Supertransient chaos in the two-dimensional complex Ginzburg-Landau equation

Nucleation of spatiotemporal structures from defect turbulence in the two-dimensional complex Ginzburg-Landau equation

We numerically investigate nucleation processes in the transient dynamics of the two-dimensional complex Ginzburg-Landau equation toward its "frozen" state with quasistationary spiral structures. We study the transition kinetics from either the defect turbulence regime or random initial configurations to the frozen state with a well-defined low density of quasistationary topological defects. Nucleation events of spiral structures are monitored using the characteristic length between the emerging shock fronts. We study two distinct situations, namely when the system is quenched either far from the transition limit or near it. In the former deeply quenched case, the average nucleation time for different system sizes is measured over many independent realizations. We employ an extrapolation method as well as a phenomenological formula to account for and eliminate finite-size effects. The nonzero (dimensionless) barrier for the nucleation of single spiral droplets in the extrapolated infinite system size limit suggests that the transition to the frozen state is discontinuous. We also investigate the nucleation of spirals for systems that are quenched close to but beyond the crossover limit and of target waves which emerge if a specific spatial inhomogeneity is introduced. In either of these cases, we observe long, "fat" tails in the distribution of nucleation times, which also supports a discontinuous transition scenario.

Lagrangian coherent structures at the onset of hyperchaos in the two-dimensional Navier-Stokes equations

We study a transition to hyperchaos in the two-dimensional incompressible Navier-Stokes equations with periodic boundary conditions and an external forcing term. Bifurcation diagrams are constructed by varying the Reynolds number, and a transition to hyperchaos (HC) is identified. Before the onset of HC, there is coexistence of two chaotic attractors and a hyperchaotic saddle. After the transition to HC, the two chaotic attractors merge with the hyperchaotic saddle, generating random switching between chaos and hyperchaos, which is responsible for intermittent bursts in the time series of energy and enstrophy. The chaotic mixing properties of the flow are characterized by detecting Lagrangian coherent structures. After the transition to HC, the flow displays complex Lagrangian patterns and an increase in the level of Lagrangian chaoticity during the bursty periods that can be predicted statistically by the hyperchaotic saddle prior to HC transition.

Transient chaotic rotating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol oscillators near a codimension-two bifurcation point

Propagating waves in a ring of unidirectionally coupled symmetric Bonhoeffer-van der Pol (BVP) oscillators were studied. The parameter values of the BVP oscillators were near a codimension-two bifurcation point around which oscillatory, monostable, and bistable states coexist. Bifurcations of periodic, quasiperiodic, and chaotic rotating waves were found in a ring of three oscillators. In rings of large numbers of oscillators with small coupling strength, transient chaotic waves were found and their duration increased exponentially with the number of oscillators. These exponential chaotic transients could be described by a coupled map model derived from the Poincaré map of a ring of three oscillators. The quasiperiodic rotating waves due to the mode interaction near the codimension-two bifurcation point were evidently responsible for the emergence of the transient chaotic rotating waves.

Transient spatiotemporal chaos is extensive in three reaction-diffusion networks

Extensive (asymptotic) spatiotemporal chaos is comprised of statistically similar subsystems that interact only weakly. A systematic study of transient spatiotemporal chaos reveals extensive system behavior in all three reaction-diffusion networks for various boundary conditions. The Lyapunov dimension, the sum of positive Lyapunov exponents, and the logarithm of the transient lifetime grow linearly with the system size. The unstable manifold of the chaotic saddle has nearly the same dimension as the saddle itself, and the stable manifold is nearly space filling.

Chaotic saddles at the onset of intermittent spatiotemporal chaos

In a recent study [Rempel and Chian, Phys. Rev. Lett. 98, 014101 (2007)], it has been shown that nonattracting chaotic sets (chaotic saddles) are responsible for intermittency in the regularized long-wave equation that undergoes a transition to spatiotemporal chaos (STC) via quasiperiodicity and temporal chaos. In the present paper, it is demonstrated that a similar mechanism is present in the damped Kuramoto-Sivashinsky equation. Prior to the onset of STC, a spatiotemporally chaotic saddle coexists with a spatially regular attractor. After the transition to STC, the chaotic saddle merges with the attractor, generating intermittent bursts of STC that dominate the post-transition dynamics.

Master stability analysis in transient spatiotemporal chaos

The asymptotic stability of spatiotemporal chaos is difficult to determine, since transient spatiotemporal chaos may be extremely long lived. A master stability analysis reveals that the asymptotic state of transient spatiotemporal chaos in the Gray-Scott system and in the Bär-Eiswirth system is characterized by negative transverse Lyapunov exponents on the attractor of the invariant synchronization manifold. The average lifetime of transient spatiotemporal chaos depends on the number of transverse directions that are unstable along a typical excitation cycle.

Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto-Sivashinsky equation

This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence.