# Chronotopic Lyapunov analysis. I. A detailed characterization of 1D systems

@article{Lepri1996ChronotopicLA, title={Chronotopic Lyapunov analysis. I. A detailed characterization of 1D systems}, author={Stefano Lepri and Antonio Politi and Alessandro Torcini}, journal={Journal of Statistical Physics}, year={1996}, volume={82}, pages={1429-1452} }

Instabilities in 1D spatially extended systems are studied with the aid of both temporal and spatial Lyapunov exponents. A suitable representation of the spectra allows a compact description of all the possible disturbances in tangent space. The analysis is carried out for chaotic and periodic spatiotemporal patterns. Singularities of the spectra and localization properties of the associated Lyapunov vectors are discussed.

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#### References

SHOWING 1-10 OF 27 REFERENCES

Chronotopic Lyapunov analysis: II. Toward a unified approach

- Physics, Mathematics
- 1996

From the analyticity properties of the equation governing infinitesimal perturbations, it is conjectured that all types of Lyapunov exponents introduced in spatially extended 1D systems can be… Expand

Periodic orbits in coupled Henon maps: Lyapunov and multifractal analysis.

- Mathematics, Medicine
- Chaos
- 1992

A powerful algorithm is implemented in a 1-d lattice of Henon maps to extract orbits which are periodic both in space and time and the arrangement of periodic orbits allows the elucidate the spatially chaotic structure of the invariant measure. Expand

Spatio-Temporal Chaos and Localization

- Physics
- 1991

Nonlinearities in the flow equations of spatially extended systems can give rise to high-dimensional deterministic chaos. This plays the role of an intrinsic source of disorder in tangent space, and… Expand

Intrinsic stochasticity with many degrees of freedom

- Mathematics
- 1984

We study a simple model equation describing a system with an infinity of degrees of freedom which displays an intrinsically chaotic behavior. Some concepts of fully developed turbulence are discussed… Expand

Local Lyapunov exponents for spatiotemporal chaos.

- Mathematics, Medicine
- Chaos
- 1993

Local Lyapunov exponents are proposed for characterization of perturbations in distributed dynamical systems with chaotic behavior. Their relation to usual and velocity-dependent exponents is… Expand

Lyapunov spectra of coupled map lattices

- Physics
- 1990

Abstract The relationship between the Lyapunov spectrum of diffusively coupled one-dimensional maps and the spectrum of the discrete Schrodinger operator is stressed. As a result, an analytic… Expand

Chaos in low-dimensional hamiltonian maps

- Physics
- 1987

Abstract Symplectic maps with more than two degrees of freedom constructed by coupling N area-preserving Chiricov-Taylor standard maps are investigated by numerical methods. We find the asymptotic… Expand

Singular Lyapunov spectra and conservation laws.

- Mathematics, Medicine
- Chaos
- 1995

We give analytic arguments and numerical evidence to show that the presence of conservation laws can produce a singularity in the spectrum of Lyapunov exponents for extended dynamical systems of low… Expand

Spatial development of chaos in nonlinear media

- Physics
- 1989

Abstract Spatial development of chaos in the media, where initial disturbances are advected (flow systems, systems with convective character of wave instability) is investigated. The problem is… Expand

Coupled map models for chaos in extended systems.

- Physics, Medicine
- Chaos
- 1992

Coupled maps with conserved quantities are introduced as models for chaos in extended systems. The long-wavelength limit of a simple one-dimensional example is investigated in detail. A Langevin… Expand