Vitaliy YATSENKO
Institute of Space Research,
40 Glushkova Avenue,
Kyiv 02022,
UKRAINE
E-mail: vitaliy_yatsenko@yahoo.com
Sandeep NAIR
L3-160, McKnight Brain Institute
PO BOX 100244
100, S. Newell Dr.
Gainesville, FL-32603
USA
E-mail: spnair@mbi.ufl.edu
Control of Symmetry by Lyapunov Exponents in Dynamical System
Abstract:
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.