A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system
Frederic Gabern Àngel Jorba
We focus on the dynamics of a small particle near the Lagrangian points of the Sun-Jupiter system. To try to account for the effect of Saturn, we develop a specific model based on the computation of a true solution of the planar three-body problem for Sun, Jupiter and Saturn, close to the real motion of these three bodies. Then, we can write the equations of motion of a fourth infinitesimal particle moving under the attraction of these three masses. Using suitable coordinates, the model is written as a time-dependent perturbation of the well-known spatial Restricted Three-Body Problem.
Then, we study the dynamics of this model near the triangular points. The tools are based on computing, up to high order, suitable normal forms and first integrals. From these expansions, it is not difficult to derive approximations to invariant tori (of dimensions 2, 3 and 4) as well as bounds on the speed of diffusion on suitable domains. We have also included some comparisons with the motion of a few Trojan asteroids in the real Solar system.
keywords: first integrals invariant tori. Trojan asteroids normal forms
On the high order approximation of the centre manifold for ODEs
Ariadna Farrés Àngel Jorba
Many times in dynamical systems one wants to understand the bounded motion around an equilibrium point. From a numerical point of view, we can take arbitrary initial conditions close to the equilibrium points, integrate the trajectories and plot them to have a rough idea of motion. If the dimension of the phase space is high, we can take suitable Poincaré sections and/or projections to visualise the dynamics. Of course, if the linear behaviour around the equilibrium point has an unstable direction, this procedure is useless as the trajectories will escape quickly. We need to get rid, in some way, of the instability of the system.
   Here we focus on equilibrium points whose linear dynamics is a cross product of one hyperbolic directions and several elliptic ones. We will compute a high order approximation of the centre manifold around the equilibrium point and use it to describe the behaviour of the system in an extended neighbourhood of this point. Our approach is based on the graph transform method. To derive an efficient algorithm we use recurrent expressions for the expansion of the non - linear terms on the equations of motion.
   Although this method does not require the system to be Hamiltonian, we have taken a Hamiltonian system as an example. We have compared its efficiency with a more classical approach for this type of systems, the Lie series method. It turns out that in this example the graph transform method is more efficient than the Lie series method. Finally, we have used this high order approximation of the centre manifold to describe the bounded motion of the system around and unstable equilibrium point.
keywords: Normaly Hyperbolic Invariant Manifolds. Graph Transfrom
Tomás Caraballo Amadeu Delshams Àngel Jorba Peter E. Kloeden Rafael Obaya
The theory of dynamical systems has undergone some spectacular and fascinating developments in the past century, as the readers of this journal are well aware, with the focus predominately on autonomous systems. There are many ways in which one could classify the work that has been done, but one that stands clearly in the forefront is the distinction between dissipative systems with their attractors and conservative systems, in particular Hamiltonian systems.
    Another classification is between autonomous and nonautonomous systems. Of course, the latter subsumes the former as special case, but with the former having special structural features, i.e., the semigroup evolution property, which has allowed an extensive and seemingly complete theory to be developed. Although not as extensive, there have also been significant developments in the past half century on nonautonomous dynamical systems, in particular the skew-product formalism involving a cocycle evolution property which generalizes the semigroup property of autonomous systems. This has been enriched in recent years by advances on random dynamical systems, which are roughly said a measure theoretic version of a skew-product flow. In particular, new concepts of random and nonautonomous attractors have been introduced and investigated.

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A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps
Àngel Jorba Joan Carles Tatjer
We introduce a scenario for the fractalization of invariant curves for a special class of quasi-periodically forced 1-D maps. In this situation, a smooth invariant curve becomes increasingly wrinkled when its Lyapunov exponent goes to zero, but it keeps being smooth as long as its exponent is negative. It is remarkable that the curve becomes so wrinkled that numerical simulations may not distinguish the curve from a strange attracting set.
    Moreover, we show that a nonreducible invariant curve with a positive Lyapunov exponent is not persistent in a general quasi-periodically forced 1-D map. We also derive some new results on the behaviour of the Lyapunov exponent of an invariant curve w.r.t. parameters.
The paper contains some numerical examples. One of them is based on the quasi-periodically forced logistic map, where we show numerically that the fractalization of an invariant curve of this system may fit into our scenario.
keywords: Lyapunov exponents reducibility.
Local study of a renormalization operator for 1D maps under quasiperiodic forcing
Àngel Jorba Pau Rabassa Joan Carles Tatjer
The authors have recently introduced an extension of the classical one dimensional (doubling) renormalization operator to the case where the one dimensional map is forced quasiperiodically. In the classic case the dynamics around the fixed point of the operator is key for understanding the bifurcations of one parameter families of one dimensional unimodal maps. Here we perform a similar study of the (linearised) dynamics around the fixed point for further application to quasiperiodically forced unimodal maps.
keywords: discretization. Quasiperiodically forced maps renormalization
Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case
Tingting Zhang Àngel Jorba Jianguo Si
In this work we consider a class of degenerate analytic maps of the form \begin{eqnarray*} \left\{ \begin{array}{l} \bar{x} =x+y^{m}+\epsilon f_1(x,y,\theta,\epsilon)+h_1(x,y,\theta,\epsilon),\\ \bar{y}=y+x^{n}+\epsilon f_2(x,y,\theta,\epsilon)+h_2(x,y,\theta,\epsilon),\\ \bar{\theta}=\theta+\omega, \end{array} \right. \end{eqnarray*} where $mn>1,n\geq m,$ $h_1 \ \mbox{and} \ h_2$ are of order $n+1$ in $z,$ and $\omega=(\omega_1,\omega_2,\ldots,\omega_{d})\in \Bbb{R}^{d}$ is a vector of rationally independent frequencies. It is shown that, under a generic non-degeneracy condition on $f$, if $\omega$ is Diophantine and $\epsilon>0$ is small enough, the map has at least one weakly hyperbolic invariant torus.
keywords: weakly hyperbolic invariant torus degenerate fixed points Quasiperiodic systems KAM iteration.
Amadeu Delshams Ernest Fontich Àngel Jorba
This volume is dedicated to Carles Simó on the occasion of his 60th anniversary and contains papers of the participants of the conference “International Conference on Dynamical Systems, Carles Simó Fest”, celebrated in S’Agaró, near Barcelona, from May 29th to June 3rd, 2006 and organized by former PhD students of him.
  &nbs Carles Simó was born in Barcelona in 1946. He studied Industrial Engineering and Mathematics at once, earning his Ph.D. in Mathematics in 1974.

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Period doubling and reducibility in the quasi-periodically forced logistic map
Àngel Jorba Pau Rabassa Joan Carles Tatjer
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, such as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and 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 a 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.
keywords: fractalization bifurcation cascades skew products Invariant curves reducibility loss.
Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss
Àngel Jorba Pau Rabassa Joan Carles Tatjer
Let $g_{\alpha}$ be a one-parameter family of one-dimensional maps with a cascade of period doubling bifurcations. Between each of these bifurcations, a superstable periodic orbit is known to exist. An example of such a family is the well-known logistic map. In this paper we deal with the effect of a quasi-periodic perturbation (with only one frequency) on this cascade. Let us call $\varepsilon$ the perturbing parameter. It is known that, if $\varepsilon$ is small enough, the superstable periodic orbits of the unperturbed map become attracting invariant curves (depending on $\alpha$ and $\varepsilon$) of the perturbed system. In this article we focus on the reducibility of these invariant curves.
    The paper shows that, under generic conditions, there are both reducible and non-reducible invariant curves depending on the values of $\alpha$ and $\varepsilon$. The curves in the space $(\alpha,\varepsilon)$ separating the reducible (or the non-reducible) regions are called reducibility loss bifurcation curves. If the map satifies an extra condition (condition satisfied by the quasi-periodically forced logistic map) then we show that, from each superattracting point of the unperturbed map, two reducibility loss bifurcation curves are born. This means that these curves are present for all the cascade.
keywords: Invariant curves bifurcation cascades.
On the accuracy of restricted three-body models for the Trojan motion
Frederic Gabern Àngel Jorba Philippe Robutel
In this note we compare the frequencies of the motion of the Trojan asteroids in the Restricted Three-Body Problem (RTBP), the Elliptic Restricted Three-Body Problem (ERTBP) and the Outer Solar System (OSS) model. The RTBP and ERTBP are well-known academic models for the motion of these asteroids, and the OSS is the standard model used for realistic simulations.
Our results are based on a systematic frequency analysis of the motion of these asteroids. The main conclusion is that both the RTBP and ERTBP are not very accurate models for the long-term dynamics, although the level of accuracy strongly depends on the selected asteroid.
keywords: Symplectic integrators frequency analysis Trojan asteroids.

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