Collective Lyapunov modes

@article{Takeuchi2013CollectiveLM,
  title={Collective Lyapunov modes},
  author={Kazumasa A. Takeuchi and Hugues Chat'e},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2013},
  volume={46}
}
  • K. Takeuchi, H. Chat'e
  • Published 24 July 2012
  • Physics, Mathematics
  • Journal of Physics A: Mathematical and Theoretical
We show, using covariant Lyapunov vectors in addition to standard Lyapunov analysis, that there exists a set of collective Lyapunov modes in large chaotic systems exhibiting collective dynamics. Associated with delocalized Lyapunov vectors, they act collectively on the trajectory and hence characterize the instability of its collective dynamics. We further develop, for globally coupled systems, a connection between these collective modes and the Lyapunov modes in the corresponding Perron… 
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References

SHOWING 1-10 OF 32 REFERENCES
Lyapunov analysis captures the collective dynamics of large chaotic systems.
We show, using generic globally coupled systems, that the collective dynamics of large chaotic systems is encoded in their Lyapunov spectra: most modes are typically localized on a few degrees of
Covariant Lyapunov vectors
Recent years have witnessed a growing interest in covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here, we review the basic results of
Characterizing dynamics with covariant Lyapunov vectors.
A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows us to address fundamental questions such as the degree of
Hyperbolicity and the effective dimension of spatially extended dissipative systems.
TLDR
Using covariant Lyapunov vectors, it is argued that N can be interpreted as the number of effective degrees of freedom, which has to be taken into account in numerical integration and control issues.
Low-Dimensional Chaos in Populations of Strongly-Coupled Noisy Maps(Oscillation, Chaos and Network Dynamics in Nonlinear Science)
We characterize the macroscopic attractor of infinite populations of noisy maps subjected to global and strong coupling by using an expansion in order parameters. We show that for any noise amplitude
Collective Chaos
An algorithm to characterize collective motion as the orbital instability at a macroscopic level presented, including the introduction of “collective Lyapunov exponent.” By applying the algorith to a
Mechanisms of extensive spatiotemporal chaos in Rayleigh–Bénard convection
TLDR
A computational investigation of a phenomenon found in nature, ‘spiral defect’ chaos in Rayleigh–Bénard convection, is reported, in which it is found that the spatiotemporal chaos in this state is extensive and characterized by about a hundred dynamical degrees of freedom.
...
...