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Singular separatrix splitting and the Melnikov method: An experimental study. (English) Zbl 0932.37012

The authors consider the family of planar standard-like maps \(F:(x,y)\mapsto (y,-x+U'(y))\), \(U(y)=\mu_0\log(1+y^2)+\varepsilon V(y)\), where \(V(y)=\sum_{n\geq 1}V_ny^{2n}\) is an even entire function. Provided that \(\mu_0+V_1\varepsilon>1\), the origin \({\mathcal O}=(0,0)\) is a hyperbolic fixed point with \(\text{Spect}[F'({\mathcal O})]=\{\exp(\pm\theta)\}\), \(\text{cosh} \theta=\mu_0+V_1\varepsilon\). Hence, the map \(F\) can be considered as a perturbation of an integrable map and two parameters \(\varepsilon\) and \(\theta>0\), which are considered as the intrinsic parameters of the map \(F\). The main goal of the paper is to show that for \(\varepsilon\neq 0\) and for a general perturbation, the separatrix splits and exactly two (transverse) primary homoclinic points, \(z^+\) and \(z^-\), appear in the quadrant \(\{x,y>0\}\). The area of the loops in connection with an even Gevrey-1 is discussed. The authors provide these results by a number of detailed numerical computations. To this end the authors use multiple-precision arithmetic and expand the local invariant curves up to very high order.

MSC:

37D30 Partially hyperbolic systems and dominated splittings
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37D05 Dynamical systems with hyperbolic orbits and sets
65L12 Finite difference and finite volume methods for ordinary differential equations
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
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