Carles Simó

In this work an exhaustive numerical and analytical investigation of the dynamics of a bi-parametric symplectic map, the so-called rational standard map, at moderate-to-large values of the amplitude parameter is addressed. After reviewing the model, a discussion concerning an analytical determination of the maximum Lyapunov exponent is provided together with thorough numerical experiments. The theoretical results are obtained in the limit of a nearly uniform distribution of the phase values. Correlations among phases lead to departures from the expected estimates. In this direction, a detailed study of the role of stable periodic islands of periods 1, 2 and 4 is included. Finally, an experimental relationship between the Lyapunov and instability times is shown, while an analytical one applies when correlations are irrelevant, which is the case, in general, for large values of the amplitude parameter.

We consider the Chirikov standard map for values of the parameter larger than but close to Greene's $k_G$. We investigate the dynamics near the golden Cantorus and study escape rates across it. Mackay [17, 19] described the behaviour of the mean of the number of iterates $\langle N_k \rangle$ to cross the Cantorus as $k\to k_G$ and showed that there exists $B<0$ so that $\langle N_k\rangle (k-k_G)^B$ becomes 1-periodic in a suitable logarithmic scale. The numerical explorations here give evidence of the shape of this periodic function and of the relation between the escape rates and the evolution of the stability islands close to the Cantorus.

We consider a family of simple flows in tori that display chaotic behavior in a wide sense. But these flows do not have homoclinic nor heteroclinic orbits. They have only a fixed point which is of parabolic type. However, the dynamics returns infinitely many times near the fixed point due to quasi-periodicity. A preliminary example is given for maps introduced in a paper containing many examples of strange attractors in [6]. Recently, a family of maps similar to the flows considered here was studied in [9]. In the present paper we consider the case of 2D tori and the extension to tori of arbitrary finite dimension. Some other facts about exceptional frequencies and behavior around parabolic fixed points are also included.

The effects of quasi-periodicity on the splitting of invariant manifolds are examined. We have found that some harmonics that could be expected to be dominant in some ranges of the perturbation parameter actually are nondominant. It is proved that, under reasonable conditions, this is due to the arithmetic properties of the frequencies.

Invariant manifolds of a periodic orbit at infinity in the planar circular RTBP are studied. To this end we consider the intersection of the manifolds with the passage through the barycentric pericenter. The intersections of the stable and unstable manifolds have a common even part, which can be seen as a displaced version of the two-body problem, and an odd part which gives rise to a splitting. The theoretical formulas obtained for a Jacobi constant $C$ large enough are compared to direct numerical computations showing improved agreement when $C$ increases. A return map to the pericenter passage is derived, and using an approximation by standard-like maps, one can make a prediction of the location of the boundaries of bounded motion. This result is compared to numerical estimates, again improving for increasing $C$. Several anomalous phenomena are described.

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 orientationreversing 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 $k \to \infty$. Elementary considerations about diffusion properties of the standard map are also presented.

In this paper we investigate numerically the following Hill’s equation $x'' + (a + bq(t))x = 0$ where $q(t) = cost + \cos\sqrt{2}t + \cos\sqrt{3}t$ is a quasi-periodic forcing with three rationally independent frequencies. It appears, also, as the eigenvalue equation of a Schrödinger operator with quasi-periodic potential.
Massive numerical computations were performed for the rotation number and the Lyapunov exponent in order to detect open and collapsed gaps, resonance tongues. Our results show that the quasi-periodic case with three independent frequencies is very different not only from the periodic analogs, but also from the case of two frequencies. Indeed, for large values of $b$ the spectrum contains open intervals at the bottom. From a dynamical point of view we numerically give evidence of the existence of open intervals of $a$, for large $b$, where the system is nonuniformly hyperbolic: the system does not have an exponential dichotomy but the Lyapunov exponent is positive. In contrast with the region with zero Lyapunov exponents, both the rotation number and the Lyapunov exponent do not seem to have square root behavior at endpoints of gaps. The rate of convergence to the rotation number and the Lyapunov exponent in the nonuniformly hyperbolic case is also seen to be different from the reducible case.

This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.

This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order $k$ variational equations, for arbitrary values of $k$, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear springpendulum problem for the values of the parameter that can not be decided using first order variational equations.

Area preserving maps close to integrable but not satisfying the twist condition are studied. The existence of invariant curves is proved, but they are no longer graphs with respect to the angular variable. Beyond the generic, codimension 1 case, several higher codimension cases are studied. Meandering curves, higher order meandering and labyrinthic curves show up. Several examples illustrate that this behavior occurs in very simple families of maps.

We present a simple method which displays a hyperbolic structure in the phase space of an area preserving map. The method is illustrated for the case of the Carleson standard map. As it follows from our experiments, the structure of the chaotic zone for the standard map is different from the one found for the systems of Anosov type.