The liapunov dimension of strange attractors

Frederickson, Paul O.; Kaplan, James L.; Yorke, Ellen; Yorke, James A.

doi:10.1016/0022-0396(83)90011-6

Cited by 483 publications

(189 citation statements)

References 9 publications

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“…In the chaotic regime the system exhibits only one positive Lyapunov exponent, 31 which means that there is only one expanding direction. An estimation of the fractal dimension of the chaotic attractor at ϭ0.029 920 06 using the KaplanYorke conjecture 37 results in the dimension DϷ2.08, which implies that in the Poincaré section the chaotic attractor is a set of dimension D p ϭDϪ1Ϸ1.08, and hence the attractor is a low-dimensional set embedded in a high-dimensional phase space. This can be seen in Fig.…”

Section: Nonlinear Dynamics Analysismentioning

confidence: 99%

Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

Rempel

Chian

Macau

et al. 2004

Chaos: An Interdisciplinary Journal of Nonlinear Science

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This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1759297͔In the past decades chaos theory has been revealed as a powerful way to explain the physics of low-dimensional dynamical systems, that is, dynamical systems with a small number of state variables. However, most problems of practical interest in physics and engineering are described by partial differential equations "PDEs…, and it is still unclear in which situations the physical mechanisms responsible for the onset of chaos in low-dimensional systems are applicable to systems described by PDEs. Several authors have tried to develop a dynamical systems theory for high-dimensional dynamical systems, frequently described by sets of coupled ordinary differential equations obtained as approximations to the original PDEs. 1-7 The simplest approach consists in first ''borrowing'' the tools developed for low-dimensional dynamical systems and applying them to spatially extended systems in regimes that display characteristics of lowdimensional chaos. Then, it is possible to develop the tools to be used in more complex regimes. In this paper we try to improve the understanding of the connection between low-dimensional chaotic systems and a certain class of spatiotemporal systems: the one that exhibits temporal chaos and spatial coherence. We focus on the development of a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems.

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Section: Nonlinear Dynamics Analysismentioning

confidence: 99%

Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

Rempel

Chian

Macau

et al. 2004

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…System (2) has Lyapunov exponents as LE 1 = 0.0696, LE 2 = 0.0359, LE 3 = 0.0002, and LE 4 = −24.5176 for the parameters listed above (see [18,19] for a detailed discussion of Lyapunov exponents of strange attractors in dynamical systems). Thus, In this paper, all the simulations are carried out by using fourth-order Runge-Kutta Method with time-step ℎ = 0.005.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a Generalized Lorenz–Stenflo Equation

2017

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Although the globally attractive sets of a hyperchaotic system have important applications in the fields of engineering, science, and technology, it is often a difficult task for the researchers to obtain the globally attractive set of the hyperchaotic systems due to the complexity of the hyperchaotic systems. Therefore, we will study the globally attractive set of a generalized hyperchaotic Lorenz-Stenflo system describing the evolution of finite amplitude acoustic gravity waves in a rotating atmosphere in this paper. Based on Lyapunov-like functional approach combining some simple inequalities, we derive the globally attractive set of this system with its parameters. The effectiveness of the proposed methods is illustrated via numerical examples.

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“…The Lyapunov dimension of the attractors of (3) according to Kaplan-Yorke conjecture is defined as [32] …”

Section: Lyapunov Exponentsmentioning

confidence: 99%

A New Nonlinear Chaotic Complex Model and Its Complex Antilag Synchronization

Mahmoud

Abood

2017

Complexity

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Another chaotic nonlinear Lü model with complex factors is covered here. We can build this riotous complex system when we add a complex nonlinear term to the third condition of the complex Lü system and think of it as if every one of the factors is mind boggling or complex. This system in real adaptation is a 6-dimensional continuous autonomous chaotic system. Different types of chaotic complex Lü system are developed. Also, another sort of synchronization is presented by us which is simple for anybody to ponder for the chaotic complex nonlinear system. This sort might be called a complex antilag synchronization (CALS). There are irregular properties for CALS and they do not exist in the literature; for example, (i) the CALS contains or fused two sorts of synchronizations (antilag synchronization ALS and lag synchronization LS); (ii) in CALS the attractors of the main and slave systems are moving opposite or similar to each other with time lag; (iii) the state variable of the main system synchronizes with a different state variable of the slave system. A scheme is intended to accomplish CALS of chaotic complex systems in light of Lyapunov function. The acquired outcomes and effectiveness can be represented by a simulation case for our new model.

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The liapunov dimension of strange attractors

Cited by 483 publications

References 9 publications

Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

Analysis of a Generalized Lorenz–Stenflo Equation

A New Nonlinear Chaotic Complex Model and Its Complex Antilag Synchronization

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