Marcus W. Beims

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.

Conservative Generalized Bifurcation Diagrams

Abstract 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 ∼ 3.9 . Results are demonstrated for the standard map and for the continuous Henon–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.

Accumulation points in nonlinear parameter lattices

In 1963 Myrberg determined a period-doubling cascade of the quadratic map to accumulate at 1.401155189… As found later, the geometric way with which model parameters approach this value has universal behavior and several characteristic exponents associated with it. In the present paper we discuss the existence of an infinite number of points characterized by the simultaneous accumulation of two or more bifurcation cascades. We present an accurate numerical determination of the vertices of a doubly infinite nonlinear lattice which lead to a point of double accumulation. In addition, we discuss the number-theoretic nature of irrationalities characterizing vertices. Novel classes of universality with characteristic exponents are conjectured to exist near points of multiple accumulations.

A generalized semiclassical expression for the eigenvalues of multiple well potentials

From the poles of a generalized semiclassical Greens function we derive expressions for the eigenvalues of 1D multiple well potentials. In the case of asymmetric and symmetric double wells, we also obtain analytical formulae for, respectively, the shift and splitting of energies. Our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy-dependent Maslov indices. We illustrate the good numerical precision of the method by discussing some case tests on double wells.

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.

Asymptotic Green functions: a generalized semiclassical approach for scattering by multiple barrier potentials

We show how quantum mechanical barrier reflection and transmission coefficients and can be obtained from asymptotic Green functions. We exemplify our results by calculating such coefficients for the Rosen-Morse (RM) potential. For multiple barrier potentials, V(x) = ∑jV (j)(x), where each V (j) goes to zero for x→±∞, we derive the asymptotic Green functions by a generalized semiclassical approximation, which is based on the usual sum over classical paths considered only in the classically allowed regions and includes local quantum effects through the individual (j) and (j). The approach is applied to double RM potentials and to Woods-Saxon barriers. We obtain analytical expressions for the transmission and reflection probabilities of these potentials which are very accurate when compared with exact numerical calculations, being much better than the usual WKB approximation. Finally we briefly discuss how to extend the present method to other kinds of potential.

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.

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.

Proliferation of stability in phase and parameter spaces of nonlinear systems

In this work, we show how the composition of maps allows us to multiply, enlarge, and move stable domains in phase and parameter spaces of discrete nonlinear systems. Using Hénon maps with distinct parameters, we generate many identical copies of isoperiodic stable structures (ISSs) in the parameter space and attractors in phase space. The equivalence of the identical ISSs is checked by the largest Lyapunov exponent analysis, and the multiplied basins of attraction become riddled. Our proliferation procedure should be applicable to any two-dimensional nonlinear system.

Alignment of Lyapunov Vectors: A Quantitative Criterion to Predict Catastrophes?

We argue that the alignment of Lyapunov vectors provides a quantitative criterion to predict catastrophes, i.e. the imminence of large-amplitude events in chaotic time-series of observables generated by sets of ordinary differential equations. Explicit predictions are reported for a Rössler oscillator and for a semiconductor laser with optoelectronic feedback.