From the Hénon conservative map to the Chirikov standard map for large parameter values

In this paper we consider conservative quadratic Hénon maps and Chirikov's standard map, and relate them in some sense. First, we present a study of some dynamical properties of orientation-preserving and orientation-reversing quadratic Hénon maps concerning the stability region, the size of the chaotic zones, its evolution with respect to parameters and the splitting of the separatrices of fixed and periodic points plus its role in the preceding aspects. Then the phase space of the standard map, for large values of the parameter $k$, is studied. There are some stable orbits which appear periodically in $k$ and are scaled somehow. Using this scaling, we show that the dynamics around these stable orbits is one of the above Hénon maps plus some small error, which tends to vanish as $k \rightarrow \infty$. Elementary considerations about diffusion properties of the standard map are also presented

Effect of Islands in Diffusive Properties of the Standard Map for Large Parameter Values

In this paper we review, based on massive, long-term, numerical simulations, the effect of islands on the statistical properties of the standard map for large parameter values. Different sources of discrepancy with respect to typical diffusion are identified, and their individual roles are compared and explained in terms of available limit models

Accelerator modes and anomalous diffusion in 3D volume-preserving maps

Angle-action maps that have a periodicity in the action direction can have accelerator modes: orbits that are periodic when projected onto the torus, but that lift to unbounded orbits in an action variable. In this paper we construct a family of volume-preserving maps, with two angles and one action, that have accelerator modes created at Hopf-one (or saddle-center-Hopf) bifurcations. Near such a bifurcation we show that there is often a bubble of invariant tori. Computations of chaotic orbits near such a bubble show that the trapping times have an algebraic decay similar to that seen around stability islands in area-preserving maps. As in the 2D case, this gives rise to anomalous diffusive properties of the action in our 3D map

Numerical integration of high-order variational equations of ODEs

This paper discusses the numerical integration of high-order variational equationsof ODEs. It is proved that, given a numerical method (say, any Runge-Kutta or Taylor method), to use automatic differentiation on this method (that is, using jet transport up to order $p$ with a time step $h$ for the numerical integration) produces exactly the same results as integrating the variational equationsup to of order $p$ with the same method and time step $h$ as before. This allows to design step-size control strategies based on error estimates of the orbit and of the jets. Finally, the paper discusses how to use jet transport to obtain power expansions of Poincaré maps (either with spatial or temporal Poincaré sections) and invariant manifolds. Some examples are provided