Evolution of Chaos Synchronization

Fractals and Choas

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These movies show the evolution of chaos synchronization in a system of coupled maps. In particular, the periodic orbit structure of the following map is shown: $$x -> cf(x), y -> cf(x) + (1-c)g(y), f(x) = a - x^2, g(y) = b - y^2$$.

In each case, $$x$$ vs. $$y$$ is plotted, and as the coupling is changed in time from $$c=0.5$$ to $$0.0$$. The yellow points depict the attractor, and the crosses are periodic orbits of periods 1-20 (cross colors refer to stability properties). By design, the map exhibits identical sychronization for $$c=1.0$$. This persists (roughly for the nonidentical case) through $$c=0.5$$. As $$c$$ is further decreased, bifurcations create additional periodic orbits outside the original synchronziation manifold. In the case of coupled identical maps, the "blowout bifucation," is immediately apparent as the attractor explodes to include these additional orbits. For the case of coupled nonidentical maps, this transition is is more gradual.

Coupled Identical Maps: $$(a, b)=(1.7, 1.7)$$.

Coupled Nonidentical Maps: $$(a, b)=(1.9, 1.45)$$.

 Author Institutional Affiliation Dept. of Physics & Astronomy, Krasnow Institute for Advanced Study George Mason University Author Email ebarreto@gmu.edu Author Postal Mail Mail Stop 2A1 George Mason University Fairfax, VA 22030 USA Tel.: (703) 993-4431 Keywords Chaos, Synchronization, Coupled Maps