Ricardo Chacón

General results on chaos suppression for biharmonically driven dissipative systems

Abstract General results concerning suppression of homoclinic (and heteroclinic) chaos are derived on the basis of a Melnikov analysis for damped, nonlinear, and low-dimensional oscillators subjected to two weak harmonic excitations (one chaos-inducing and the other chaos-suppressing). Analytical expressions are deduced for the intervals of initial phase difference between the two excitations for which chaotic dynamics can be eliminated. It is demonstrated that 0,π/2,π,3π/2 are, in general, the only optimal values of such phase differences, in the sense that they allow the widest amplitude ranges for the chaos-suppressing excitation.

TAMING CHAOS IN A DRIVEN JOSEPHSON JUNCTION

We present analytical and numerical results concerning the inhibition of chaos in a single driven Josephson junction by means of an additional weak resonant perturbation. From Melnikov analysis, we theoretically find parameter-space regions, associated with the chaos-suppressing perturbation, where chaotic states can be suppressed. In particular, we test analytical expressions for the intervals of initial phase difference between the two excitations for which chaotic dynamics can be eliminated. All the theoretical predictions are in overall good agreement with numerical results obtained by simulation.

ROLE OF PARAMETRIC RESONANCE IN THE INHIBITION OF CHAOTIC ESCAPE FROM A POTENTIAL WELL

Abstract This paper shows how a periodic parametric modulation can inhibit chaotic escape of a driven oscillator from the cubic potential well that typically models a metastable system close to a fold. Melnikov analysis shows that, depending on its amplitude, period, and initial phase, a periodic parametric modulation of the linear potential term suppresses chaotic escape when certain resonance conditions are met. In particular, it is shown that chaotic escape suppression is impossible under a period-1 parametric perturbation. The effect of nonlinear damping on the inhibition scenario is also studied.

Melnikov method approach to control of homoclinic/heteroclinic chaos by weak harmonic excitations

A review on the application of Melnikovs method to control homoclinic and heteroclinic chaos in low-dimensional, non-autonomous and dissipative oscillator systems by weak harmonic excitations is presented, including diverse applications, such as chaotic escape from a potential well, chaotic solitons in Frenkel–Kontorova chains and chaotic-charged particles in the field of an electrostatic wave packet.

Comparison between parametric excitation and additional forcing terms as chaos-suppressing perturbations

Abstract The inhibition of chaos in the damped, driven one-well Duffing oscillator by application of weak parametric excitations or small additional forcings is studied theoretically by means of Melnikovs analysis. It is demonstrated that the “maximum survival” of the symmetries, under chaos-suppressing excitations, leads to the optimal values of the initial phase differences between the primary chaos-inducing and chaos-suppressing excitations.

Chaos and geometrical resonance in the damped pendulum subjected to periodic pulses

The chaotic behavior of a damped pendulum driven by a periodic string of pulses is studied by means of Melnikov’s analysis. The reduction of homoclinic chaos, in the asymptotic case of infinite period driving, is explained in terms of geometrical resonance.

Role of nonlinear dissipation in the suppression of chaotic escape from a potential well

The inhibition of chaotic escape from a universal escape oscillator due to a periodic parametric perturbation of the quadratic potential term is studied theoretically by means of Poincare–Melnikov–Arnold analysis, and the predictions are tested against numerical simulations based on a high-resolution grid of initial conditions. It is shown that chaotic escape suppression is impossible under period-1 and period-2 parametric perturbations. The role of a nonlinear damping term, proportional to the nth power of the velocity, on the inhibition scenario is also discussed.

Inhibition of chaotic escape from a potential well using small parametric modulations

It is shown theoretically for the first time that, depending on its period, amplitude, and initial phase, a periodic parametric modulation can suppress a chaotic escape from a potential well. The instance of the Helmholtz oscillator is used to demonstrate, by means of Melnikov’s method, that parametric modulations of the linear or quadratic potential terms inhibit chaotic escape when certain resonance conditions are met.

INHIBITION OF CHAOTIC ESCAPE BY AN ADDITIONAL DRIVEN TERM

In this paper, we are devoted to the problem of escaping from a potential well which is present in a great number of physical situations. We use the Helmholtz oscillator as a model for those situations and consider the behavior of the oscillator under an additional driven perturbation. The Melnikov analysis reveals it as an adequate method. Some comparisons are made with the perturbations of the oscillator on the linear and quadratic terms.

Towards AC-induced optimum control of dynamical localization

It is shown that dynamical localization (quantum suppression of classical diffusion) in the context of ultracold atoms in periodically shaken optical lattices subjected to time-periodic modulations having equidistant zeros depends on the impulse transmitted by the external modulation over half-period rather than on the modulation amplitude. This result provides a useful principle for optimally controlling dynamical localization in general periodic systems, which is capable of experimental realization.