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### Open Access Journals

DCDS-B

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.

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.

DCDS-B

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.

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.

DCDS

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.

For more information please click the “Full Text” above.

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.

For more information please click the “Full Text” above.

keywords:

DCDS-B

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.

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.

DCDS-S

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.

DCDS

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.

DCDS-B

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.

For the full preface, please click the Full Text "PDF" button above.

&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.

For the full preface, please click the Full Text "PDF" button above.

keywords:

DCDS-B

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.

DCDS

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.

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.

DCDS

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.

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.

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