Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

@article{Tantet2017ResonancesIA,
  title={Resonances in a Chaotic Attractor Crisis of the Lorenz Flow},
  author={Alexis Tantet and Valerio Lucarini and Henk A. Dijkstra},
  journal={Journal of Statistical Physics},
  year={2017},
  volume={170},
  pages={584-616}
}
Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems. However, it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a… 
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References

SHOWING 1-10 OF 129 REFERENCES
Crisis of the Chaotic Attractor of a Climate Model: A Transfer Operator Approach
The destruction of a chaotic attractor leading to rough changes in the dynamics of a dynamical system is studied. Local bifurcations are characterised by a single or a pair of characteristic
Behaviour of Lyapunov exponents near crisis points in the dissipative standard map
We numerically study the behaviour of the largest Lyapunov characteristic exponent λ1 in dependence on a control parameter in the 2D standard map with dissipation. In order to investigate the
From attractor to chaotic saddle: a tale of transverse instability
Suppose that a dynamical system possesses an invariant submanifold, and the restriction of the system to this submanifold has a chaotic attractor A. Under which conditions is A an attractor for the
Ergodic theory of chaos and strange attractors
Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the
Liouvillian dynamics of the Hopf bifurcation.
  • P. Gaspard, S. Tasaki
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2001
TLDR
Two-dimensional vector fields undergoing a Hopf bifurcation are studied in a Liouville-equation approach and degeneracy in the resonance spectrum is shown to yield a Jordan-block structure in the spectral decomposition.
Maximal Lyapunov exponent at crises.
  • Mehra, Ramaswamy
  • Physics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1996
TLDR
The largest Lyapunov exponent has universal behavior, showing abrupt variation as a function of the control parameter as the system passes through the crisis point, either in the value itself, in the case of an attractsor-widening crisis, or in the slope, for an attractor-merging crisis.
A review of linear response theory for general differentiable dynamical systems
The classical theory of linear response applies to statistical mechanics close to equilibrium. Away from equilibrium, one may describe the microscopic time evolution by a general differentiable
Diffusion, effusion, and chaotic scattering: An exactly solvable Liouvillian dynamics
We study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area-preserving mapping of an infinitely extended strip onto itself. This exactly solvable mapping
...
...