Velocity-dependent Lyapunov exponents as a measure of chaos for open-flow systems

Deissler, Robert J.; Kaneko, Kunihiko

doi:10.1016/0375-9601(87)90581-0

Cited by 137 publications

(102 citation statements)

References 27 publications

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“…There the instability arising from the off-diagonal element [13] in the Jacobi matrix may lead to a novel dynamical behavior. Such instability is also seen in the convective instability in open flow problem [14], where the instability due to the off-diagonal element leads to a rich variety of dynamics [15]. The strange non-chaotic attractor in our model is also due to the instability by the off-diagonal element, as is characterized by the phase sensitivity of the amplitude.…”

Section: Summary and Discussionmentioning

confidence: 64%

Fractalization of a torus as a strange nonchaotic attractor

1996

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Fractalization of torus and its transition to chaos in a quasi-periodically forced logistic map is re-investigated in relation with a strange nonchaotic attractor, with the aid of functional equation for the invariant curve. Existence of fractal torus in an interval in parameter space is confirmed by the length and the number of extrema of the torus attractor, as well as the Fourier mode analysis. Mechanisms of the onset of fractal torus and the transition to chaos are studied in connection with the intermittency.

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Section: Summary and Discussionmentioning

confidence: 64%

Fractalization of a torus as a strange nonchaotic attractor

1996

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“…For the maps considered in this section the identity V p = V L is always fulfilled. As shown in [6], in a closed system with symmetric coupling Λ = Λ(v) has typically a concave shape, with the maximum located at v = 0 (in particular λ = Λ(v = 0)). An approximate expression can be obtained for Λ(v), by substituting i = vt in (14) and by comparing it with (15) :…”

Section: A Linear Mechanismsmentioning

confidence: 97%

“…by following the perturbation along the world line i = vt one can easily measures the corresponding comoving Lyapunov exponent Λ(v) (for more details see [6,22]). The information propagation velocity is the maximal velocity for which a disturbance still propagates without being damped.…”

Section: A Linear Mechanismsmentioning

confidence: 99%

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Linear and nonlinear information flow in spatially extended systems

Cencini

Torcini

2001

Phys. Rev. E

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Infinitesimal and finite amplitude error propagation in spatially extended systems are numerically and theoretically investigated. The information transport in these systems can be characterized in terms of the propagation velocity of perturbations Vp. A linear stability analysis is sufficient to capture all the relevant aspects associated to propagation of infinitesimal disturbances. In particular, this analysis gives the propagation velocity VL of infinitesimal errors. If linear mechanisms prevail on the nonlinear ones Vp = VL. On the contrary, if nonlinear effects are predominant finite amplitude disturbances can eventually propagate faster than infinitesimal ones (i.e. Vp > VL). The finite size Lyapunov exponent can be successfully employed to discriminate the linear or nonlinear origin of information flow. A generalization of finite size Lyapunov exponent to a comoving reference frame allows to state a marginal stability criterion able to provide Vp both in the linear and in the nonlinear case. Strong analogies are found between information spreading and propagation of fronts connecting steady states in reaction-diffusion systems. The analysis of the common characteristics of these two phenomena leads to a better understanding of the role played by linear and nonlinear mechanisms for the flow of information in spatially extended systems.

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“…We start by briefly recalling the concept of convective Lyapunov exponents [26]. Given a unidimensional lattice model in the stationary regime, let us introduce a δ-like perturbation at time t = 0 in the origin i = 0 and imagine to monitor the perturbation amplitude w i (t).…”

Section: Relationship With Deterministic Chaosmentioning

confidence: 99%

Nonlinear Dynamics and Chaos: Advances and Perspectives

Thiel¹,

Kurths²,

Romano³

et al. 2010

Understanding Complex Systems

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Abstract. Stable chaos is a generalization of the chaotic behaviour exhibited by cellular automata to continuous-variable systems and it owes its name to an underlying irregular and yet linearly stable dynamics. In this review we discuss analogies and differences with the usual deterministic chaos and introduce several tools for its characterization. Some examples of transitions from ordered behavior to stable chaos are also analyzed to further clarify the underlying dynamical properties. Finally, two models are specifically discussed: the diatomic hard-point gas chain and a network of globally coupled neurons.

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Velocity-dependent Lyapunov exponents as a measure of chaos for open-flow systems

Cited by 137 publications

References 27 publications

Fractalization of a torus as a strange nonchaotic attractor

Fractalization of a torus as a strange nonchaotic attractor

Linear and nonlinear information flow in spatially extended systems

Nonlinear Dynamics and Chaos: Advances and Perspectives

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