Noise robustness condition for chaotic maps with piecewise constant invariant density

Chaotic maps represent an effective method of generating random-like sequences, that combines the benefits of relying on simple, causal models with good unpredictability. However, since chaotic maps behavior is generally strongly dependent on unavoidable implementation errors and external perturbations, the possibility of guaranteeing map statistical robustness is of great practical concern. Here we present a technique to guarantee the independence of the first-order statistics of external perturbations, modeled as an additive, map-independent random variable. The developed criterion applies to a quite general class of maps.

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Constructing Multi-Branches Complete Chaotic Maps That Preserve Specified Invariant Density

Discrete Dynamics in Nature and Society ◽

10.1155/2009/378761 ◽

2009 ◽

Vol 2009 ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

Weihong Huang

Keyword(s):

Invariant Measure ◽

Chaotic Maps ◽

Invariant Density

A general formulation for multi-branches complete chaotic maps that preserve a specified invariant density is provided and an implication relationship among this class of maps is revealed. Such relationship helps to derive a whole family of complete chaotic maps that preserve not only the same invariant measure but also the same degree of chaos in terms of Lyapunov component.

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Lyapunov exponents and the natural invariant density determination of chaotic maps: an iterative maximum entropy ansatz

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8113/43/12/125103 ◽

2010 ◽

Vol 43 (12) ◽

pp. 125103 ◽

Cited By ~ 9

Author(s):

Parthapratim Biswas ◽

Hironori Shimoyama ◽

Lawrence R Mead

Keyword(s):

Maximum Entropy ◽

Lyapunov Exponents ◽

Chaotic Maps ◽

Invariant Density ◽

Density Determination

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SYNTHESIS OF PIECEWISE-LINEAR CHAOTIC MAPS: INVARIANT DENSITIES, AUTOCORRELATIONS, AND SWITCHING

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408021646 ◽

2008 ◽

Vol 18 (08) ◽

pp. 2169-2189 ◽

Cited By ~ 7

Author(s):

ALAN ROGERS ◽

ROBERT SHORTEN ◽

DANIEL M. HEFFERNAN ◽

DAVID NAUGHTON

Keyword(s):

Matrix Method ◽

Chaotic Maps ◽

Piecewise Linear ◽

Image Generation ◽

Two Dimensional ◽

Invariant Density ◽

Interesting Application ◽

The Matrix ◽

Simple Matrix ◽

Invariant Densities

In this paper, we give a review of the Inverse Frobenius–Perron problem (IFPP): how to create chaotic maps with desired invariant densities. After describing some existing methods for solving the IFPP, we present a new and simple matrix method of doing this. We show how the invariant density and the autocorrelation properties of the maps can be controlled independently. We also give some fundamental results on switching between a number of different chaotic maps and the effect this has on the overall invariant density of the system. The invariant density of the switched system can be controlled by varying the probabilities of choosing each individual map. Finally, we present an interesting application of the matrix method to image generation, by synthesizing a two-dimensional map, which when iterated, generates a well-known image.

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