Stability of heteroclinic cycles in ring graphs.

Networks of interacting nodes connected by edges arise in almost every branch of scientific inquiry. The connectivity structure of the network can force the existence of invariant subspaces, which would not arise in generic dynamical systems. These invariant subspaces can result in the appearance of robust heteroclinic cycles, which would otherwise be structurally unstable. Typically, the dynamics near a stable heteroclinic cycle is non-ergodic: mean residence times near the fixed points in the cycle are undefined, and there is a persistent slowing down. In this paper, we examine ring graphs with nearest-neighbor or nearest- m-neighbor coupling and show that there exist classes of heteroclinic cycles in the phase space of the dynamics. We show that there is always at least one heteroclinic cycle that can be asymptotically stable, and, thus, the attracting dynamics of the network are expected to be non-ergodic. We conjecture that much of this behavior persists in less structured networks and as such, non-ergodic behavior is somehow typical.

Deceleration of one-dimensional mixing by discontinuous mappings.

We present a computational study of a simple one-dimensional map with dynamics composed of stretching, permutations of equally sized cells, and diffusion. We observe that the combination of the aforementioned dynamics results in eigenmodes with long-time exponential decay rates. The decay rate of the eigenmodes is shown to be dependent on the choice of permutation and changes nonmonotonically with the diffusion coefficient for many of the permutations. The global mixing rate of the map M in the limit of vanishing diffusivity approximates well the decay rates of the eigenmodes for small diffusivity, however this global mixing rate does not bound the rates for all values of the diffusion coefficient. This counterintuitively predicts a deceleration in the asymptotic mixing rate with an increasing diffusivity rate. The implications of the results on finite time mixing are discussed.

Rate of chaotic mixing and boundary behavior.

We discuss rigorous results on the rate of mixing for an idealized model of a class of fluid mixing device. These show that the decay of correlations of a scalar field is governed by the presence of boundaries in the domain, and in particular by the behavior of the modeled fluid at such boundaries.

DNA microarrays: design principles for maximizing ergodic, chaotic mixing.

In this article we show that models of flows in DNA microarrays generated by pulsed source-sink pairs can be studied as linked twist maps. The significance of this is that it enables us to relate the flow to mathematically precise notions of chaotic mixing that can be realized through specific design criteria. We apply these techniques to three different mixing protocols, two of which have been previously described in the literature, and we are able to isolate the features of each mixer that lead to "good" or "bad" mixing. Based on this, we propose a new design to generate a "well-mixed" flow in a DNA microarray.

Cycling chaotic attractors in two models for dynamics with invariant subspaces.

Nonergodic attractors can robustly appear in symmetric systems as structurally stable cycles between saddle-type invariant sets. These saddles may be chaotic giving rise to "cycling chaos." The robustness of such attractors appears by virtue of the fact that the connections are robust within some invariant subspace. We consider two previously studied examples and examine these in detail for a number of effects: (i) presence of internal symmetries within the chaotic saddles, (ii) phase-resetting, where only a limited set of connecting trajectories between saddles are possible, and (iii) multistability of periodic orbits near bifurcation to cycling attractors. The first model consists of three cyclically coupled Lorenz equations and was investigated first by Dellnitz et al. [Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 1243-1247 (1995)]. We show that one can find a "false phase-resetting" effect here due to the presence of a skew product structure for the dynamics in an invariant subspace; we verify this by considering a more general bi-directional coupling. The presence of internal symmetries of the chaotic saddles means that the set of connections can never be clean in this system, that is, there will always be transversely repelling orbits within the saddles that are transversely attracting on average. Nonetheless we argue that "anomalous connections" are rare. The second model we consider is an approximate return mapping near the stable manifold of a saddle in a cycling attractor from a magnetoconvection problem previously investigated by two of the authors. Near resonance, we show that the model genuinely is phase-resetting, and there are indeed stable periodic orbits of arbitrarily long period close to resonance, as previously conjectured. We examine the set of nearby periodic orbits in both parameter and phase space and show that their structure appears to be much more complicated than previously suspected. In particular, the basins of attraction of the periodic orbits appear to be pseudo-riddled in the terminology of Lai [Physica D 150, 1-13 (2001)].

Phase resetting effects for robust cycles between chaotic sets.

In the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated, owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena, including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. In this paper we introduce and discuss an instructive example of an ordinary differential equation where one can observe and analyze robust cycling behavior. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a Rössler system), and/or saddle equilibria. For this model, we distinguish between cycling that includes phase resetting connections (where there is only one connecting trajectory) and more general non(phase) resetting cases, where there may be an infinite number (even a continuum) of connections. In the nonresetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability, whereas more general cases may give rise to "stuck on" cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase resetting and connection selection.

Infinities of stable periodic orbits in systems of coupled oscillators.

We consider the dynamical behavior of coupled oscillators with robust heteroclinic cycles between saddles that may be periodic or chaotic. We differentiate attracting cycles into types that we call phase resetting and free running depending on whether the cycle approaches a given saddle along one or many trajectories. At loss of stability of attracting cycling, we show in a phase-resetting example the existence of an infinite family of stable periodic orbits that accumulate on the cycling, whereas for a free-running example loss of stability of the cycling gives rise to a single quasiperiodic or chaotic attractor.