Relation between fractal dimension and spatial correlation length for extensive chaos

Egolf, David A.; Greenside, Henry

doi:10.1038/369129a0

Cited by 90 publications

(67 citation statements)

References 15 publications

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“…A clear signature of the subwavelength chaotic degree of freedom was not found using our spatial diagnostics. Our findings that the chaotic length scale ␦ is smaller than, and not correlated with, the pattern's long-range spatial order are in agreement with previous findings using the complex Ginzburg-Landau equation 30,31 and coupled map lattices. 16 A physical understanding of the chaotic length scale with respect to the spatial features of the patterns remains an open challenge and further study would be useful.…”

Section: B Variation Of the Fractal Dimension With Forcingsupporting

confidence: 82%

Extensive chaos in the Lorenz-96 model

Karimi

Paul

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

145

123

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We explore the high-dimensional chaotic dynamics of the Lorenz-96 model by computing the variation of the fractal dimension with system parameters. The Lorenz-96 model is a continuous in time and discrete in space model first proposed by Lorenz to study fundamental issues regarding the forecasting of spatially extended chaotic systems such as the atmosphere. First, we explore the spatiotemporal chaos limit by increasing the system size while holding the magnitude of the external forcing constant. Second, we explore the strong driving limit by increasing the external forcing while holding the system size fixed. As the system size is increased for small values of the forcing we find dynamical states that alternate between periodic and chaotic dynamics. The windows of chaos are extensive, on average, with relative deviations from extensivity on the order of 20%. For intermediate values of the forcing we find chaotic dynamics for all system sizes past a critical value. The fractal dimension exhibits a maximum deviation from extensivity on the order of 5% for small changes in system size and the deviation from extensivity decreases nonmonotonically with increasing system size. The length scale describing the deviations from extensivity is consistent with the natural chaotic length scale in support of the suggestion that deviations from extensivity are due to the addition of chaotic degrees of freedom as the system size is increased. We find that each wavelength of the deviation from extensive chaos contains on the order of two chaotic degrees of freedom. As the forcing is increased, at constant system size, the dimension density grows monotonically and saturates at a value less than unity. We use this to quantify the decreasing size of chaotic degrees of freedom with increased forcing which we compare with spatial features of the patterns. © 2010 American Institute of Physics. ͓doi:10.1063/1.3496397͔We explore fundamental questions regarding the composition and description of spatiotemporal chaos in highdimensional systems. The Lorenz-96 model is used for its phenomenological relevance to fluid convection and atmospheric dynamics. We compute the variation of the fractal dimension over a wide range of system parameters, for very long-times, and over many initial conditions. We explore two limits: the "spatiotemporal chaos" limit where the system size is increased while holding the external forcing constant and the "strong driving" limit where the external forcing is increased for a system of fixed size. The variations in the fractal dimension with system parameters are used to provide estimates for characteristic length and time scales describing the chaotic dynamics. These findings are directly compared with experimentally accessible diagnostics of the pattern dynamics to provide new physical insights into the underlying features of high-dimensional chaos.

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Section: B Variation Of the Fractal Dimension With Forcingsupporting

confidence: 82%

Extensive chaos in the Lorenz-96 model

Karimi

Paul

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

145

123

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“…Recent work [3][4][5][6] indicates that such lattices often exhibit "non-trivial" cooperative behavior demonstrating a rich variety of phase transitions. Such behavior is "non-trivial" because the chaotic dynamics of these lattices with short-scale interaction must exhibit extensive chaos: the number of Lyapunov exponents increases with the size of the lattice (see, for example, [1,2]). Actually, the cooperative behavior of chaotic lattices depends strongly on the strength of the local connection, variations of which reveal a rich diversity of synchronization.…”

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confidence: 99%

“…-The analysis of the behavior of large assemblies of chaotic elements has been the subject of recent investigations, and is of interest both from the fundamental [1][2][3][4][5] and modeling points of view [6][7][8][9]. In particular, a network of chaotic elements is currently a very popular ingredient of information processing [7].…”

mentioning

confidence: 99%

Clusters of synchronization and bistability in lattices of chaotic neurons

Huerta¹,

Bazhenov²,

Rabinovich³

1998

Europhys. Lett.

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General, theoretical, and mathematical biophysics (including logic of biosystems, quantum biology, and relevant aspects of thermodynamics, information theory, cybernetics, and bionics). PACS. 87.22As -General theory and phenomenology.Abstract. -We investigated different regimes of synchronization in large lattices of chaotic spiking-bursting neurons. The lattices exhibit the following features: developed spatio-temporal disorder with no synchronization; spatial clusters of bursting synchronization; homogeneous bursting synchronization; and complete chaotic synchronization. We observed a bistable synchronization phenomenon in a wide region of the control parameter space. The bistability exists for homogeneous bursting synchronization with long-range correlation and spatial clusters of partial synchronization. The bistable regime appears in lattices with a size larger than the space-scale of these clusters.Introduction. -The analysis of the behavior of large assemblies of chaotic elements has been the subject of recent investigations, and is of interest both from the fundamental [1][2][3][4][5] and modeling points of view [6][7][8][9]. In particular, a network of chaotic elements is currently a very popular ingredient of information processing [7].We report in this letter on the collective dynamics of chaotic neurons with local interactions. Recent work [3][4][5][6] indicates that such lattices often exhibit "non-trivial" cooperative behavior demonstrating a rich variety of phase transitions. Such behavior is "non-trivial" because the chaotic dynamics of these lattices with short-scale interaction must exhibit extensive chaos: the number of Lyapunov exponents increases with the size of the lattice (see, for example, [1,2]). Actually, the cooperative behavior of chaotic lattices depends strongly on the strength of the local connection, variations of which reveal a rich diversity of synchronization.

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“…Equation (1) has been investigated recently by some authors and there has been considerable progress in understanding its chaotic dynamics [3][4][5]. It has been found that there are at least two types of a chaotic state in this system.…”

mentioning

confidence: 99%

“…PACS numbers: 05.45.+b One of the simplest models of spatially extended dissipative systems showing spatio-temporal chaos is the one-dimensional complex Ginzburg-Landau equation [1] It describes a small and slowly varying complex field A(x, t), called an order parameter, which is a measure of an oscillatory instability of a one-dimensional medium in the vicinity of a Hopf bifurcation [2]. Equation (1) has been investigated recently by some authors and there has been considerable progress in understanding its chaotic dynamics [3][4][5]. It has been found that there are at least two types of a chaotic state in this system.…”

mentioning

confidence: 99%

Turbulence in the One-Dimensional Complex Ginzburg-Landau Equation

Zubrzycki¹

1995

Acta Phys. Pol. A

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The transition to phase and amplitude turbulent states in the one-dimensional complex Ginzburg-Landau equation is investigated by a numerical simulation. In order to describe existing attractors quantitatively, the largest Lyapunov exponent is estimated. The largest Lyapunov exponent is positive but small for the phase turbulence and it is much larger for the amplitude turbulence. Moreover, it seems independent on the system size. PACS numbers: 05.45.+b One of the simplest models of spatially extended dissipative systems showing spatio-temporal chaos is the one-dimensional complex Ginzburg-Landau equation [1] It describes a small and slowly varying complex field A(x, t), called an order parameter, which is a measure of an oscillatory instability of a one-dimensional medium in the vicinity of a Hopf bifurcation [2]. Equation (1) has been investigated recently by some authors and there has been considerable progress in understanding its chaotic dynamics [3][4][5]. It has been found that there are at least two types of a chaotic state in this system. The first is the "phase turbulent" state with the field magnitude |A| almost constant in space and time. The threshold of the phase turbulence in the (a, Q) parameters space is the zero value of 1 -k a,6. The phase turbulence appears for negative 1 + ag. On the other hand as long as 1 + αβ is positive a homogeneous solution of Eq. (1) for p> 0, is linearly stable. The second chaotic state is the "amplitude turbulent" state, called also a "defect chaos", involving large amplitude fluctuations and isolated "defects" with the zero field A value. The aim of this paper is to investigate by the numerical simulation a process of the transition to the phase and amplitude turbulence. In order to describe quantitatively the chaotic state in the two phases, the largest Lyapunov exponent was determined. A still open question is whether (925)

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Relation between fractal dimension and spatial correlation length for extensive chaos

Cited by 90 publications

References 15 publications

Extensive chaos in the Lorenz-96 model

Extensive chaos in the Lorenz-96 model

Clusters of synchronization and bistability in lattices of chaotic neurons

Turbulence in the One-Dimensional Complex Ginzburg-Landau Equation

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