Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation

@article{Delshams2011TransversalityOH,
  title={Transversality of homoclinic orbits to hyperbolic equilibria in a Hamiltonian system, via the Hamilton–Jacobi equation},
  author={Amadeu Delshams and Pere Guti'errez and Juan R. Pacha},
  journal={Physica D: Nonlinear Phenomena},
  year={2011},
  volume={243},
  pages={64-85}
}
Abstract We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside… 

Figures from this paper

Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations
TLDR
A time-dependent perturbation is applied to a mechanical system consisting of n-penduli and a d-degree-of-freedom rotator, and an explicit formula for the Melnikov vector is provided in terms of convergent improper integrals of the perturbations along homoclinic orbits of the unperturbed system.
Asymptotic trajectories of KAM torus
In this paper we construct a certain type of nearly integrable systems of two and a half degrees of freedom: \[H(p,q,t)=h(p)+\epsilon f(p,q,t),\quad (q,p)\in T^{*}\mathbb{T}^2,t\in

References

SHOWING 1-10 OF 35 REFERENCES
Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel’nikov method
We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n +
Homoclinic orbits to invariant tori near a homoclinic orbit to center-center-saddle equilibrium
We consider a perturbation of an integrable Hamiltonian vector field with three degrees of freedom with a center–center–saddle equilibrium having a homoclinic orbit or loop. With the help of a
Exponentially small splitting for whiskered tori in Hamiltonian systems: Continuation of transverse homoclinic orbits
We consider an example of singular or weakly hyperbolic Hamiltonian, with 3 degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The model
Homoclinic orbits to invariant tori in Hamiltonian systems
We consider a perturbation of an integrable Hamiltonian system which possesses invariant tori with coincident whiskers (like some rotators and a pendulum). Our goal is to measure the splitting
Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems
TLDR
A geometric approach is used closely related to the Lagrangian properties of the whiskers, to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar function of n phases, which is called Melnikov potential.
Melnikov Potential for Exact Symplectic Maps
Abstract:The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called
Universal homoclinic bifurcations and chaos near double resonances
We study the dynamics near the intersection of a weaker and a stronger resonance inn-degree-of-freedom, nearly integrable Hamiltonian systems. For a truncated normal form we show the existence of
A new method for measuring the splitting of invariant manifolds
We study the so-called Generalized Arnol'd Model (a weakly hyperbolic near-integrable Hamiltonian system), with d+1 degrees of freedom (d⩾2), in the case where the perturbative term does not affect a
On the existence of separatrix loops in four-dimensional systems similar to the integrable hamiltonian systems
Abstract A method analogous to the V.K. Mel'nikov method /1/ is used to derive the conditions of existence of separatrix loops of the saddle-focus type singularity, for the systems similar to the
Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results
In this paper we study 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
...
1
2
3
4
...