arXiv subjects

Cesar Manchein

Publications and source records attributed to Cesar Manchein.

32 records · Page 2Linked to original sources

arXiv2013

Bifurcation structures and transient chaos in a four-dimensional Chua model

A four-dimensional four-parameter Chua model with cubic nonlinearity is studied applying numerical continuation and numerical solutions methods. Regarding numerical solution methods, its dynamics is characterized on Lyapunov and isoperiodic diagrams and regarding numerical continuation method, the bifurcation curves are obtained. Combining both methods the bifurcation structures of the model were obtained with the possibility to describe the {\it shrimp}-shaped domains and their endoskeletons. We study the effect of a parameter that controls the dimension of the system leading the model to present transient chaos with its corresponding basin of attraction being riddled.

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arXiv2013

Comment on "Lyapunov statistics and mixing rates for intermittent systems"

In Pires {\it et al.} [Phys. Rev. E 84, 066210 (2011)] intermittent maps are considered, and the tight relationship between correlation decay of smooth observables and large deviations estimates, as for instance employed in Artuso and Manchein [Phys. Rev. E 80, 036210 (2009)], is questioned. We try to clarify the problem, and provide rigorous arguments and an analytic estimate that disprove the objections raised in Pires {\it et al.} [Phys. Rev. E 84, 066210 (2011)] when ergodic systems are considered.

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arXiv2012

Conservative Generalized Bifurcation Diagrams

Bifurcation cascades in conservative systems are shown to exhibit a generalized diagram, which contains all relevant informations regarding the location of periodic orbits (resonances), their width (island size), irrational tori and the infinite higher-order resonances, showing the intricate way they are born. Contraction rates for islands sizes, along period-doubling bifurcations, are estimated to be $α_I\sim 3.9$. Results are demonstrated for the standard map and for the continuous Hénon-Heiles potential. The methods used here are very suitable to find periodic orbits in conservative systems, and to characterize the regular, mixed or chaotic dynamics as the nonlinear parameter is varied.

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arXiv2012

Temperature Resistant Optimal Ratchet Transport

Stable periodic structures containing optimal ratchet transport, recently found in the parameter space dissipation versus ratchet parameter [PRL 106, 234101 (2011)], are shown to be resistant to reasonable temperatures, reinforcing the expectation that they are essential to explain the optimal ratchet transport in nature. Critical temperatures for their destruction, valid from the overdamping to close to the conservative limits, are obtained numerically and shown to be connected to the current efficiency, given here analytically. Results are demonstrated for a discrete ratchet model and generalized to the Langevin equation with an additional external oscillating force.

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arXiv2012

Estimating hyperbolicity of chaotic bidimensional maps

We apply to bidimensional chaotic maps the numerical method proposed by Ginelli et al. to approximate the associated Oseledets splitting, i.e. the set of linear subspaces spanned by the so called covariant Lyapunov vectors (CLV) and corresponding to the Lyapunov spectrum. These subspaces are the analog of linearized invariant manifolds for non-periodic points, so the angles between them can be used to quantify the degree of hyperbolicity of generic orbits; however, being such splitting non invariant under smooth transformations of phase space, it is interesting to investigate the properties of transversality when coordinates change, e.g. to study it in distinct dynamical systems. To illustrate this issue on the Chirikov-Taylor standard map we compare the probability densities of transversality for two different coordinate systems; these are connected by a linear transformation that deforms splitting angles through phase space, changing also the probability density of almost-zero angles although complete tangencies are in fact invariant. This is completely due to the PDF transformation law and strongly suggests that any statistical inference from such distributions must be generally taken with care.

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arXiv2012

Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities the chaotic stripes are the initial condition inside which Arnold diffusion occurs.

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arXiv2011

Generic Structures in Parameter Space and Ratchet Transport

This work reports the existence of Isoperiodic Stable Ratchet Transport Structures in the parameter spaces dissipation versus spatial asymmetry and versus phase of a ratchet model. Such structures were found [Phys. Rev. Lett. 106, 234101 (2011)] in the parameter space dissipation versus amplitude of the ratchet potential and they appear to have generic shapes and to align themselves along preferred directions in the parameter space. Since the ratchet current is usually larger inside these structures, this allows us to make general statements about the relevant parameters combination to obtain an efficient ratchet current. Results of the present work give further evidences of the suggested generic properties of the isoperiodic stable structures in the context of ratchet transport.

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arXiv2011

Ratchet transport and periodic structures in parameter space

Ratchet models are prominent candidates to describe the transport phenomenum in nature in the absence of external bias. This work analyzes the parameter space of a discrete ratchet model and gives direct connections between chaotic domains and a family of isoperiodic stable structures with the ratchet current. The isoperiodic structures appear along preferred direction in the parameter space giving a guide to follow the current, which usually increases inside the structures but is independent of the corresponding period. One of such structures has the shrimp-shaped form which is known to be an universal structure in the parameter space of dissipative systems. Currents in parameter space provide a direct measure of the momentum asymmetry of the multistable and chaotic attractors times the size of the corresponding basin of attraction. Transport structures are shown to exist in the parameter space of the Langevin equation with an external oscillating force.

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arXiv2009

Instability statistics and mixing rates

We claim that looking at probability distributions of \emph{finite time} largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and Poincaré recurrences in the -quite delicate- case of dynamical systems with weak chaotic properties.

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arXiv2009

Dissipation effects in the ratchetlike Fermi acceleration

Ac driven asymmetric pulses can be used to control the Fermi acceleration between three different motions, ${\bf A:}$ the {\it accelerated} mode; ${\bf D:}$ the {\it decelerated} mode and ${\bf H:}$ the {\it hyperaccelerated} mode. In this work we show that dissipation strongly affects the particles velocity, reducing the possibility for an accurate control of the dynamics. The saturation time, where the mean velocity starts to be constant due to dissipation, decays with a power law $\simγ^{-β}$, where $γ$ is the dissipation parameter and $β$ is close to 1. The value of the saturated mean velocity also decays with a power law with exponent $β\sim0.6$ for the case ${\bf H}$, and $β\sim0.3$ for the case ${\bf A}$. In the case ${\bf D}$ this velocity is almost constant for small dissipations.

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arXiv2009

Chaotic motion at the emergence of the time averaged energy decay

A system plus environment conservative model is used to characterize the nonlinear dynamics when the time averaged energy for the system particle starts to decay. The system particle dynamics is regular for low values of the $N$ environment oscillators and becomes chaotic in the interval $13\le N\le15$, where the system time averaged energy starts to decay. To characterize the nonlinear motion we estimate the Lyapunov exponent (LE), determine the power spectrum and the Kaplan-Yorke dimension. For much larger values of $N$ the energy of the system particle is completely transferred to the environment and the corresponding LEs decrease. Numerical evidences show the connection between the variations of the {\it amplitude} of the particles energy time oscillation with the time averaged energy decay and trapped trajectories.

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arXiv2009

Origin of chaos in soft interactions and signatures of nonergodicity

The emergence of chaotic motion is discussed for hard-point like and soft collisions between two particles in a one-dimensional box. It is known that ergodicity may be obtained in hard-point like collisions for specific mass ratios $γ=m2/m1$ of the two particles and that Lyapunov exponents are zero. However, if a Yukawa interaction between the particles is introduced, we show analytically that positive Lyapunov exponents are generated due to double collisions close to the walls. While the largest finite-time Lyapunov exponent changes smoothly with $γ$, the number of occurrences of the most probable one, extracted from the distribution of finite-time Lyapunov exponents over initial conditions, reveals details about the phase-space dynamics. In particular, the influence of the integrable and pseudointegrable dynamics without Yukawa interaction for specific mass ratios can be clearly identified and demonstrates the sensitivity of the finite-time Lyapunov exponents as a phase-space probe. Being not restricted to two-dimensional problems such as Poincaré sections, the number of occurrences of the most probable Lyapunov exponents suggests itself as a suitable tool to characterize phase-space dynamics in higher dimensions. This is shown for the problem of two interacting particles in a circular billiard.

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arXiv2009

Gauss map and Lyapunov exponents of interacting particles in a billiard

We show that the Lyapunov exponent (LE) of periodic orbits with Lebesgue measure zero from the Gauss map can be used to determine the main qualitative behavior of the LE of a Hamiltonian system. The Hamiltonian system is a one-dimensional box with two particles interacting via a Yukawa potential and does not possess Kolmogorov-Arnold-Moser (KAM) curves. In our case the Gauss map is applied to the mass ratio $γ= m_2/m_1$ between particles. Besides the main qualitative behavior, some unexpected peaks in the $γ$ dependence of the mean LE and the appearance of 'stickness' in phase space can also be understand via LE from the Gauss map. This shows a nice example of the relation between the "instability" of the continued fraction representation of a number with the stability of non-periodic curves (no KAM curves) from the physical model. Our results also confirm the intuition that pseudo-integrable systems with more complicated invariant surfaces of the flow (higher genus) should be more unstable under perturbation.

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arXiv2009

Ratchetlike pulse controlling the Fermi deceleration and hyperacceleration

Using an ac driven asymmetric pulse we show how the Fermi acceleration (deceleration) can be controlled. A {\it deformed} sawtooth (Ratchetlike) pulse representing the moving wall in the static Fermi-Ulam model is considered. The time integral from the pulse over one period of oscillation must be negative to obtain deceleration and positive to obtain hyperacceleration. We show that while the decelerated case is chaotic, for the hyperaccelerated case the Lyapunov exponents converge to zero. Numerical simulations indicate that the hyperaccelerated case is ergodic in velocity space. Switching between different pulse deformations we are able to control the particle acceleration. Results should be valid for any pulse for which the time integral can be manipulated between positive and negative values.

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