Spatiotemporal chaos in terms of unstable recurrent patterns

  title={Spatiotemporal chaos in terms of unstable recurrent patterns},
  author={Freddy Bugge Christiansen and Predrag Cvitanovi{\'c} and Vakhtang Putkaradze},
Spatiotemporally chaotic dynamics of a Kuramoto - Sivashinsky system is described by means of an infinite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as an accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics. 

Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics.

  • Y. LanP. Cvitanović
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2008
We undertake an exploration of recurrent patterns in the antisymmetric subspace of the one-dimensional Kuramoto-Sivashinsky system. For a small but already rather "turbulent" system, the long-time

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

A novel technique is described 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 Chaos attractor.

Intermittent chaos driven by nonlinear Alfvén waves

Abstract. We investigate the relevance of chaotic saddles and unstable periodic orbits at the onset of intermittent chaos in the phase dynamics of nonlinear Alfven waves by using the

Intermittency induced by attractor-merging crisis in the Kuramoto-Sivashinsky equation.

  • E. RempelA. Chian
  • Physics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2005
An attractor-merging crisis in a spatially extended system exemplified by the Kuramoto-Sivashinsky equation is characterized, with the origin of this crisis-induced intermittency explained in terms of alternate switching between two chaotic saddles embedded in the merged chaotic attractor.

Inferring symbolic dynamics of chaotic flows from persistence.

This work adapts a topological data analysis technique known as persistent homology for the characterization of state space projections of chaotic trajectories and periodic orbits for deducing the symbolic dynamics of time series data obtained from high-dimensional chaotic attractors.

Chaotic saddles at the onset of intermittent spatiotemporal chaos.

It is demonstrated that a similar mechanism is present in the damped Kuramoto-Sivashinsky equation that undergoes a transition to spatiotemporal chaos (STC) via quasiperiodicity and temporal chaos.

Spatiotemporally periodic solutions by variational methods

The intriguing “Hopf’s last hope”— proposal for a theory of turbulence has significant physical meaning. The problem is how to describe chaotic/turbulent spatiotemporal patterns. Typical system



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In a spontaneously bursting neuronal network in vitro, chaos can be demonstrated by the presence of unstable fixed-point behaviour. Chaos control techniques can increase the periodicity of such

Symmetry decomposition of chaotic dynamics

Discrete symmetries of dynamical flows give rise to relations between periodic orbits, reduce the dynamics to a fundamental domain, and lead to factorizations of zeta functions. These factorizations

Deterministic nonperiodic flow

Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with


The onset of space-time chaos is studied on the basis of a Galilean invariant model that exhibits the essential characteristics of the phenomenon. By keeping the linear part of the model extremely

Symmetry of attractors and the Karhunen-Loegve decomposition

Recent fluid dynamics experiments [13, 10, 4] have shown that the symmetry of attractors can manifest itself through the existence of spatially regular patterns in the time average of an appropriate

Chaos: a mixed metaphor for turbulence

  • E. Spiegel
  • Physics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1987
There are special circumstances when the equations of fluid mechanics can be asymptotically reduced to third- or higher-order differential equations that admit chaotic solutions. For physically

Recycling of strange sets: I. Cycle expansions

The strange sets which arise in deterministic low-dimensional dynamical systems are analysed in terms of (unstable) cycles and their eigenvalues. The general formalism of cycle expansions is


The aperiodic behavior of the solution of the equation of motion derived previously (1966) when considering a model thermomechanical oscillator is examined. Periodic solutions of this equation are

Invariant measurement of strange sets in terms of cycles.

We argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible, but is also a theoretically optimal measurement of the invariant properties of a dynamical

Persistent Propagation of Concentration Waves in Dissipative Media Far from Thermal Equilibrium

The origin of persistent wave propagation through medium of reactwn-diffusion type 1s explored. Our theory is based on a generalized time-dependent Ginzburg-Landau equation for a complex field W,