Institute of Space Research,
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L3-160, McKnight Brain Institute
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Control of Symmetry by Lyapunov Exponents in Dynamical System
In this paper we present theoretical and numerical results for systems with local and global symmetry. Recent results in control theory have demonstrated that control can lead to symmetry breaking in chaotic systems with a simple type of symmetry. In our work we analyze controllability of Lyapunov exponents using continuous control functions. We show that, by controlling Lyapunov exponents, a chaotic attractor lying in some invariant subspace can be made unstable with respect to perturbations transverse to the invariant subspace. Furthermore, a symmetry-increasing bifurcation can occur, after which the attractor possesses the system symmetry. We demonstrate control of local Lyapunov exponents for the control of symmetry in nonlinear dynamical systems. We also study the effect of noise in the system. It is shown that the small-amplitude noise can restore the symmetry in the attractor after the bifurcation and that the average time for trajectories to switch between the symmetry-broken components of the attractor scales algebraically with the noise amplitude. We demonstrate the relation between Lyapunov exponents, order parameters (Haken, 1983, 1988) and symmetry using a simple physical system and discuss the applicability of our approach to the study of state transitions in the epileptic brain.