Period doubling and reducibility in the quasi-periodically forced logistic map

We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and its reducibility. This reveals that the parameter values for which the invariant curve doubles its period are contained in regions of the parameter space where the invariant curve is reducible. Then we present two additional studies to explain this fact. In first place we consider the images and the preimages of the critical set (the set where the derivative of the map w.r.t the non-periodic coordinate is equal to zero). Studying these sets we construct constrains in the parameter space for the reducibility of the invariant curve. In second place we consider the reducibility loss of the invariant curve as codimension one bifurcation and we study its interaction with the period doubling bifurcation. This reveals that, if the reducibility loss and the period doubling bifurcation curves meet, they do it in a tangent way

Numerical computation of normal forms around some periodic orbits of the restricted three body problem

In this paper we introduce a general methodology for computing (numerically) the normal form around a periodic orbit of an autonomous analytic Hamiltonian system. The process follows two steps. First, we expand the Hamiltonian in suitable coordinates around the orbit and second, we perform a standard normal form scheme, based on the Lie series method. This scheme is carried out up to some finite order and, neglecting the remainder, we obtain an accurate description of the dynamics in a (small enough) neighbourhood of the orbit. In particular, we obtain the invariant tori that generalize the elliptic directions of the periodic orbit. On the other hand, bounding the remainder one obtains lower estimates for the diffusion time around the orbit. This procedure is applied to an elliptic periodic orbit of the spatial Restricted Three Body Problem. The selected orbit belongs to the Lyapunov family associated to the vertical oscillation of the equilibrium point $L_5$. The mass parameter $\mu$ has been chosen such that $L_5$ is unstable but the periodic orbit is still stable. This allows to show the existence of regions of effective stability near $L_5$ for values of $\mu$ bigger that the Routh critical value. The computations have been done using formal expansions with numerical coefficients

On the persistence of lower dimensional invariant tori under quasiperiodic perturbations

In this work we consider time dependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the effect that this kind of perturbations has on lower dimensional invariant tori. Our results show that, under standard conditions of analyticity, nondegeneracy and nonresonance, most of these tori survive, adding the frequencies of the perturbation to the ones they already have. The paper also contains estimates on the amount of surviving tori. The worst situation happens when the initial tori are normally elliptic. In this case, a torus (identified by the vector of intrinsic frequencies) can be continued with respect to a perturbative parameter $\epsilon\in[0,\epsilon_0]$, except for a set of $\epsilon$ of measure exponentially small with $\epsilon_0$. In case that $\epsilon$ is fixed (and sufficiently small), we prove the existence of invariant tori for every vector of frequencies close to the one of the initial torus, except for a set of frequencies of measure exponentially small with the distance to the unperturbed torus. As a particular case, if the perturbation is autonomous, these results also give the same kind of estimates on the measure of destroyed tori. Finally, these results are applied to some problems of celestial mechanics, in order to help in the description of the phase space of some concrete models

Invariant curves near Hamiltonian-Hopf bifurcations of 4D symplectic maps

In this paper we give a numerical description of the neighbourhood of a fixed point of a symplectic map undergoing a transition from linear stability to complex instability, i.e., the so called Hamiltonian-Hopf bifurcation. We have considered both the direct and inverse cases. The study is based on the numerical computation of the Lyapunov families of invariant curves near the fixed point. We show how these families, jointly with their invariant manifolds and the invariant manifolds of the fixed point organise the phase space around the bifurcation

On the normal behaviour of partially elliptic lower dimensional tori of hamiltonian systems

The purpose of this paper is to study the dynamics near a reducible lower dimensional invariant tori of a finite-dimensional autonomous Hamiltonian system with $\ell$ degrees of freedom. We will focus in the case in which the torus has (some) elliptic directions. First, let us assume that the torus is totally elliptic. In this case, it is shown that the diffusion time (the time to move away from the torus) is exponentially big with the initial distance to the torus. The result is valid, in particular, when the torus is of maximal dimension and when it is of dimension 0 (elliptic point). In the maximal dimension case, our results coincide with previous ones. In the zero dimension case, our results improve the existing bounds in the literature. Let us assume now that the torus (of dimension $r$, $0\le r<\ell$) is partially elliptic (let us call $m_e$ to the number of these directions). In this case we show that, given a fixed number of elliptic directions (let us call $m_1\le m_e$ to this number), there exist a Cantor family of invariant tori of dimension $r+m_1$, that generalize the linear oscillations corresponding to these elliptic directions. Moreover, the Lebesgue measure of the complementary of this Cantor set (in the frequency space \RR^{r+m_1}) is proven to be exponentially small with the distance to the initial torus. This is a sort of ``Cantorian central manifold'' theorem, in which the central manifold is completely filled up by invariant tori and it is uniquely defined. The proof of these results is based on the construction of suitable normal forms around the initial torus