Characteristic Lyapunov vectors in chaotic time-delayed systems

Pazó, Diego; López, Juan M.

doi:10.1103/physreve.82.056201

Cited by 15 publications

(19 citation statements)

References 28 publications

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“…It is very instructive to check the theoretical predictions in two-dimensional models since the KPZ critical exponents depend on the spatial dimension. Current numerical capabilities allow us to study a two-dimensional lattice composed by L 2 coupled logistic maps (on a torus geometry) (20) where f [u(t)] ≡ 4u(t)[1 − u(t)] and the sum is over the set N of nearest neighbors.…”

Section: B Two-dimensional Systems (D = 2)mentioning

confidence: 99%

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Universal scaling of Lyapunov-exponent fluctuations in space-time chaos

2013

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Finite-time Lyapunov exponents of generic chaotic dynamical systems fluctuate in time. These fluctuations are due to the different degree of stability across the accessible phase space. A recent numerical study of spatially extended systems has revealed that the diffusion coefficient D of the Lyapunov exponents (LEs) exhibits a nontrivial scaling behavior, D(L)~L(-γ), with the system size L. Here, we show that the wandering exponent γ can be expressed in terms of the roughening exponents associated with the corresponding "Lyapunov surface." Our theoretical predictions are supported by the numerical analysis of several spatially extended systems. In particular, we find that the wandering exponent of the first LE is universal: in view of the known relationship with the Kardar-Parisi-Zhang equation, γ can be expressed in terms of known critical exponents. Furthermore, our simulations reveal that the bulk of the spectrum exhibits a clearly different behavior and suggest that it belongs to a possibly unique universality class, which has, however, yet to be identified.

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Section: B Two-dimensional Systems (D = 2)mentioning

confidence: 99%

“…(Color online) Rescaled variance χ 2 in the Mackey-Glass model, Eq. (19), with a = 0.1 and b = 0.2 (like in[20]). The slope of the straight line is 1/3.…”

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confidence: 99%

Universal scaling of Lyapunov-exponent fluctuations in space-time chaos

2013

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“…for u ≫ 1. The validity of this relationship in the context of Lyapunov dynamics in extended dissipative dynamical has been repeatedly investigated [9,10,12,21,22], showing that the leading Lyapunov vector falls within the universality class of KPZ dynamics [11]. The observable χ 2 defined in Eq.…”

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confidence: 96%

“…As a result, the same "physics" can be found in two significantly different contexts. In particular, the universality class of roughening phenomena identified by the Kardar-Parisi-Zhang (KPZ) equation [11] includes also the perturbation evolution in spatially extended chaotic systems [9,10,12].In spite of its broadness, the KPZ universality class does not encompass Hamiltonian models [13,14]. Preliminary studies revealed different critical properties and attributed the anomalous scaling to non-better specified long-range correlations [13].…”

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Diverging Fluctuations of the Lyapunov Exponents

2016

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We show that in generic one-dimensional Hamiltonian lattices the diffusion coefficient of the maximum Lyapunov exponent diverges in the thermodynamic limit. We trace this back to the long-range correlations associated with the evolution of the hydrodynamic modes. In the case of normal heat transport, the divergence is even stronger, leading to the breakdown of the usual single-function Family-Vicsek scaling ansatz. A similar scenario is expected to arise in the evolution of rough interfaces in the presence of suitably correlated background noise. DOI: 10.1103/PhysRevLett.117.034101 Lyapunov exponents (LEs) are dynamical invariants that provide a detailed characterization of low-dimensional as well as spatiotemporal chaos [1]: they indeed allow estimating the fractal dimension, the Kolmogorov-Sinai entropy, and allow us to ascertain the extensivity of the underlying dynamical regime. LEs are average quantities, defined as the infinite-time limit of the so-called finite-time Lyapunov exponents (FTLEs). Interestingly, also the temporal fluctuations of FTLEs carry important information that is ultimately encoded in yet another invariant: a suitable large deviation function. Fluctuations help to shed light on important phenomena such as intermittency, strange nonchaotic attractors, and stable chaos [1].In dissipative systems with many degrees of freedom, the fluctuations of the largest FTLE have been investigated in various numerical setups such as a shell model for the energy cascade in turbulence [2], a cellular automaton [3], molecular dynamics simulations [4], coupled-map lattice models [5,6], and a variety of continuous-time models [7,8]. In particular, in spatially extended systems like those in Refs. [5,6,8], the dynamics of Lyapunov vectors, i.e., perturbation fields, is formally equivalent to the evolution of rough interfaces in a noisy environment, the LE corresponding to the velocity of the interface [9,10]. This relationship is essentially based on the interpretation of the logarithm of the local amplitude of the perturbation with the height hðx; tÞ of a suitable interface. As a result, the same "physics" can be found in two significantly different contexts. In particular, the universality class of roughening phenomena identified by the Kardar-ParisiZhang (KPZ) equation [11] includes also the perturbation evolution in spatially extended chaotic systems [9,10,12].In spite of its broadness, the KPZ universality class does not encompass Hamiltonian models [13,14]. Preliminary studies revealed different critical properties and generically attributed the anomalous scaling to long-range correlations [13]. Later on, powerful methods for the characterization of large deviations revealed that extreme fluctuations of the FTLE in the classical Fermi-Pasta-Ulam (FPU-β) chain [15] correspond to atypical solutions of solitonlike and chaotic-breather dynamics [6,16]. However, it is not clear to what extent they are responsible for the anomalous non-KPZ behavior.In this Letter we study the diffusion coefficient D of t...

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Systems with Time-delay Feedback

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Characteristic Lyapunov vectors in chaotic time-delayed systems

Cited by 15 publications

References 28 publications

Universal scaling of Lyapunov-exponent fluctuations in space-time chaos

Universal scaling of Lyapunov-exponent fluctuations in space-time chaos

Diverging Fluctuations of the Lyapunov Exponents

Systems with Time-delay Feedback

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