Multistability in dynamical systems induced by weak periodic perturbations

Abstract

It is shown that weak resonant perturbations at subharmonic frequencies can induce multistability in a wide class of nonlinear systems, which display the period-doubling route into chaos or possess isolated subharmonic branches. The number of attractors induced depends on the subharmonic frequency, amplitude, and phase of periodic perturbations, as well as an initial dynamical state of nonlinear systems. Experimental and numerical evidences are given on the basis of a loss-modulated CO2 laser.

Self-Similarities in the Frequency-Amplitude Space of a Loss-ModulatedCO2Laser

“…1 and 2, and previously known to exist only in the parameter space of discrete-time dynamical systems [14 -17]. Although there are theoretical grounds to expect them to be present [17], thus far all attempts to uncover shrimps in flows, i.e., in continuous-time dynamical systems modeled with sets of differential equations, have failed to produce them [18].The single-mode dynamics of the loss-modulated CO 2 laser involves two coupled degrees of freedom and a timedependent parameter which we write, as usual [3,6,7],Here, u is proportional to the radiation density, z and z 0 are the gain and unsaturated gain in the medium, respectively, denotes the transit time of the light in the laser cavity, is the gain decay rate, and k kt represents the total cavity losses. The losses are modulated periodically as follows, kt k 0 1 a cos2ft;…”

Periodic precursors of nonlinear dynamical transitions

“…The main interest in the periodically driven dynamical system has been focused on the response of a nonlinear system, which is near the onset of dynamical instabilities, to small periodic perturbations, small-signal amplification of bifurcating system [1][2][3], periodic multistability [4,5], control of chaos and spatiotemporal patterns by global or local periodic forcing [6][7][8][9][10][11][12][13], and other periodic driving induced behaviors in excitable or oscillatory systems [14,15]. All those efforts have been dedicated to understanding how the dynamical features of a nonlinear system change as a function of the amplitude and frequency of the periodic modulation.…”

Steering multiattractors to overcome parameter inaccuracy and noise effects

“…If yes, you achieved the needed extension of the available parameter which induce the same dynamics. If not, techniques to generated multiattractors in continuous systems [52] might be used. In this case, start again in step (i).…”