Decay of the distance autocorrelation and Lyapunov exponents.

This work presents numerical evidence that for discrete dynamical systems with one positive Lyapunov exponent the decay of the distance autocorrelation is always related to the Lyapunov exponent. Distinct decay laws for the distance autocorrelation are observed for different systems, namely, exponential decays for the quadratic map, logarithmic for the Hénon map, and power-law for the conservative standard map. In all these cases the decay exponent is close to the positive Lyapunov exponent. For hyperchaotic conservative systems the power-law decay of the distance autocorrelation is not directly related to any Lyapunov exponent.

Controlling ratchet transport via a finite kicked environment.

We study the effects of a finite kicked environment (bath) composed of N harmonic oscillators on the particle transport in a weakly dissipative quasisymmetric potential system. The small spatial asymmetry is responsible for the appearance of directed particle transport without a net bias, known as the ratchet transport. The whole dynamics is governed by a generalized map where dissipation in the system emerges due to its interaction with the kicked environment. Distinct spectral densities are imposed to the bath oscillators and play an essential role in such models. By changing the functional form of the spectral density, we observe that the transport can be optimized or even suppressed. We show evidences that the transport optimization is related to stability properties of periodic points of the ratchet system and depends on the bath temperature. In a Markovian approach, transport can be increased or suppressed depending on the bath influence.

Dissipative dynamics in a finite chaotic environment: Relationship between damping rate and Lyapunov exponent.

We consider the energy flow between a classical one-dimensional harmonic oscillator and a set of N two-dimensional chaotic oscillators, which represents the finite environment. Using linear response theory we obtain an analytical effective equation for the system harmonic oscillator, which includes a frequency dependent dissipation, a shift, and memory effects. The damping rate is expressed in terms of the environment mean Lyapunov exponent. A good agreement is shown by comparing theoretical and numerical results, even for environments with mixed (regular and chaotic) motion. Resonance between system and environment frequencies is shown to be more efficient to generate dissipation than larger mean Lyapunov exponents or a larger number of bath chaotic oscillators.

Characterizing weak chaos using time series of Lyapunov exponents.

We investigate chaos in mixed-phase-space Hamiltonian systems using time series of the finite-time Lyapunov exponents. The methodology we propose uses the number of Lyapunov exponents close to zero to define regimes of ordered (stickiness), semiordered (or semichaotic), and strongly chaotic motion. The dynamics is then investigated looking at the consecutive time spent in each regime, the transition between different regimes, and the regions in the phase space associated to them. Applying our methodology to a chain of coupled standard maps we obtain (i) that it allows for an improved numerical characterization of stickiness in high-dimensional Hamiltonian systems, when compared to the previous analyses based on the distribution of recurrence times; (ii) that the transition probabilities between different regimes are determined by the phase-space volume associated to the corresponding regions; and (iii) the dependence of the Lyapunov exponents with the coupling strength.

Quantum-classical transition and quantum activation of ratchet currents in the parameter space.

The quantum ratchet current is studied in the parameter space of the dissipative kicked rotor model coupled to a zero-temperature quantum environment. We show that vacuum fluctuations blur the generic isoperiodic stable structures found in the classical case. Such structures tend to survive when a measure of statistical dependence between the quantum and classical currents are displayed in the parameter space. In addition, we show that quantum fluctuations can be used to overcome transport barriers in the phase space. Related quantum ratchet current activation regions are spotted in the parameter space. Results are discussed based on quantum, semiclassical, and classical calculations. While the semiclassical dynamics involves vacuum fluctuations, the classical map is driven by thermal noise.

Finite kicked environments and the fluctuation-dissipation relation.

In this work we derive a generalized map for a system coupled to a kicked environment composed of a finite number N of uncoupled harmonic oscillators. Dissipation is introduced via the interaction between system and environment which is switched on and off simultaneously (kicks) at regular time intervals. It is shown that kicked environments naturally generate a non-Markovian rotated dynamics, describe more complicated system-environment couplings which involve position and momentum, and satisfy an unusual fluctuation-dissipation relation. As an example, the motion of a kicked Brownian particle is discussed.

Temperature resistant optimal ratchet transport.

Stable periodic structures containing optimal ratchet transport, recently found in the parameter space dissipation versus ratchet parameter by [A. Celestino et al. Phys. Rev. Lett. 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. A region where thermal activation of the rachet current takes place is also found, and its underlying mechanism is unveiled. Results are demonstrated for a discrete ratchet model and generalized to the Langevin equation with an additional external oscillating force.

Three unequal masses on a ring and soft triangular billiards.

The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the masses ratio between particles and softness ratio of the particles interaction. The transition from soft to hard interactions can be appropriately explored using potentials for which the corresponding equations of motion are well defined in the hard wall limit. Numerical examples are shown for the soft Toda-like interaction and the error function.

Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems.

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) 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.

Intrinsic stickiness and chaos in open integrable billiards: tiny border effects.

Rounding border effects at the escape point of open integrable billiards are analyzed via the escape-time statistics and emission angles. The model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form. Stickiness, chaos, and self-similar structures for the escape times and emission angles are generated inside "backgammon" like stripes of initial conditions. These stripes are born at the boundary between two different emission angles but with the same escape times and when rounding effects increase they start to overlap generating a very rich dynamics. Tiny rounded borders (around 0.1% from the whole billiard size) are shown to be sufficient to generate the sticky motion with power-law decay γ(esc)=1.27, while borders larger than 10% are enough to produce escape times related to the chaotic motion. Escape exponents in the interval 1<γ(esc)<2 are generated due to marginal unstable periodic orbits trapping alternately (in time) regular and chaotic trajectories.

Ratchet transport and periodic structures in parameter space.

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, where larger currents are usually observed inside, appear along preferred direction in the parameter space giving a guide to follow the current. 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.

Soft wall effects on interacting particles in billiards.

The effect of physically realizable wall potentials (soft walls) on the dynamics of two interacting particles in a one-dimensional (1D) billiard is examined numerically. The 1D walls are modeled by the error function and the transition from hard to soft walls can be analyzed continuously by varying the softness parameter sigma . For sigma-->0 the 1D hard wall limit is obtained and the corresponding wall force on the particles is the delta function. In this limit the interacting particle dynamics agrees with previous results obtained for the 1D hard walls. We show that the two interacting particles in the 1D soft walls model is equivalent to one particle inside a soft right triangular billiard. Very small values of sigma substantiously change the dynamics inside the billiard and the mean finite-time Lyapunov exponent decreases significantly as the consequence of regular islands which appear due to the low-energy double collisions (simultaneous particle-particle-1D wall collisions). The rise of regular islands and sticky trajectories induced by the 1D wall softness is quantified by the number of occurrences of the most probable finite-time Lyapunov exponent. On the other hand, chaotic motion in the system appears due to the high-energy double collisions. In general we observe that the mean finite-time Lyapunov exponent decreases when sigma increases, but the number of occurrences of the most probable finite-time Lyapunov exponent increases, meaning that the phase-space dynamics tends to be more ergodiclike. Our results suggest that the transport efficiency of interacting particles and heat conduction in periodic structures modeled by billiards will strongly be affected by the smoothness of physically realizable walls.

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 gamma=m(2)/m(1) 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 gamma , 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.