On the Lyapunov dimension of the attractor of the Chirikov dissipative mapping

2005 ◽  
pp. 15-28 ◽  
Author(s):  
G. A. Leonov ◽  
M. S. Poltinnikova
Keyword(s):  
Lyapunov Dimension

scholarly journals Qualitative Study of a 4D Chaos Financial System

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Fuchen Zhang ◽  
Gaoxiang Yang ◽  
Yong Zhang ◽  
Xiaofeng Liao ◽  
Guangyun Zhang
Keyword(s):  
Hausdorff Dimension ◽  
Chaos Synchronization ◽  
Chaos Control ◽  
Dynamical Behavior ◽  
Lyapunov Stability Theory ◽  
Attraction Domain ◽  
Ultimate Boundedness ◽  
Lyapunov Dimension ◽  
Simulation Results ◽  
Ultimate Bound

Some dynamics of a new 4D chaotic system describing the dynamical behavior of the finance are considered. Ultimate boundedness and global attraction domain are obtained according to Lyapunov stability theory. These results are useful in estimating the Lyapunov dimension of attractors, Hausdorff dimension of attractors, chaos control, and chaos synchronization. We will also present some simulation results. Furthermore, the volumes of the ultimate bound set and the global exponential attractive set are obtained.

THE NONEQUIVALENCE AND DIMENSION FORMULA FOR ATTRACTORS OF LORENZ-TYPE SYSTEMS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350200 ◽  
Author(s):  
YUMING CHEN ◽  
QIGUI YANG
Keyword(s):  
Upper Bound ◽  
Analytical Formula ◽  
Type Systems ◽  
Lyapunov Dimension ◽  
Dimension Formula ◽  
Lü System ◽  
Suitable Parameter ◽  
Lu System

In this paper, Lü system with a set of chaotic parameters is proved to be smoothly nonequivalent to Chen and Lorenz systems with any parameter. The analytical formula for the upper bound of Lyapunov dimension of attractors in Lorenz-type systems are presented under some suitable parameter conditions. These properties studied in this paper may contribute to a better understanding of the Lorenz-type systems.

ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS

1996 ◽  
Vol 06 (05) ◽  
pp. 919-948 ◽  
Author(s):  
D. TURAEV
Keyword(s):  
Lyapunov Exponents ◽  
Center Manifold ◽  
Critical Dimension ◽  
Lyapunov Dimension ◽  
Stable And Unstable Manifolds ◽  
Local Bifurcations ◽  
Non Local ◽  
Unstable Manifolds ◽  
The Difference ◽  
Manifold Theory

An analogue of the center manifold theory is proposed for non-local bifurcations of homo- and heteroclinic contours. In contrast with the local bifurcation theory it is shown that the dimension of non-local bifurcational problems is determined by the three different integers: the geometrical dimension dg which is equal to the dimension of a non-local analogue of the center manifold, the critical dimension dc which is equal to the difference between the dimension of phase space and the sum of dimensions of leaves of associated strong-stable and strong-unstable foliations, and the Lyapunov dimension dL which is equal to the maximal possible number of zero Lyapunov exponents for the orbits arising at the bifurcation. For a wide class of bifurcational problems (the so-called semi-local bifurcations) these three values are shown to be effectively computed. For the orbits arising at the bifurcations, effective restrictions for the maximal and minimal numbers of positive and negative Lyapunov exponents (correspondingly, for the maximal and minimal possible dimensions of the stable and unstable manifolds) are obtained, involving the values dc and dL. A connection with the problem of hyperchaos is discussed.

CHAOTIC HIERARCHY IN HIGH DIMENSIONS

2000 ◽  
Vol 14 (24) ◽  
pp. 2511-2527 ◽  
Author(s):  
D. E. POSTNOV ◽  
A. G. BALANOV ◽  
O. V. SOSNOVTSEVA ◽  
E. MOSEKILDE
Keyword(s):  
Discrete Time ◽  
Chaotic Behavior ◽  
Higher Order ◽  
Chaotic Attractors ◽  
Lyapunov Dimension ◽  
High Dimensions ◽  
Time Model ◽  
Discrete Time Model ◽  
The Creation

The paper suggests a new mechanism for the development of higher-order chaos in accordance with the concept of a chaotic hierarchy. A discrete-time model is proposed which demonstrates how the creation of coexisting chaotic attractors combined with boundary crises can produce a continued growth of the Lyapunov dimension of the resulting chaotic behavior.

scholarly journals KOLMOGOROV CAPACITY AND LYAPUNOV DIMENSION OF STRANGE ATTRACTORS OF FORCED BRUSSELATOR

1984 ◽  
Vol 33 (9) ◽  
pp. 1246
Author(s):  
WANG GUANG-RUI ◽  
CHEN SHI-GANG ◽  
HAO BAI-LIN
Keyword(s):  
Strange Attractors ◽  
Lyapunov Dimension

scholarly journals An Innovative Way to Generate Hamiltonian Energy of a New Hyperchaotic Complex Nonlinear Model and Its Control

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Kholod M. Abualnaja
Keyword(s):  
Fixed Point ◽  
Active Control ◽  
Nonlinear Model ◽  
Control Method ◽  
Complex Variables ◽  
Bifurcation Diagrams ◽  
Lyapunov Dimension ◽  
Innovative Method ◽  
Modern System ◽  
Active Control Method

We are implementing a new Rabinovich hyperchaotic structure with complex variables in this research. This modern system is a real, autonomous hyperchaotic, and 8-dimensional continuous structure. Some of the characteristics of this system, as well as for invariance, dissipation, balance, and stability, are technically analyzed. Some other properties are also studied numerically, such as Lyapunov exponents, Lyapunov dimension, bifurcation diagrams, and chaotic actions. Hamiltonian energy is being studied and applying by using the innovative method. Via active control method, we inhibit our system’s hyperchaotic behavior. The new system’s hyperchaotic solutions are transformed into its unstable, trivial fixed point. The validity of the findings obtained is illustrated by an example of numerics and reenactment. Numerical results are plotted to display the variables of the state after and before the control to demonstrate that the control is being achieved.

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