Chaotic Spatiotemporal Patterns Research Articles

Spatially "chaotic" solutions in reaction-convection models and their bifurcations to moving waves.

The emergence of stationary spatially multiperiodic or even spatially chaotic patterns is analyzed for a simple model of convection, reaction, and conduction in a cross-flow reactor. Spatial patterns emerge much like dynamic temporal patterns in a mixed system of the same kinetics. Moving waves are formed in an unbounded system but they are transformed into stationary spatially inhomogeneous patterns in a bounded system. The sequence of period doubling bifurcations is determined numerically. The incorporation of a slow nondiffusing inhibitor leads to chaotic spatiotemporal patterns.

Control of chaotic patterns in a Josephson junction model

The effect of an applied rf signal on the dynamics of a large-area Josephson junction is examined. The problem of controlling spatiotemporal chaotic patterns induced by the external magnetic field is addressed. Chaos control is conducted by a weak spatially distributed force.

Periodic, quasiperiodic and chaotic spatiotemporal patterns in a tubular catalytic reactor with periodic flow reversal

Periodic, quasiperiodic and chaotic symmetric and unsymmetric spatiotemporal temperature patterns arise in the heterogeneous (two-phase) model of the catalytic reactor with periodic flow reversal. Coexistence of various types of stable solutions was observed. Chaotic solutions arising via the torus breaking route were also observed. The transitions among individual types of spatiotemporal patterns and their development are studied by means of Poincaré maps of the associated model represented as a one-parameter family of discrete dynamical systems derived from the simulation results of the original continuous model.

Analysis of complex and chaotic patterns near a codimension-2 Turing-Hopf point in a reaction-diffusion model

We study a reaction-diffusion system of activator-inhibitor type, and characterize the resulting complex and chaotic spatio-temporal patterns by their Lyapunov spectrum and by a Karhunen-Loève decomposition into empirical orthogonal eigenmodes. Different periodic patterns corresponding to localized Hopf and Turing modes, and mixed modes including subharmonic spatio-temporal spiking are found near a codimension-2 bifurcation point. The asymptotic patterns are preceded by transient spatio-temporal chaos, before the system abruptly locks into a periodic state in space and time. The Karhunen-Loève decomposition is shown to be a powerful tool for extracting detailed quantitative information of complex space-time data.

Pattern formation and spatiotemporal chaos in the presence of boundaries

Experimentally obtained time averages of disordered spatiotemporal chaotic patterns of electroconvection in a nematic liquid crystal reveal an ordered structure due to the boundaries of the pattern-forming system. The instantaneous snapshots and the time averages are characterized in terms of amplitude, wave number, and correlations in space and time. The averages have a significantly larger correlation length than that of the snapshots. Time averages reveal a wave number that differs from that of the underlying snapshots. Quantization effects within the regime of spatiotemporal chaos are found for the correlation length and for the estimated angle of coherent zigzag structures.

Spatiotemporal periodic and chaotic patterns in a two-dimensional coupled map lattice system

The pattern dynamics of a two-dimensional coupled map lattice system is studied using both analytical and numerical calculations. Two interesting spatiotemporal periodic patterns are solved analytically. Their stability boundaries for small system size are obtained by linear stability analysis. As the system sizes mismatch the spatial periodicity of the patterns, a frozen random chaotic defect cluster and a slow random propagated chaotic defect string appear.

Reaction-diffusion system with Brusselator kinetics: Control of a quasiperiodic route to chaos.

A methodology for controlling complex dynamics and chaos in distributed parameter systems is discussed. The reaction-diffusion system with Brusselator kinetics, where the torus-doubling route to chaos exists in a defined range of parameter values, is used as an example. Poincar\'e maps and singular value decomposition are used for characterization of quasiperiodic and chaotic attractors and for the identification of dominant modes. Tested modal feedback control schemes based on identified dominant spatial modes confirm the possibility of stabilization of simple quasiperiodic trajectories in the complex quasiperiodic or chaotic spatiotemporal patterns.

Chaotic patterns in a Josephson junction model.

The effect of an applied rf signal on the dynamics of a large area Josephson junction is examined by means of a model based on the sine-Gordon equation. In particular the problem of characterizing spatiotemporal chaotic patterns induced by the external harmonic varying magnetic field is addressed. In general it is found that instantaneous spatiotemporal patterns, i.e., patterns confined to one period of oscillation of the external field correlates poorly. Also the correlation between instantaneous patterns and average patterns is poor. However, the correlation between patterns obtained as averages over, e.g., ten or more consecutive instantaneous spatiotemporal patterns is high. Furthermore, averaged spatiotemporal patterns obtained for different values of the magnetic field amplitude correlate well. The average potential and kinetic energies of spatiotemporal patterns are calculated. In general the energies in the chaotic regions are lower than in regions with periodic response.

Regular and Chaotic Spatiotemporal Patterns in a Model of Slime Mold Aggregation11The project supported by National Natural Science Foundation of China

In this paper we proposed a spatial modulated two-variable Martiel–Goldbeter model to describe the complex spatiotemporal disorder dynamical behavior during development of Dictyostelium discoideum strain FR17. As the nonlinear modulated parameter A and diffusion coefficient varied, the system shows: 1) multiperiodic phase, 2) co-existence phase of chaotic and multi-periodic state, 3) spatiotemporal chaotic phase, 4) co-existence phase of chaotic, multi-periodic and steady state, and 5) co-existence phase of chaotic and steady state. These phases can be described by spatiotemporal power spectra, pattern distribution function and Lyapunov spectra. We believed that the complex spatiotemporal disorder dynamical behavior during development of Dictyostelium discoideum strain FR17 is a spatiotemporal chaotic state.

Time averaging of chaotic spatiotemporal wave patterns.

Chaotic spatiotemporal patterns can have highly ordered time averages. This fact is demonstrated for Faraday waves by averaging shadowgraph images. The averages exhibit box quantization effects, and their spatial form is similar to that predicted for patterns near onset, despite the large fluctuations. Patches of the disordered instantaneous patterns retain significant phase coherence with the ordered averages.