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Complex Dynamics in Coupled Period-Doubling Systems;

Quasiperiodicity, Mode-Lockings, Torus Doublings, and Chaos [PPT]

  It is well known that two coupled oscillators with competing frequencies exhibit quasiperiodicity, mode-lockings, and chaos. However, such interesting dynamical behaviors may also arise when two identical oscillators exhibiting period-doubling transitions to chaos are coupled symmetrically. For this case, the period-doubling transition to chaos in each subsystem may be replaced by a Quasiperiodic Transition to Chaos. The reason for this replacement is that at an early stage of the period-doubling cascade, a symmetric asynchronous periodic orbit with a time shift of half a period (i.e., the so-called 180$^{\circ}$ out-of-phase "antiphase" orbit) loses its stability through a Hopf Bifurcation instead of a period-doubling bifurcation. 

  First, we investigate these interesting phenomena in two coupled logistic maps. The Hopf bifurcation of the symmetric asynchronous periodic orbit occurs when its Floquet multipliers are $e^{\pm 2\pi i \alpha}$. For each irrational $\alpha$, a symmetric quasiperiodic attractor, surrounding the unstable periodic orbit, emerges. On the other hand, for rational $\alpha (=r/s)$, the Hopf Bifurcation may create one Symmetry-Conserved pair of symmetric periodic attractor and saddle or two Symmetry-Broken pairs of asymmetric periodic attractors and saddles according to whether $r$ is even or odd. These symmetric and asymmetric periodic attractors exist in the Mode-Locked Arnold tongues, emanating from the Hopf bifurcation line in the parameter space. Bifurcations in these Arnold's tongues are also interesting. Unlike the circle-map case, secondary Hopf bifurcations may occur inside the primary Arnold tongue. When two neighboring tongues overlap, quasiperiodicities between are broken and chaos occurs, as in the circle map.

  However, another interesting Torus-Doubling phenomenon that may be observed in the symmetrically-coupled period-doubling oscillators does not occur in the 2D coupled logistic maps. Hence, in addition to the coupled 1D maps, we also study two symmetrically coupled Henon maps, that are more real models for the coupled oscillators. Each Henon map has a constant Jacobian $b$ controlling the "degree" of dissipation. As $b$ decreases to zero, the Henon map becomes strongly dissipative, and it reduced to the logistic map. With increasing $b$ from 0, we investigate the dynamical behaviors of the coupled Henon maps. It is thus found that when $b$ is larger than a threshold value, torus doublings are observed in the coupled Henon maps, in contrast to the coupled logistic maps. Furthermore, bifurcation structures in the Arnold's tongues also change as $b$ passes a threshold value.

  To check the robustness of these rich dynamical behaviors in the symmetrically coupled systems, we also introduce a general coupled 1D map with one additional parameter $\alpha$ tuning the degree of the asymmetry of the system. The case of $\alpha=0$ corresponds to the symmetric-coupling case, while the coupling for a non-zero $\alpha$ becomes asymmetric. The unidirectionally-coupled case of $\alpha=1$ corresponds to the extreme case of asymmetric coupling. By changing $\alpha$ from 0 to 1, we investigate the effect of asymmetry. Finally, an experimental work to confirm the theoretical results will also be performed in symmetrically coupled oscillators consisting of parametrically forced pendulums, Duffing oscillators, and Rossler oscillators 

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