We consider two stable heteroclinic cycles rotating in opposite directions, coupled via diffusive terms. A complete synchronization in this system is impossible, and numerical exploration shows that chaos is abundant at low levels of coupling. With increase of coupling strength, several symmetry-changing transitions are observed, and finally a stable periodic orbit appears via an inverse period-doubling cascade. To reveal the behavior at extremely small couplings, a piecewise-constant model for the dynamics is suggested. Within this model we construct a Poincaré map for a chaotic state numerically, it appears to be an expanding non-invertable circle map thus confirming abundance of chaos in the small coupling limit. We also show that within the piecewise-constant description, there is a set of periodic solutions with different phase shifts between subsystems, due to dead zones in the coupling.
Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of periodic, closed, or homoclinic (heteroclinic) orbits. The interferences embedded in such orbit sums are governed by classical action functions and Maslov indices. For chaotic systems, the relative actions of such orbits can be expressed in terms of phase space areas bounded by segments of stable and unstable manifolds, and Moser invariant curves. This also generates direct relations between periodic orbits and homoclinic (heteroclinic) orbit actions. Simpler, explicit approximate expressions following f…
In this paper an approach to generate hidden attractors based on piecewise linear (PWL) systems is studied. The approach consists of the coexistence of self-excited attrators and hidden attractors, i.e., the equilibria of the system are immersed in the basin of attraction of the seft-excited attractors, so hidden attractors appear around these self-excited attractors. The approach starts by generating a double-scroll attractor based on two equilibria presented heteroclinic orbits. Then, two equilibria are added to the system, so biestability is generated by displaying two self-excited attracto…
Recent studies suggest that unstable recurrent solutions of the Navier-Stokes equation provide new insights into dynamics of turbulent flows. In this study, we compute an extensive network of dynamical connections between such solutions in a weakly turbulent quasi-two-dimensional Kolmogorov flow that lies in the inversion-symmetric subspace. In particular, we find numerous isolated heteroclinic connections between different types of solutions -- equilibria, periodic, and quasi-periodic orbits -- as well as continua of connections forming higher-dimensional connecting manifolds. We also compute…
Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos), While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Her…
