The Smale Horseshoe

Synchronization of chaotic oscillators has become well characterized by errors which shrink relative to a synchronization manifold. This manifold is the identity function in the case of identical systems, or some other slow manifold in the case of generalized synchronizaton in the case of nonidentical components. On the other hand, since many decades beginning with the Smale horseshoe, chaotic oscillators can be well understood in terms of symbolic dynamics as information producing processes. We study here the synchronization of a pair of chaotic oscillators as a process for sharing information bearing bits transferred between each other, by measuring the transfer entropy tracked as the global system transitions to the synchronization state. Further, we present for the first time the notion of transfer entropy in the measure theoretic setting of transfer operators.

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Introduction to Applied Nonlinear Dynamical Systems and Chaos - Texts in Applied Mathematics ◽

10.1007/0-387-21749-5_24 ◽

2006 ◽

pp. 555-575

Keyword(s):

Smale Horseshoe

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Smale horseshoe maps by 5-adic fractions

Chinese Science Bulletin ◽

10.1007/bf02882485 ◽

1997 ◽

Vol 42 (10) ◽

pp. 799-803

Author(s):

Jiehua Mai

Keyword(s):

Smale Horseshoe

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A NEW AND SIMPLE CHAOS TOY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127402005480 ◽

2002 ◽

Vol 12 (08) ◽

pp. 1843-1857 ◽

Cited By ~ 2

Author(s):

ERIK M. BOLLT ◽

AARON KLEBANOFF

Keyword(s):

Ball Bearing ◽

Nonlinear Oscillations ◽

Equations Of Motion ◽

Future Application ◽

Lagrangian Mechanics ◽

Smale Horseshoe ◽

Melnikov's Method ◽

Melnikov’S Method ◽

Bicycle Wheel

We present two new, and perhaps the simplest yet, mechanical chaos demonstrations. They are both designed based on a recipe of competing nonlinear oscillations. One of these devices is simple enough that using the provided description, it can be built using a bicycle wheel, a piece of wood routed with an elliptical track, and a ball bearing. We provide a thorough Lagrangian mechanics based derivation of equations of motion, and a proof of chaos based on showing the existence of an embedded Smale horseshoe using Melnikov's method. We conclude with discussion of a future application.

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PIECEWISE TWO-DIMENSIONAL MAPS AND APPLICATIONS TO CELLULAR NEURAL NETWORKS

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740401062x ◽

2004 ◽

Vol 14 (07) ◽

pp. 2223-2228 ◽

Cited By ~ 5

Author(s):

HSIN-MEI CHANG ◽

JONG JUANG

Keyword(s):

Neural Networks ◽

Cellular Neural Networks ◽

Two Dimensional ◽

Smale Horseshoe ◽

Linear Map ◽

One Dimensional ◽

Spatial Chaos

Of concern is a two-dimensional map T of the form T(x,y)=(y,F(y)-bx). Here F is a three-piece linear map. In this paper, we first prove a theorem which states that a semiconjugate condition for T implies the existence of Smale horseshoe. Second, the theorem is applied to show the spatial chaos of one-dimensional Cellular Neural Networks. We improve a result of Hsu [2000].

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Suppression of Smale horseshoe structure via secondary perturbations in pendulum systems

Pramana ◽

10.1007/bf02830103 ◽

1999 ◽

Vol 52 (4) ◽

pp. 375-387 ◽

Cited By ~ 4

Author(s):

K. I. Thomas ◽

G. Ambika

Keyword(s):

Smale Horseshoe

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Study on Chaotic Behavior of the Restricted Four-Body Problem with an Equilateral Triangle Configuration

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417500262 ◽

2017 ◽

Vol 27 (02) ◽

pp. 1750026 ◽

Cited By ~ 4

Author(s):

Xuhua Cheng ◽

Zhikun She

Keyword(s):

Equilateral Triangle ◽

Body Problem ◽

Chaotic Behavior ◽

Homoclinic Orbits ◽

Simple Zero ◽

Integral Function ◽

Smale Horseshoe ◽

Melnikov Integral ◽

Transversal Homoclinic Orbits ◽

Four Body Problem

In this paper, the chaotic behavior of a planar restricted four-body problem with an equilateral triangle configuration is analytically studied. Firstly, according to the perturbation method of Melnikov, the planar restricted four-body problem is regarded as a perturbation of the two-body model. Then, we show that the Melnikov integral function has a simple zero, arriving at the existence of transversal homoclinic orbits. Afterwards, since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle, we alternatively construct an invertible map [Formula: see text] and check that [Formula: see text] is a Smale horseshoe map, showing that our restricted four-body problem possesses chaotic behavior of the Smale horseshoe type.

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