On the accumulation of separatrices by invariant circles

Let f be a smooth symplectic diffeomorphism of ${\mathbb R}^2$ admitting a (non-split) separatrix associated to a hyperbolic fixed point. We prove that if f is a perturbation of the time-1 map of a symplectic autonomous vector field, this separatrix is accumulated by a positive measure set of invariant circles. However, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles.

Aubry-Mather type at the Double Resonance

This chapter proves Aubry-Mather type for the double resonance regime. It begins by considering the “non-critical energy case” and showing that the cohomologies as chosen are of Aubry-Mather type. The proof consists of two cases. In the first case, the chapter uses the almost verticality of the cylinder, and the idea is similar to the proof of Theorem 9.3. It applies the a priori Lipschitz estimates for the Aubry sets. In the second case, the chapter uses the strong Lipschitz estimate for the energy, and the idea is similar to the proof of Theorem 11.1. It then looks at the construction of the local coordinates. This is done separately near the hyperbolic fixed point (local) and away from it (global).

From homoclinics to quasi-periodic solutions for ordinary differential equations

We consider the quasi-periodic solutions bifurcated from a degenerate homoclinic solution. Assume that the unperturbed system has a homoclinic solution and a hyperbolic fixed point. The bifurcation function for the existence of a quasi-periodic solution of the perturbed system is obtained by functional analysis methods. The zeros of the bifurcation function correspond to the existence of the quasi-periodic solution at the non-zero parameter values. Some solvable conditions of the bifurcation equations are investigated. Two examples are given to illustrate the results.

HIGH-ORDER BISECTION METHOD FOR COMPUTING INVARIANT MANIFOLDS OF TWO-DIMENSIONAL MAPS

We describe an efficient and accurate numerical method for computing smooth approximations to invariant manifolds of planar maps, based on geometric modeling ideas from Computer Aided Geometric Design (CAGD). The unstable manifold of a hyperbolic fixed point is modeled by a piecewise Bézier interpolant (a Catmull–Rom spline) and properties of such curves are used to define a rule for adaptively adding points to ensure that the approximation resolves the manifold to within a specified tolerance. Numerical tests on a variety of example mappings demonstrate that the new method produces a manifold of a given accuracy with far fewer calls to the map, compared with previous methods. A brief introduction to the relevant ideas from CAGD is provided.