High-precision computations of divergent asymptotic series and homoclinic phenomena
Vassili Gelfreich,Carles Simó +1 more
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In this article, asymptotic expansions for the exponentially small splitting of separatrices of area preserving maps combining analytical and numerical points of view are studied using high-precision arithmetic, which involves up to several thousands of decimal digits.Abstract:
We study asymptotic expansions for the exponentially small splitting of separatrices of area preserving maps combining analytical and numerical points of view. Using analytic information, we conjecture the basis of functions of an asymptotic expansion and then extract actual values of the coefficients of the asymptotic series numerically. The computations are performed with high-precision arithmetic, which involves up to several thousands of decimal digits. This approach allows us to obtain information which is usually considered to be out of reach of numerical methods. In particular, we use our results to test that the asymptotic series are Gevrey-1 and to study positions and types of singularities of their Borel transform. Our examples are based on generalisations of the standard and Henon maps.read more
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High-Precision Computation: Mathematical Physics and Dynamics
TL;DR: It is concluded that high-precision arithmetic facilities are now an indispensable component of a modern large-scale scientific computing environment.
Journal ArticleDOI
Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps
Carles Simó,A. Vieiro +1 more
TL;DR: In this article, the authors consider a one-parameter family of area preserving maps in a neighbourhood of an elliptic fixed point and provide accurate formulae describing the splittings of these manifolds as a function of the parameter and the relative values of these magnitudes as well as geometric properties.
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Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results
TL;DR: In this paper, the authors studied the problem of exponentially small splitting of separatrices of one degree of freedom classical Hamiltonian systems with a non-autonomous perturbation which is fast and periodic in time.
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Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift–Hohenberg equation as an example
TL;DR: In this article, an asymptotic expansion for a homoclinic invariant is proposed, which quantitatively describes the transversality of the invariant manifolds.
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The inner equation for generic analytic unfoldings of the Hopf-zero singularity
Inmaculada Baldomá,Tere M. Seara +1 more
TL;DR: In this paper, the inner system associated to the Hopf-zero bifurcation problem is derived, which is independent on the unfolding parameter, and the existence of two solutions of this system related with the stable and unstable manifolds of the unfolding is proved.