# 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…

## 2 Citations

Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations

- Mathematics, PhysicsJ. Nonlinear Sci.
- 2018

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

- Mathematics
- 2013

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

- Mathematics
- 2010

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

- Mathematics
- 2005

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

- Mathematics
- 2003

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

- Mathematics
- 1998

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

- Mathematics, Computer ScienceJ. Nonlinear Sci.
- 2000

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

- Mathematics
- 1997

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

- Mathematics
- 1997

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

- Mathematics
- 2001

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

- Mathematics
- 1983

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

- Mathematics
- 2012

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 …