Exponentially small splitting of separatrices beyond Melnikov analysis: Rigorous results

DOI10.1016/j.jde.2012.09.003zbMath1271.34050arXiv1201.5152OpenAlexW2064641691MaRDI QIDQ1759899

Inmaculada Baldomá, Ernest Fontich, Marcel Guardia, Teresa M. Seara

Publication date: 22 November 2012

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1201.5152

Mathematics Subject Classification ID

Related Items (18)

Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity. I ⋮ Normal forms, stability and splitting of invariant manifolds. I: Gevrey Hamiltonians ⋮ Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity. II: The generic case ⋮ Breakdown of homoclinic orbits to \(L_3\) in the RPC3BP. I: Complex singularities and the inner equation ⋮ Critical velocity in kink-defect interaction models: rigorous results ⋮ Stationary phase methods and the splitting of separatrices ⋮ Response Solutions for Quasi-Periodically Forced, Dissipative Wave Equations ⋮ Separatrix splitting at a Hamiltonian \(0^2i\omega\) bifurcation ⋮ Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio ⋮ Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton-Jacobi equation ⋮ On the asymptotic wavenumber of spiral waves in $\lambda -\omega $ systems ⋮ Breakdown of homoclinic orbits to \(L_3\) in the RPC3BP. II: An asymptotic formula ⋮ Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies ⋮ Exponentially small splitting: a direct approach ⋮ Oscillatory motions for the restricted planar circular three body problem ⋮ Exponentially Small Splitting of Separatrices and Transversality Associated to Whiskered Tori with Quadratic Frequency Ratio ⋮ Symbolic dynamics in the restricted elliptic isosceles three body problem ⋮ High order Melnikov method: pendulums

Cites Work

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• Exponentially small splitting for the pendulum: A classical problem revisited
• Resurgence of inner solutions for perturbations of the McMillan map
• Exponentially small splitting of separatrices in the perturbed McMillan map
• Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity
• The inner equation for generic analytic unfoldings of the Hopf-zero singularity
• High-precision computations of divergent asymptotic series and homoclinic phenomena
• Splitting of separatrices for the Chirikov standard map
• The separation of motions in systems with rapidly rotating phase
• Multiphase averaging for classical systems. With applications to adiabatic theorems. Transl. from the French by H. S. Dumas
• An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum
• Hyperbolic tori and asymptotic surfaces in Hamiltonian systems
• Splitting of separatrices for a pendulum with rapidly oscillating suspension point
• Separatrix splitting for systems with three time scales
• A proof of the exponentially small transversality of the separatrices for the standard map
• Parabolic orbits in the elliptic restricted three-body problem
• Exponentially small splitting of separatrices under fast quasiperiodic forcing
• Melnikov potential for exact symplectic maps
• Exponentially small splitting of separatrices for perturbed integrable standard-like maps
• Resurgence in a Hamilton-Jacobi equation.
• Oscillatory integrals and phenomena beyond all algebraic orders with applications to homoclinic orbits in reversible systems
• Separatrix splitting for a high-frequency perturbation of the pendulum.
• Exponentially small splitting of separatrices in a weakly hyperbolic case
• Exponentially small splitting for whiskered tori in Hamiltonian systems: Flow-box coordinates and upper bounds
• Oscillatory solutions in the planar restricted three-body problem
• Splitting potential and the Poincaré-Melnikov method for whiskered tori in Hamiltonian systems
• Résurgence paramétrique et exponentielle petitesse de l'écart des séparatrices du pendule rapidement forcé. (Parametric resurgence and exponential smallness of the splitting of the separatrices of the rapidly forced pendulum.)
• Rapidly forced planar vector fields and splitting of separatrices
• Transcendentally small transversality in the rapidly forced pendulum
• A new method for measuring the splitting of invariant manifolds
• Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift–Hohenberg equation as an example
• Asymptotics beyond All Orders in a Model of Crystal Growth
• Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps
• Reference systems for splitting of separatrices
• Adiabatic Invariant of the Harmonic Oscillator, Complex Matching and Resurgence
• Singular Separatrix Splitting and the Melnikov Method: An Experimental Study
• Biasymptotic solutions of perturbed integrable Hamiltonian systems
• TWISTLESS KAM TORI, QUASI FLAT HOMOCLINIC INTERSECTIONS, AND OTHER CANCELLATIONS IN THE PERTURBATION SERIES OF CERTAIN COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS: A REVIEW
• Exponentially small splittings in Hamiltonian systems
• Exponentially small splitting of invariant manifolds of parabolic points
• Exponentially small upper bounds for the splitting of separatrices for high frequency periodic perturbations
• On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems
• Discretization of homoclinic orbits, rapid forcing and “invisible” chaos
• The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems
• Borel summation and splitting of separatrices for the Hénon map
• Melnikov method and exponentially small splitting of separatrices

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