Modeling velocity in gradient flows with coupled-map lattices with advection

A downstream drift into chaos: Asymmetric dispersal in a classic density dependent population model

In the ocean, propagules with a planktonic stage are typically dispersed some distance downstream of the parent generation, introducing an asymmetry to the dispersal. Ocean-dwelling species have also been shown to exhibit chaotic population dynamics. Therefore, we must better understand chaotic population dynamics under the influence of asymmetrical dispersal. Here, we examine a density-dependent population in a current, where the current has both a mean and stochastic component. In our finite domain, the current moves offspring in the downstream direction. This system displays a rich variety of dynamics from chaotic to steady-state, depending on the mean distance the offspring are moved downstream, the diffusive spread of the offspring, and the domain size. We find that asymmetric dispersal can act as a stabilizing or destabilizing mechanism, depending on the size of the mean dispersal distance relative to the other system parameters. As the strength of the current increases, the system can experience period-halving bifurcation cascades. Thus, we show that stability of chaotic aquatic populations is directly tied to the strength of the ocean current in their environment, and our model predicts increased prevalence of chaos with decreasing dispersal distance. Climate change is likely to alter the dispersal patterns of many species, and so our results have implications for conservation and management of said species. We discuss the management implications, particularly of exploited species.

Spatial chaos of traveling waves has a given velocity

We study the complexity of stable waves in unidirectional bistable coupled map lattices as a test tube to spatial chaos of traveling patterns in open flows. Numerical calculations reveal that, grouping patterns into sets according to their velocity, at most one set of waves has positive topological entropy for fixed parameters. By using symbolic dynamics and shadowing, we analytically determine velocity-dependent parameter domains of existence of pattern families with positive entropy. These arguments provide a method to exhibit chaotic sets of stable waves with arbitrary velocity in extended systems.

Desynchronized wave patterns in synchronized chaotic regions of coupled map lattices

We analyze the size limits of coupled map lattices with diffusive coupling at the crossover of low-dimensional to high-dimensional chaos. We investigate the existence of standing-wave-type periodic patterns, within the low-dimensional limit, in addition to the stable synchronous chaotic states depending upon the initial conditions. Further, we bring out a controlling mechanism to explain the emergence of standing-wave patterns in the coupled map lattices. Finally, we give an analytic expression in terms of the unstable periodic orbits of the isolated map to represent the standing-wave patterns.

Pattern formation in diffusive-advective coupled map lattices

We investigate pattern formation and evolution in coupled map lattices when advection is incorporated, in addition to the usual diffusive term. All patterns may be suitably grouped into five classes: three periodic, supporting static patterns and traveling waves, and two nonperiodic. Relative frequencies are determined as a function of all model parameters: diffusion, advection, local nonlinearity, and lattice size. Advection plays an important role in coupled map lattices, being capable of considerably altering pattern evolution. For instance, advection may induce synchronization, making chaotic patterns evolve periodically. As a byproduct we describe a practical algorithm for classifying generic pattern evolutions and for measuring velocities of traveling waves.

Inducing coherence in networks of bistable maps by varying the interaction range

Ordinarily, two different topologies have been used to model spatiotemporal chaos and to study complexity in networks of maps: one where sites interact only with nearest neighbors (e.g., the standard diffusive coupling) and one where sites interact with all sites in the network (global coupling). Here we investigate intermediate regimes considering the interaction range as a free tunable parameter. The synchronization behavior normally seen in globally coupled maps is found to set in for interaction ranges considerably smaller than the system size. In addition, we analytically derive stability conditions for the onset of coherent states (full synchronization) from which the minimum interaction range needed to induce coherence in homogeneously coupled maps can be determined. Such conditions are also obtained for inhomogeneous situations when the coupling strength decreases linearly with the distance. The characteristic range for the onset of coherence is studied in detail as a function of model parameters.