Phase portraits of separable Hamiltonian systems

XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matem´ atica Aplicada Sevilla, 24-28 septiembre 2007 (pp. 1–8) Phase portraits of separable Hamiltonian systems A. Guillamon1, Ch. Pantazi2, 1Dept. de Matem`atica Aplicada I, Universitat Polit`ecnica de Catalunya, Dr. Mara´on n.44-50, 08028, Barcelona, Catalonia, Spain,E-mail: [email protected]. 2Dept. de Matem`atica Aplicada I, Universitat Polit`ecnica de Catalunya, Avda. Diagonal 647 08028 Barcelona E-mails: [email protected]. Resumen We study some generalizations of potential Hamiltonian systems (H(x, y) = y2+ F(x)) with one degree of freedom. In particular, we are interested in Hamiltonian systems with Hamiltonian functions of type H(x, y) = F(x) + G(y) arising in applied mechanical problems. We present an algorithm to plot the phase portrait (include the behavior at infinity) of any Hamiltonian system of type H(x, y) = F(x)+G(y), where Fand Gare arbitrary polynomials. We are able to give the full description in the Poincar´e disk according to the graphs of Fand G, extending the well-known method for the “finite”phase portrait of potential systems. 1. Introduction The algorithm to plot the finite phase portrait of potential systems is a well-known and classical example of qualitative theory of ordinary differential equations. The clue to be able to topologically classify this type of Hamiltonian systems is, obviously, the simplicity of its energy function, H(x, y) = y2 2+F(x), with F∈C1(IR) (see Figure 1). The fact that H“depends”basically on F(x) allows to relate the phase portrait of {x0=−Hy(x, y), y0=Hx(x, y)}with the graph of the one-variable function F(x). This kind of reduction was also explored in [2] for Hamiltonian systems of type H(r, θ) = r2/2 + rn+1 g(θ). The aim of this work is to extend this type of results to Hamiltonian systems depending of more than one function. We have chosen a natural generalization of potential systems, those with energy function H(x, y) = F(x) + G(y), which covers a wide range 1 Ch. Pantazi, A. Guillamon F .... . . Figura 1: Finite representation of the phase portrait of a potential systems; minima of F coincide with center points while maxima coincide with saddles. of conservative physical models. We provide an algorithm to obtain the global (along the paper, “global”means that includes the behaviour at infinity) phase portrait of polynomial Hamiltonian systems of the form x0=−G0(y), y0=F0(x),(1) where Fand Gare arbitrary polynomials. We will refer to this vector field as XH. Our description includes the classical classification of finite critical points of smooth potential systems. To fix notation, we write: F0(x) = anxn+· · · +a1x, an6= 0; G0(y) = bmym+· · · +b1y, bm6= 0.(2) In the next two sections we provide the results that support the algorithm. For the sake of space, this document only presents the main stream to apply the algorithm, but does not cover all the proofs and details. 2. Bounded dynamics Next two results are useful to determine the behaviour of bounded orbits of the system (1). Proposition 1 (Finite critical points) Let P= (x0, y0)be a finite singular point of system (1). Then, (a) Pis a saddle if and only if Fhas a maximum (resp., a minimum) at x0and Ghas a minimum (resp., a maximum) at y0. 2 Phase portraits of separable Hamiltonian systems (b) Pis a center point if either Fhas a maximum at x0and Ghas a maximum at y0 or Fhas a minimum at x0and Ghas a minimum at y0. (c1) Pis a cusp point if either Fhas an inflection point at x0or Ghas an inflection point at y0. (c2) If Fhas an inflection point at x0and Ghas an inflection point at y0, then Pis a critical point formed by the union of two hyperbolic sectors. Saddle points will be denoted by S; centers arising from two maxima will be denoted by C−, while those coming from two minima will be denoted by C+. All other types of finite critical points will be denoted by D. The bounded separatrices of a saddle organize themselves according to the rules of the following proposition. Proposition 2 (Bounded separatrices) Any center of type C+is embraced by the proximal separatrices of the neighbouring saddle S∗which energy level satisfies H(S∗) = m´ın{S:H(S)> H(C+)}. Any center of type C−is embraced by the proximal separatrices of the neighbouring saddle S∗which energy level satisfies H(S∗) = m´ax{S:H(S)< H(C+)}. If the four separatrices of a saddle Sembrace the period annuli of a set of centers C:= {C1, . . . , Cn}, then C ∪ S∪Wu,s(S)can be treated as a new center with the same energy level than S. 3. Unbounded dynamics In order to study the behaviour of system (1) near infinity we use Poincar´e compactification (see for instance [3]). If we call {X, Y, Z}the coordinates in which the sphere is expressed, the equator lies on Z= 0. Along the paper we are going to use both charts (U1, F1) ({X > 0}) and (U2, F2) ({Y > 0}) of the vector field (1) extended to the sphere. Additionally, we denote by (V1, G1) ({X < 0}) and (V2, G2) ({Y < 0}) the two charts on the opposite side of (U1, F1) and (U2, F2) respectively. Depending on the sign and parity of n−mwe will use either chart, as it can be appreciated from the following cases: Case 1: n > m. In the (U1, F1) chart the vector field takes the form      z1 0=zn 2³z1G0(z1 z2) + F0(1 z2)´ =an+an−1z2+· · · +a1zn−1 2+bmzm+1 1zn−m 2+···+b1z2 1zn−1 2, z2 0=zn+1 2G0(z1 z2) = zn−m+1 2(bmzm 1+bm−1zm−1 1z2+· · · +b1z1zm−1 2). (3) 3 Ch. Pantazi, A. Guillamon We note that there are no critical points on {z2= 0}since an6= 0. In the (U2, F2) chart the vector field can be written as      z0 1=−zn 2³G0(1 z2) + z1F0(z1 z2)´ =−bmzn−m 2− · · · − b1zn−1 2−anzn+1 1− · · · − a1z2 1zn−1 2, z0 2=−zn+1 2F0(z1 z2) = −anzn 1z2− · · · − a1z1zn 2, (4) which has a unique critical point, z1=z2= 0, on {z2= 0}. Case 2: n=m On the (U1, F1) chart, the vector field becomes ½z0 1= (an+bnzn+1 1)+(an−1+bn−1zn 1)z2+· · · + (a1+b1z2 1)zn−1 2, z0 2=z2(bnzn 1+· · · +b1z1).(5) We note that critical points at infinity are given by zn+1 1=−an bn. On the (U2, F2) chart the vector field is written ½z1 0=−bn−bn−1z2− · · · − b1zn−1 2−anzn+1 1−an−1zn 1z2− · · · − a1z2 1zn−1 2, z0 2=−z2(anzn 1+···+a1z1zn−1 2),(6) and the critical points at infinity are given by zn+1 1=−bn an. When nis odd and bn an<0, both (5) and (6) have 4 critical points at infinity, which coincide; if bn an>0, there are no critical points at infinity. When nis even, both (5) and (6) have 2 critical points at infinity, which coincide. Case 3: n < m. After an appropriate change of variables, this case is the equivalent to Case 1. Without loss of generality, then, we will omit its study and concentrate on system (6), defined on chart (U2, F2). One of the crucial steps to obtain the global phase portrait is to determine which separatrices tend (both for positive and negative times) to critical points at infinity. For this purpose, we need to distinguish special maxima and minima of functions Fand G, which apply for the case that S= (x∗, y∗) is a saddle: Definition 1 Given a critical point S= (x∗, y∗)of system (1), we say that: x∗satisfies the property l(resp., L) if F(x∗)is a minimum (resp., maximum) of F and F(x∗)< F(x)(resp., F(x∗)> F(x)) for each x < x∗. x∗satisfies the property r(resp., R) if F(x∗)is a minimum (resp., maximum) of F and F(x∗)< F(x)(resp., F(x∗)> F(x)) for each x > x∗. y∗satisfies the property d(resp., D) if G(y∗)is a minimum (resp., maximum) of G and G(y∗)< G(y)(resp., G(y∗)> G(y)) for each y < y∗. y∗satisfies the property u(resp., U) if G(y∗)is a minimum (resp., maximum) of G and G(y∗)< G(y)(resp., G(y∗)> G(y)) for each y > y∗. Additionally, we define the following properties: 4 Phase portraits of separable Hamiltonian systems The saddle S∗= (x∗, y∗)satisfies property (h1) if x∗satisfies both the properties rand l. Analogously, we will say that it satisfies property (H1) if x∗satisfies both the properties Rand L. The saddle S∗= (x∗, y∗)satisfies property (h2u) (resp., (H2U), (H2d) or (H2D)) if y∗satisfies property u(resp. U,dor D) and its maximal (resp., maximal, minimal or minimal) with respect to this property. Finally, we say that (x(t), y(t)) is an upper (resp., lower) separatrix of the saddle Sif there exists ² > 0such that for |y(t)−y∗|< ², we have y(t)> y∗(resp., y(t)< y∗). Next result (and the corollaries not shown here) gives the behavior of the main separatrices that reach or come from a critical point at infinity. Proposition 3 Let S= (x∗, y∗)be a finite saddle point. Then: If Ssatisfies the properties (h1) and (H2U), or (H1) and (H2u), the upper separatrices of Sform an elliptic sector Eat +∞(that is, the critical point of the vector field on chart (U2, F2)on {z2= 0}). If Ssatisfies the properties (h1) and (H2D), or (H1) and (H2d), the lower separatrices of Sform an elliptic sector Eat −∞ (that is, the critical point of the vector field on chart (V2, G2)on {z2= 0}), see Figure 4. Finally, for the infinite critical points we can use Hartmann Theorem in case they are hyperbolic. However, since the degrees n,mand the coefficients anand bmgive all the information about the index of the vector field on R2, we can deduce the index of the infinite critical points from Poincar´e-Hopf Theorem. Then, with the help of Proposition 3 and additional reasonings, we can give the topological classification of all infinite critical points. More precisely the information about the finite index allows to establish the following result: Proposition 4 Suppose first that n−mis odd and let q∞a critical point on the equator of the Poincar´e sphere. Then, If q∞does not have elliptic sectors, q∞is a node. If q∞has one elliptic sectors, q∞is the union of this elliptic sector and a hyperbolic one, each of them lying on a different hemisphere of the Poincar´e sphere (see Figure 2)(a). Suppose now that n−mis even. Then: If iR2=−1,q∞is the union of two elliptic sectors (see Figure 2)(b). If iR2= 1,q∞is the union of two hyperbolic sectors (see Figure 2)(c). 5 Ch. Pantazi, A. Guillamon . . Elliptic and Two elliptic hyperbolic hyperbolic sectors sectors sectors Figura 2: Different types of critical points on the equator of the Poincar´e sphere. 4. Algorithm Combining propositions 1, 2, 3 and 4 one can describe the complete phase portrait on the Poincar´e sphere. In Figures 3 and 4 we give a couple of examples, explaining the whole procedure to build up the global phase portrait. Agradecimientos Partially supported by the DGES grant number MTM2005-06098-C02-1, the CONACIT grant number 2005SGR-986 and the MCyT/FEDER grant MTM2006-00478. Referencias [1] S. Diliberto,On systems of ordinary differential equations, Contributions to the Theory of Nonlinear Oscillators 1(1950), 1–38. [2] A. Gasull, A. Guillamon, V. Ma˜ nosa Phase portrait of Hamiltonian systems with homogeneous nonlinearities, Nonlinear Analysis: Theory, Methods & Applications 42 (2000), 679–707. [3] J. Sotomayor,Li¸coes de equa¸co´es diferenciais ordin´arias, IMPA, Rio de Janerio, 1979. 6 Phase portraits of separable Hamiltonian systems F’=(x-6)(x-1)x(x-3)(x+1) 2 G’=y(y-9)(y-4) .. .. . . . . . . . . C C S S S S C C SC CS + + + + - - 461 10 7 12 9 11 8 3 5 2 .. . . . . . . . . . . .. . . .. . . S8 S9 S5 S6 S3 + S10 - . . . . S8S9 S5S6 . . Figura 3: The saddles S10 and S3become S− 10 and S+ 3respectively, see Proposition 2. There is a connection between each of the saddles S9,S8,S6,S5and a critical point at infinity. Critical points at infinity are nodes, see Proposition 4. 7 Ch. Pantazi, A. Guillamon F‘=(x-4)(x-3)(x+5) G’=(y-1)(y+2) ... ... CSC SCS ++ - 143 265 . . . .. CS C ++ ... SC S - Figura 4: The saddle S2satisfies the properties (h1) and (H2D). From Proposition 3, the lower separatrices of the saddle S2form an elliptic sector at −∞ and so S2becomes S+ 2. Then, the saddle S4embraces the “closed heteroclinic loop”S+ 2and so S4becomes S+ 4. At the end, the saddle S5embraces S+ 4. Additionally, the singular point at infinity is a union of elliptic and hyperbolic sectors, see also Figure 2(a). 8