Cumulant Analysis of Strange Attractors: Theory and Applications

Kontorovich, Valeri; Lovtchikova, Z.

doi:10.1007/978-3-642-04227-0_4

Cited by 17 publications

(38 citation statements)

References 24 publications

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“…where the local characteristics, here denoted as K 2 (x) and K 1 (x) for Eq. (6), are K 1 (x)=f (x), K 2 (x)=K, and ξ(t) is AWGN with unitary intensity [5,20,23]. 1 If the input signal for filtering is:…”

Section: Stochastic Differential Equation Of the First Order (Sde-1)mentioning

confidence: 99%

Ubiquitous Filtering for Nonlinear Problems

Kontorovich¹,

Ramos-Alarcon²

2020

Research Advances in Chaos Theory

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This chapter develops and extends the general theoretical results, previously published in the chapter "Nonlinear filtering of weak chaotic signals", and presents detailed implementations of a computationally simple, robust (filtering fidelity almost insensitive to changes of the desired input signal properties) and rather precise approach for the filtering of weak signals of different physical nature (biological, seismic, voice, etc.) in presence of white Gaussian noise. The implementations rely on non-linear filtering techniques that in general can be considered as either one-moment or multi-moment, in the sense that they operate with a single sample (instantaneous fashion) or with various adjacent samples (non-instantaneous fashion). Chaotic modeling of the real input signals allows achieving an almost ubiquitous filtering approach with a computationally simple implementation. Application of the linearization strategies (for both one and twomoment filtering) provide, additionally, "invariance" of the processing algorithms to variations on the nature and statistics of the input signals.

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Section: Stochastic Differential Equation Of the First Order (Sde-1)mentioning

confidence: 99%

Ubiquitous Filtering for Nonlinear Problems

Kontorovich¹,

Ramos-Alarcon²

2020

Research Advances in Chaos Theory

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show abstract

“…) for solution of SDE (2), one can operate with the linear Kolmogorov-Fokker-Plank (FPK) operator and its adjoin inverse Kolmogorov operator L + (x) [3]:…”

Section: Degenerated Cumulant Equations For Two-moment Cumulantsmentioning

confidence: 99%

“…[6]. Recently the authors proposed an effective approximate approach for solving statistical problems of chaos based on the cumulant calculus and named it "degenerated cumulant equations" [1][2][3][4], which can be regarded as a generalization of the approach founded and developed by A. N. Malakhov [7,8] for the case of chaotic signals.…”

Section: Introductionmentioning

confidence: 99%

“…It was shown (see [1,3] and the references therein) that the application of the "cumulant concept" to chaotic signals provides a rather simple approximate method for the calculus of many statistical parameters that are relevant for applications (see [1][2][3][4] for example).…”

Section: Introductionmentioning

confidence: 99%

“…are not well known tools in the engineering statistical analysis. Therefore, to fill this gap, a thorough explanation of all those issues was introduced in [3,7,8], which can help the interested reader in the process of getting familiar with the topic.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Correlation Properties of Chaos: Cumulant Approach

Kontorovich

Lovtchikova²,

Ramos-Alarcon

2010

MCA

Self Cite

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The current trend in the statistical analysis of chaos shows certain gaps particularly regarding the engineering applications. This paper, which is a sequel of previous publications from the authors [1-5], develops an application of the cumulant approach to the analysis of the covariance properties of chaotic signals. A general approach for the analysis of two-moment cumulants is considered, particular emphasis is made in the covariance function and the third order cumulant behavior. The cumulant functions of the Lorenz and Chua strange attractors are considered as examples.

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Nonlinear Filtering of Chaos for Real Time Applications

Kontorovich

Lovtchikova

2013

Selected Topics in Nonlinear Dynamics and Theoretical Electrical Engineering

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Cumulant Analysis of Strange Attractors: Theory and Applications

Cited by 17 publications

References 24 publications

Ubiquitous Filtering for Nonlinear Problems

Ubiquitous Filtering for Nonlinear Problems

Correlation Properties of Chaos: Cumulant Approach

Nonlinear Filtering of Chaos for Real Time Applications

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