Dynamics of two coupled chaotic systems driven by external signals

Abstract

Abstract. Setting-up a controlled or synchronized state in a space-time chaotic structure targeting an unstable periodic orbit is a key feature of many problems in high dimensional physical, electronics, biological and ecological systems (among others (2000)]. In that case the flow was generated in a small container with inhomogeneous heating in order to have a single roll structure produced by a Bénard-Marangoni instability [E.L. Koshmieder, Bénard Cells and Taylor Vortices (Cambridge University Press, 1993)]. Phase synchronization was achieved by a small amplitude signal injected at a subharmonic frequency obtained from the measured Fourier temperature spectrum. In this work, we analyze numerically the effects of driving two previously synchronized chaotic oscillators by an external signal. The numerical system represents a convective experiment in a small container with square symmetry, where boundary layer instabilities are coupled by a common flow. This work is an attempt to control this situation and overcome some difficulties to select useful frequency values for the driving force, analyzing the influence of different harmonic injection signals on the synchronization in a system composed by two identical chaotic Takens-Bogdanov equations (TBA and TBB) bidirectionally coupled.

Frozen dynamics and synchronization through a secondary symmetry-breaking bifurcation

“…We may ask how such different systems show similar dynamical responses (e.g., generation and spatiotemporal distribution of defects) when they undergo an instability towards a "new state." Precedents on synchronization are found in the pioneering work by Winfree [1] and Kuramoto [2] and in the research on collective behavior in life sciences like neurology [3], ecology [4], and networks [5][6][7][8][9][10]. However, joint experimental [11][12][13][14][15] and theoretical [16][17][18] efforts have been scarce, and more research must be devoted to the understanding of synchronization processes in complex systems from the networks approach.…”

Control and synchronization of hyperchaotic states in mathematical models of Bènard-Marangoni convective experiments

“…6(a). The existence of this common shape in the Fourier spectrum was observed in Takens-Bodanov 28,30 and in Chen and Lü hyperchaotic systems. 35 The Fourier…”

“…To overcome this problem, we calculate a histogram of a data file like x(t) to see the most relevant characteristic periods that appear in the system as described in Ref. 30. Periods are obtained from a long signal data file measuring the time between two maximum or minimum first neighbors in the signal.…”

“…29 but keeping their synchronization or dynamic possibilities as in Ref. 30. Following the latter work, in this paper, we use a method to choose the control frequencies based on histograms obtained from the signals observed.…”

“…As a resume, in this work, we extend the methods used in Ref. 30 to two hyperchaotic models ("Chen" and "Lü") that are related to the Lorenz system but with different topologies and dynamics, with the aim to check the possibility of the existence of generic behaviors in four dimensional systems. We present the results of self-synchronization by coupling two equal oscillators and the attempts to control the coupled systems by injecting small signals (less than 1% in amplitude of the variable x) and modifying their frequencies.…”