## 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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