Replicate periodic windows in the parameter space of driven oscillators

Abstract

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have periodicity and pattern similar to stable and unstable periodic orbits already existent for the unperturbed oscillator. These features indicate that the reported replicate periodic windows are associated with chaos control of the considered oscillators.

Proliferation of stability in phase and parameter spaces of nonlinear systems

“…Indications for the possible duplication of structures in parameter space was given for the composition of two quadratic coupled maps in the context of chaos suppression [24]. The replication of a shrimp-like ISSs was observed in a continuous oscillator [8], but its origin remained unknown. This work extends previous results for one-dimensional systems [25] to the non-trivial two-dimensional case.…”

“…ISSs were found in many systems, and we would like to mention some of them. In theoretical [1] and experimental [2] electronic circuits, continuous systems [3][4][5][6][7][8][9], maps [3,[10][11][12][13][14][15] lasers models [16], cancer models [17], classical [18][19][20] and quantum ratchet systems [21][22][23]. For the description of nature processes it is essential to discover generic properties for parameter combinations in nonlinear dynamical systems which can be applied to any realistic situation, independent of the specific physical system.…”

Nonlinear Behavior of a Suspended Particle in Single-Axis Acoustic Levitators

“…4 (B) and Both questions can be evaluated by calculating the Lyapunov exponents onto the space of the parameters. Such Lyapunov exponents are standard measures used to discriminate between chaos and periodicity and, the projection of Lyapunov exponents onto the space of the parameters is called Lyapunov diagrams [17][18][19] .…”

Non-feedback technique to directly control multistability in nonlinear oscillators by dual-frequency driving

“…4 below), which are standard ingredients for bifurcation diagrams; furthermore, the maximum bubble radius and the maximum absolute value of the bubble wall velocity, which are important for applications; finally, the period, the Lyapunov exponent and the winding number of the attractors found, quantities that are essential for a detailed analysis of bifurcation structures. A strategy to represent the results of parametric studies involving high-dimensional parameter spaces consists in creating high-resolution bi-parametric plots, a rapidly spreading technique in the investigation of nonlinear systems with a high-dimensional parameter space [39][40][41][42][43][44][45][46][47][48][49]. The system studied here, a bubble in water with dual-frequency acoustic excitation, has a four-dimensional driving parameter space (P A1 , P A2 , ω R1 , ω R2 ).…”