scholarly journals Chaotic Dynamics in Nonautonomous Maps: Application to the Nonautonomous Hénon Map

2015 ◽  
Vol 25 (12) ◽  
pp. 1550172 ◽  
Author(s):  
Francisco Balibrea-Iniesta ◽  
Carlos Lopesino ◽  
Stephen Wiggins ◽  
Ana M. Mancho
Keyword(s):  
Chaotic Dynamics ◽  
Sufficient Conditions ◽  
Invariant Set ◽  
Precise Definition ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Hénon Map ◽  
Henon Map ◽  
Definition Of

In this paper, we analyze chaotic dynamics for two-dimensional nonautonomous maps through the use of a nonautonomous version of the Conley–Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley–Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous Hénon map and give sufficient conditions, in terms of the parameters defining the map, for the nonautonomous Hénon map to have a hyperbolic chaotic invariant set.

Relation between Topological Entropies and Fractal Dimensions of Chaotic Attractors of the Hénon Map

2012 ◽  
Vol 569 ◽  
pp. 447-450
Author(s):  
Xiao Zhou Chen ◽  
Liang Lin Xiong ◽  
Long Li
Keyword(s):  
Linear Relation ◽  
Chaotic Dynamics ◽  
Fractal Dimensions ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Hénon Map ◽  
Henon Map ◽  
Efficient Approach ◽  
Integral Relationship

In two-dimensional chaotic dynamics, relationship between fractal dimensions and topological entropies is an important issue to understand the chaotic attractors of Hénon map. we proposed a efficient approach for the estimation of topological entropies through the study on the integral relationship between fractal dimensions and topological entropies. Our result found that there is an approximate linear relation between their topological entropies and fractal dimensions.

On isotropic rank 1 convex functions

1999 ◽  
Vol 129 (5) ◽  
pp. 1081-1105 ◽  
Author(s):  
Miroslav Šilhavý
Keyword(s):  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Singular Values ◽  
Invariant Function ◽  
Two Dimensional ◽  
Rank 1 Convexity ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Rotationally Invariant ◽  
Positive Determinant

Let f be a rotationally invariant function defined on the set Lin+ of all tensors with positive determinant on a vector space of arbitrary dimension. Necessary and sufficient conditions are given for the rank 1 convexity of f in terms of its representation through the singular values. For the global rank 1 convexity on Lin+, the result is a generalization of a two-dimensional result of Aubert. Generally, the inequality on contains products of singular values of the type encountered in the definition of polyconvexity, but is weaker. It is also shown that the rank 1 convexity is equivalent to a restricted ordinary convexity when f is expressed in terms of signed invariants of the deformation.

Bifurcations of one- and two-dimensional maps

1984 ◽  
Vol 311 (1515) ◽  
pp. 43-102 ◽  
Keyword(s):  
Periodic Orbits ◽  
Period Doubling ◽  
Jacobian Determinant ◽  
Two Dimensional ◽  
Hénon Map ◽  
Stable And Unstable Manifolds ◽  
Henon Map ◽  
One Dimensional ◽  
The Family ◽  
Area Preserving

We study the qualitative dynamics of two-parameter families of planar maps of the form F^e(x, y) = (y, -ex+f(y)), where f :R -> R is a C 3 map with a single critical point and negative Schwarzian derivative. The prototype of such maps is the family f(y) = u —y 2 or (in different coordinates) f(y) = Ay(1 —y), in which case F^ e is the Henon map. The maps F e have constant Jacobian determinant e and, as e -> 0, collapse to the family f^. The behaviour of such one-dimensional families is quite well understood, and we are able to use their bifurcation structures and information on their non-wandering sets to obtain results on both local and global bifurcations of F/ ue , for small e . Moreover, we are able to extend these results to the area preserving family F/u. 1 , thereby obtaining (partial) bifurcation sets in the (/u, e)-plane. Among our conclusions we find that the bifurcation sequence for periodic orbits, which is restricted by Sarkovskii’s theorem and the kneading theory for one-dimensional maps, is quite different for two-dimensional families. In particular, certain periodic orbits that appear at the end of the one-dimensional sequence appear at the beginning of the area preserving sequence, and infinitely many families of saddle node and period doubling bifurcation curves cross each other in the ( /u, e ) -parameter plane between e = 0 and e = 1. We obtain these results from a study of the homoclinic bifurcations (tangencies of stable and unstable manifolds) of F /u.e and of the associated sequences of periodic bifurcations that accumulate on them. We illustrate our results with some numerical computations for the orientation-preserving Henon map.

THREE-DIMENSIONAL HÉNON-LIKE MAPS AND WILD LORENZ-LIKE ATTRACTORS

2005 ◽  
Vol 15 (11) ◽  
pp. 3493-3508 ◽  
Author(s):  
S. V. GONCHENKO ◽  
I. I. OVSYANNIKOV ◽  
C. SIMÓ ◽  
D. TURAEV
Keyword(s):  
Lyapunov Exponent ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Maximal Lyapunov Exponent ◽  
Hénon Map ◽  
Henon Map ◽  
Quadratic Map ◽  
Different Types ◽  
Natural Way

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a natural way in unfoldings of several types of homoclinic bifurcations). Among other observations, we have evidence that there are different types of Lorenz-like attractor domains in the parameter space of the 3D Hénon map. In all cases the maximal Lyapunov exponent, Λ1, is positive. Concerning the next Lyapunov exponent, Λ2, there are open domains where it is definitely positive, others where it is definitely negative and, finally, domains where it cannot be distinguished numerically from zero (i.e. |Λ2| < ρ, where ρ is some tolerance ranging between 10-5 and 10-6). Furthermore, several other types of interesting attractors have been found in this family of 3D Hénon maps.

Digital Henon Sequences Generation and its Analysis

2014 ◽  
Vol 981 ◽  
pp. 793-796 ◽  
Author(s):  
Bing Bing Song ◽  
Jing Pan ◽  
Qun Ding
Keyword(s):  
Numerical Results ◽  
Statistical Properties ◽  
Two Dimensional ◽  
Hénon Map ◽  
Digital Output ◽  
Balance Test ◽  
Henon Map ◽  
Output Sequence ◽  
Dsp Builder ◽  
Run Test

In this paper, the typical two-dimensional Henon map is studied. Firstly, the model of Henon map is proposed based on DSP Builder platform in Simulink library, so it can generate digital output sequence of Henon map. Then, its statistical properties are analyzed for such output sequences, including balance test, run test and autocorrelation test. Finally, the numerical results show that such digital Henon sequences have good pseudo-randomness.

SCENE SEGMENTATION OF THE CHAOTIC OSCILLATOR NETWORK

2000 ◽  
Vol 10 (07) ◽  
pp. 1697-1708 ◽  
Author(s):  
L. ZHAO ◽  
E. E. N. MACAU ◽  
N. OMAR
Keyword(s):  
Theoretical Prediction ◽  
Computer Simulations ◽  
Chaotic Dynamics ◽  
Chaotic Synchronization ◽  
Chaotic Oscillator ◽  
Two Dimensional ◽  
Scene Segmentation ◽  
Oscillatory Correlation ◽  
Oscillator Network ◽  
Definition Of

A chaotic oscillatory correlation network for scene segmentation is presented. It is a two-dimensional array with locally coupled chaotic elements. It offers a mechanism to escape from the synchrony–desynchrony dilemma. As a result, this model has unbounded capacity of segmentation. Chaotic dynamics and chaotic synchronization in the model are analyzed. Desynchronization property is guaranteed by the definition of chaos. Computer simulations confirm the theoretical prediction.

On the Mode Splitting of Degenerate Mechanical Systems Containing Cracks

1993 ◽  
Vol 60 (4) ◽  
pp. 929-935 ◽  
Author(s):  
I. Y. Shen ◽  
C. D. Mote
Keyword(s):  
Orthogonal Transformation ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Vibration Modes ◽  
Two Dimensional ◽  
Frequency Shifts ◽  
Numerical Examples ◽  
Perfect System ◽  
Mode Splitting ◽  
Circular Domains

This paper presents sufficient conditions governing mode splitting in a two-dimensional, degenerate, mechanical system whose eigensolutions satisfy the Helmholtz equation. When cracks are introduced into such systems, a pair of repeated vibration modes may remain repeated or become distinct (termed split modes) depending on the location and geometry of the cracks. Two types of split modes can occur. Split modes of the first kind are a pair of split modes in which one mode undergoes a frequency shift but the other does not. In contrast, split modes of the second kind are a pair of split modes in which both modes undergo frequency shifts. A sufficient condition for split modes of the first kind is derived through an orthogonal transformation of repeated eigenmodes of the perfect system. Sufficient conditions for repeated modes and split modes of the second kind are derived through an asymptotic analysis. Numerical examples on square and circular domains illustrate the analytical predictions on mode splitting.

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