A comparison of correlation and Lyapunov dimensions

En este trabajo se presentan 10 nuevos sistemas autónomos no lineales caóticos simples. Estos sistemas se encontraron utilizando el método Monte Carlo y tienen la característica de tener uno de sus puntos de equilibrio asintóticamente estables. Estos nuevos sistemas no tienen caos en el sentido de Shilnikov, pero sus diagramas de bifurcación muestran una ruta de periodo doble hacia el caos. Se calculó también la dimensión de Kaplan-Yorke, dando como resultado en un rango de 2-3.

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“…There is a relationship between Lyapunov exponents and the Kaplan-Yorke dimension (Chlouverakis and Sprott, 2005) given by: …”

Section: M10mentioning

confidence: 99%

Puntos de equilibrio asintóticamente estables en nuevos sistemas caóticos

Casas-García

Quezada-Téllez

Carrillo-Moreno

et al. 2016

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“…It was done following the KaplanYorke definition which establishes a conjecture about the fractal dimension of the attractor and the Lyapunov spectrum [30]. It can be defined as the fractional dimension in which a cluster of initial conditions will neither expand nor contract as it evolves in time [4]. The Lyapunov dimension D L can be calculated according to [30]:…”

Section: Poincaré Map and Lyapunov Exponentsmentioning

confidence: 99%

A modified Chua chaotic oscillator and its application to secure communications

Zapateiro

Acho

Vidal

2014

Applied Mathematics and Computation

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In this paper, a new modified Chua oscillator is introduced. The original Chua oscillator is well known for its simple implementation and mathematical modeling. A modification of the oscillator is proposed in order to facilitate the synchronization and the encryption and decryption scheme. The modification consists in changing the nonlinear term of the original oscillator to a smooth and bounded nonlinear function. A bifurcation diagram, a Poincaré map and the Lyapunov exponents are presented as proofs of chaoticity of the newly modified oscillator. An application to secure communications is proposed in which two channels are used. Numerical simulations are performed in order to analyze the communication system.

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“…While studying a three-dimensional dynamical system Chlouverakis and Sprott (2004) propose to fit S j with a parabola. The result is:…”

Section: A54 Liapounoff Dimension and Chlouverakis-sprott Conjecturementioning

confidence: 99%

“…According to Chlouverakis and Sprott (2004), Kaplan-Yorke conjecture is a good approximation for a strange attractor the fractal dimension of which is close to an integer and the proposed quadratic modification provides a significative difference when the fractal dimension is close to an odd halfinteger.…”

Section: A54 Liapounoff Dimension and Chlouverakis-sprott Conjecturementioning

confidence: 99%

Back Matter

2009

World Scientific Series on Nonlinear Science Series A

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In this work a new approach which consists in applying Differential Geometry to Dynamical Systems and called Flow Curvature Method has been presented. By considering the trajectory curve, integral of any n-dimensional dynamical system, as a curve in Euclidean n-space, the curvature of the trajectory curve, i.e. curvature of the flow has been analytically computed enabling thus to define a manifold called: flow curvature manifold. It has been stated that, since such manifold only involves the time derivatives of the velocity vector field and so, contains information about the dynamics of the system, it enables to find again the main features of the dynamical system studied. Thus, fixed points stability, invariant sets, centre manifold , normal forms, local bifurcations, slow invariant manifold and integrability of any n-dimensional dynamical systems have been deduced from the flow curvature manifold, i.e. according to the Flow Curvature Method . The concepts of global invariance and local invariance has been (re)defined from Darboux invariance theorem. So, it has been stated that flow curvature manifold also enabled to "detect" linear invariant manifolds of any n-dimensional dynamical systems which may be used to build first integrals of these systems. For nonlinear invariant manifolds identity between flow curvature manifold and the so-called extatic manifolds has also been stated. It has been established that the Flow Curvature Method enabled to easily compute the coefficients of the centre manifold approximation of any n-dimensional dynamical systems according to global invariance of the flow curvature manifold. Then, a link between normal forms of dynamical systems and "normal forms" of flow curvature manifold has been highlighted. Such a link enabled to directly compute the normal form of a dynamical system starting from its flow curvature manifold . 265

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A comparison of correlation and Lyapunov dimensions

Cited by 69 publications

References 13 publications

Puntos de equilibrio asintóticamente estables en nuevos sistemas caóticos

Puntos de equilibrio asintóticamente estables en nuevos sistemas caóticos

A modified Chua chaotic oscillator and its application to secure communications

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