Singular Separatrix Splitting and the Melnikov Method: An Experimental Study

We consider families of analytic area-preserving maps depending on two parameters: the perturbation strength " and the characteristic exponent h of the origin. For " = 0, these maps are integrable with a separatrix to the origin, whereas they asymptote to ows with homoclinic connections as h ! 0. For xed " 6= 0 and small h, we show that these connections break up. The area of the lobes of the resultant turnstile is given asymptotically by " exp( =h) (h), where (h) is an even Gevrey-1 function such that (0) 6= 0 and the radius of convergence of its Borel transform is 2 . As " ! 0, the function " tends to an entire function . This function 0 agrees with the one provided by the Melnikov theory, which cannot be applied directly, due to the exponentially small size of the lobe area with respect to h. These results are supported by detailed numerical computations; we use an expensive multiple-precision arithmetic and expand the local invariant curves up to very high order.

[1]  A. Neishtadt.  The separation of motions in systems with rapidly rotating phase , 1984 .

[2]  Tere M. Seara,et al.  Splitting of Separatrices in Hamiltonian Systems and Symplectic Maps , 1999 .

[3]  Tere M. Seara,et al.  Exponentially Small Splitting in Hamiltonian Systems , 1994 .

[4]  Y B Suris.  On the complex separatrices of some standard-like maps , 1994 .

[5]  Carles Simó,et al.  Hamiltonian systems with three or more degrees of freedom , 1999 .

[6]  Leo P. Kadanoff,et al.  The break-up of a heteroclinic connection in a volume preserving mapping , 1993 .

[7]  C. Simó.  On the analytical and numerical approximation of invariant manifolds. , 1990 .

[8]  Bernold Fiedler,et al.  Discretization of homoclinic orbits, rapid forcing, and "invisible" chaos , 1996 .

[9]  Amadeu Delshams,et al.  Melnikov Potential for Exact Symplectic Maps , 1997 .

[10]  K. Mallick,et al.  Exponentially small splitting of separatrices, matching in the complex plane and Borel summation , 1993 .

[11]  Héctor E. Lomelí,et al.  Perturbations of elliptic billiards , 1996 .

[12]  A. Valdés,et al.  Exponentially Small Splitting of Separatrices for Perturbed Integrable Standard-Like Maps , 1998 .

[13]  David Sauzin,et al.  Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé , 1995 .

[14]  Jerrold E. Marsden,et al.  Exponentially Small Estimates for Separatrix Splittings , 1991 .

[15]  L. P. Kadanoff,et al.  Beyond all orders: Singular perturbations in a mapping , 1992 .

[16]  M. B. Tabanov.  Separatrices splitting for Birkhoff's billiard in symmetric convex domain, closed to an ellipse. , 1994, Chaos.

[17]  Robert W. Easton,et al.  Transport through chaos , 1991 .

[18]  V. F. Lazutkin,et al.  Splitting of separatrices for standard and semistandard mappings , 1989 .

[19]  Amadeu Delshams Valdés,et al.  Splitting of separatrices in Hamiltonian systems and symplectic maps , 1997 .

[20]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[21]  Tere M. Seara,et al.  Splitting of Separatrices in Hamiltonian Systems with one and a half Degrees of Freedom , 1997 .

[22]  V. F. Lazutkin,et al.  Exponentially small splittings in Hamiltonian systems. , 1991, Chaos.

[23]  Alan Weinstein,et al.  Lagrangian Submanifolds and Hamiltonian Systems , 1973 .

[24]  Donald E. Knuth.  The Art of Computer Programming 2 / Seminumerical Algorithms , 1971 .

[25]  Vassiliĭ G. Gelfreich.  Conjugation to a shift and the splitting of invariant manifolds , 1997 .

[26]  James D. Meiss,et al.  Heteroclinic orbits and Flux in a perturbed integrable Suris map , 1996, math/9604231.

[27]  Philippe Levallois,et al.  Calcul d'une fonction de Melnikov et de ses zeros pour une perturbation algebrique du billard elliptique , 1997, Ergodic Theory and Dynamical Systems.

[28]  Ernest Fontich.  Rapidly Forced Planar Vector Fields and Splitting of Separatrices , 1995 .

[29]  Y. Suris,et al.  Integrable mappings of the standard type , 1989 .

[30]  James D. Meiss,et al.  Transport in Hamiltonian systems , 1984 .

[31]  Amadeu Delshams Valdés,et al.  Poincaré-Melnikov-Arnold method for analytic planar maps , 1995 .

[32]  James D. Meiss,et al.  Resonances in area-preserving maps , 1987 .

[33]  Wolf-Jürgen Beyn,et al.  The Numerical Computation of Homoclinic Orbits for Maps , 1997 .

[34]  Amadeu Delshams,et al.  Poincaré - Melnikov - Arnold method for analytic planar maps , 1996 .

[35]  Carles Simó,et al.  The splitting of separatrices for analytic diffeomorphisms , 1990, Ergodic Theory and Dynamical Systems.

[36]  P. Levallois,et al.  Séparation des séparatrices du billard elliptique pour une perturbation algébrique et symétrique de l'ellipse , 1993 .

[37]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[38]  Reinhard Schäfke,et al.  Gevrey separation of fast and slow variables , 1996 .

[39]  V. F. Lazutkin,et al.  A refined formula for the separatrix splitting for the standard map , 1994 .

[40]  Michal Fečkan.  On the existence of homoclinic points , 1991 .

[41]  J. Meiss.  Symplectic maps, variational principles, and transport , 1992 .

[42]  Eduardo Tabacman.  Variational computation of homoclinic orbits for twist maps , 1995 .

[43]  Dmitry V. Treschev,et al.  An averaging method for Hamiltonian systems, exponentially close to integrable ones. , 1996, Chaos.