,... ,x neαn µt). The last equality is due to the fact that Fis a linear quasi-homogeneous vector field with weights α1,α 2,... ,α n. 90 A. Cima, A. Gasull, F. Ma˜ nosas Now let xbe such that xi(t)=xiemitwith mi=cαiis a solution of ˙x=F(x). If c= 0, then xi(t)=xifor all t∈R, and all the points of Lxare critical points of ˙x=F(x). If c= 0, then x i(t)=Fi(x(t)) implies cαixiecαit=Fi(x1ecα1t,x 2ecα2t,... ,x necαnt)=ecαitFi(x). Hence αixi=(1/c)Fi(x). Then, from (4), we see that Lxis an invariant straight line. Remark 2.4. There are some systems which have solutions of exponential type and they are not quasi-homogeneous. For instance, the system ˙x=−x+xy −x2y2+2x2y ˙y=y+x2−x3y−2xy2 has the solution x(t)=x0et,y(t)=(1/x0)e−tand it is not a quasihomogeneous vector field. If Fis a linear quasi-homogeneous vector field with weights α1,α 2,... ,α nand we look for the solutions of exponential type we have to solve the system of equations Fi(x)=cαixi,i=1,2,... ,n where cis a real number. This is a system of nquasi-homogeneous equations. It is easy to see that the set of solutions of this type of systems, either, it reduces to the origin or, if xis a solution of the system then each point in Lxis also a solution. Solving the above system we can find the invariant “semi straight lines”. Proposition 2.5. Let Fbe a linear quasi-homogeneous vector field and consider the differential system ˙x=F(x). Then the integration of ˙x=F(x)in Rnreduces to the integration of a system in Rn−1. Proof: Consider the transformation (outside x1=0) y1=x1, yj=xα1 jx−αj 1,j≥2, and using the homogeneity of Fwe have that Fi(x)=Fi(x1,x α2 α1 1y 1 α1 2,... ,x αn α1 1y 1 α1 n)=x αi α1 1˜ F(y2,y 3,... ,y n). Polynomial Markus-Yamabe counterexamples 91 Then, the system ˙x=F(x) reduces to:    ˙y1=y1˜ F1(y2,y 3,... ,y n) ˙yj=α1y α1−1 α1 j˜ Fj(y2,y 3,... ,y n)−αjyj˜ F1(y2,y 3,... ,y n),j≥2. Hence, we have that the last n−1 equations only depend on the last n−1 variables and the first one has separate variables if we already know yj(t) for j=2,3,... ,n. Remark 2.6. Let Fbe a linear quasi-homogeneous vector field and consider the discrete dynamical system generated by F. Then, it is easy to obtain similar results to Propositions 2.1, 2.2 and 2.4. In particular the points lying in invariant “semi straight lines” (notice that now invariant means F(Lx)⊂Lx) have also solutions of exponential type (i.e., x(m)= (x(0) 1am 1,x (0) 2am 2,... ,x (0) nam n) for some constants a1,a 2,... ,a n∈R). In order to find the invariant straight lines we have to solve the system of equations Fi(x)=λαixi where λis a real positive number. 3. Construction of counterexamples In this section we will give a classes of polynomial maps which satisfy simultaneously the hypothesis of the MYC and the hypothesis of the DMYQ. Lemma 3.1. (See [CGM]). Let Fbe a polynomial map from Rn into itself such that for any x∈Rn,JF(x)has all its eigenvalues with modulus less than one. Then the characteristic polynomial of (DF)xis independent on x. Due to Lemma 3.1 we start with simple maps with characteristic polynomial independent on x. Particularly we consider maps of type F(x)=λI +N, where λ∈R,Iis the identity map and Nis a nilpotent map (i.e., (DN)xhas all its eigenvalues equal zero at each x∈Rn). Then, (DF)xhas all its eigenvalues equal λat each x∈Rn. Hence, we will consider λ<0 (resp. |λ|<1) for the continous (resp. discrete) problem. Assume that n= 2. Then Nnilpotent implies that ∂N1 ∂x +∂N2 ∂y ≡0 and ∂N1 ∂x ∂N2 ∂y −∂N1 ∂y ∂N2 ∂x ≡0. 92 A. Cima, A. Gasull, F. Ma˜ nosas So, it exists some H(x, y) with N1(x, y)=−∂H ∂y (x, y), N2=∂H ∂x (x, y) and the hessian of His identically zero. From a classical result of differential geometry we have that these type of maps can be written, through an affine transformation, as H(u, v)=u+g(v) (see [C] and also [D]). Hence, H(x, y)=αx+βy+γ+g(ax+by+c) provide us a family of MY examples in R2: F(x, y)=(λx −bf(ax +by +c),λy+af(ax +by +c)) where f=g, which can be extended to Rnfor n≥3by F(x)=(λx1−b(x3)f(u),λx 2+a(x3)f(u),λx 3,... ,λx n) with u=a(x3)x1+b(x3)x2+c(x3) and a, b, c arbitrary smooth functions of x3. Taking a(x3)=axl 3,b(x3)=bxm 3and f(u)=uk(a, b ∈R,k, l,m ∈N) we obtain the following: Theorem 3.2. The family of maps F(x)=(λx1−bxm 3(ax1xl 3+bx2xm 3)k, λx2+axl 3(ax1xl 3+bx2xm 3)k,λx 3,... ,λx n) satisfy the following properties: (1) For all a, b, λ ∈R,k,l,m ∈Nthey are linear quasi-homogeneous with weights (α1,α 2,... ,α n)=(m+kl,l +km,1−k,... ,1−k). (2) For all λ∈Rwith λ<0(resp. |λ|<1) they satisfy the hypothesis of the MYC (resp. the DMYQ). (3) For all λ∈Rwith λ<0(resp. |λ|<1)kan even number, l,k, l− m∈Ndifferent from zero and for all a, b ∈Rthe differential system ˙x=F(x)(resp., the discret dynamical system generated by F) has unbounded orbits. Proof: The proof of (1) and (2) is straightforward. In order to see (3) we begin considering the dynamical system ˙x=F(x). The system of equations Fi(x)=cαixiwrites as: (5)      λx1−bxm 3(ax1xl 3+bx2xm 3)k=c(m+kl)x1 λx2+axl 3(ax1xl 3+bx2xm 3)k=c(l+km)x2 λxi=c(1 −k)xi,i=3,4,... ,n Polynomial Markus-Yamabe counterexamples 93 which gives the solutions (6)                  c=λ 1−k>0, x1=−bB aA x2xm−l 3 xk−1 2xl+km 3=−B aA bλ(l−m)k where A=λ(1−k)−(m+kl) 1−kand B=λ(1−k)−(l+km) 1−k. It is clear that system (5) has always real solutions. Let ¯x1,¯x2and ¯x3be one of them and let ¯xibe arbitrary real numbers for i=4,5,... ,n. Then        x1(t)=¯x1eλ(m+kl) 1−kt x2(t)=¯x2eλ(l+km) 1−kt xi(t)=¯xieλt,i=3,4,... ,n is a solution of ˙x=F(x). And it is clear that x1(t), x2(t)→∞as t→∞. Now consider the discret dynamical system generated by F. From Remark 2.6 we have to solve the system of equations Fi(x)=µαixi, i.e.,      λx1−bxm 3(ax1xl 3+bx2xm 3)k=µm+klx1 λx2+axl 3(ax1xl 3+bx2xm 3)k=µl+kmx2 λxi=µ1−kxi,i=3,4,... ,n which gives the solutions (7)              µ=λ1 1−k x1=−bD aC x2xm−l 3 xk−1 2xl+km 3=−D aC b(C−D)k where C=λ−λm+kl 1−kand D=λ−λl+km 1−k. Notice that |µ|>1. As before take ¯x1,¯x2and ¯x3a solution of (7) and ¯xiarbitrary real numbers for i=4,5,... ,n. Then            x(n) 1=¯x1λm+kl 1−kn x(n) 2=¯x2λl+km 1−kn x(n) i=¯xi(λ)n,i=3,4,... ,n gives the complete orbit which begins in ¯x=(¯x1,¯x2,... ,¯xn) and clearly x(n) 1,x(n) 2→∞as n→∞. 100 A. Cima, A. Gasull, F. Ma˜ nosas [MY] L. Markus and H. Yamabe, Global stability criteria for differential systems, Osaka Math. Journal 12 (2) (1960), 305–317. [S] J. P. La Salle, The stability of Dynamical Systems, CBMS-NSF Regional Conference Series in Applied Math. 25, SIAM 1976, 81 pages, 2nd printing 1993. Anna Cima: Departament de Matem`atica Aplicada II E. T. S. d’Enginyers Industrials Universitat Polit`ecnica de Catalunya Colom 11 08222 Terrassa (Barcelona) SPAIN e-mail: [email protected]c.es Armengol Gasull: Departament de Matem`atiques Universitat Aut`onoma de Barcelona 08193 Bellaterra (Barcelona) SPAIN e-mail: [email protected] Francesc Ma˜nosas: Departament de Matem`atiques Universitat Aut`onoma de Barcelona 08193 Bellaterra (Barcelona) SPAIN e-mail: man[email protected] Rebut el 30 de Novembre de 1996