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Keywords = Benettin method

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18 pages, 999 KiB  
Article
On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities
by Alexander N. Pchelintsev
Mathematics 2021, 9(17), 2057; https://doi.org/10.3390/math9172057 - 26 Aug 2021
Cited by 3 | Viewed by 1924
Abstract
This article discusses the search procedure for Poincaré recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system, using a previously developed high-precision numerical method. For the resulting limiting solution, the Lyapunov exponents are calculated, using the modified Benettin’s algorithm [...] Read more.
This article discusses the search procedure for Poincaré recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system, using a previously developed high-precision numerical method. For the resulting limiting solution, the Lyapunov exponents are calculated, using the modified Benettin’s algorithm to study the stability of the found regime and confirm the type of attractor. Full article
(This article belongs to the Special Issue Nonlinear Dynamics)
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Figure 1

Figure 1
<p>Projection of the cycle of the system (<a href="#FD1-mathematics-09-02057" class="html-disp-formula">1</a>) with the period <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>≈</mo> <mn>0.3655</mn> </mrow> </semantics></math> into the space <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Projection of the cycle of the system (<a href="#FD1-mathematics-09-02057" class="html-disp-formula">1</a>) with the period <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>≈</mo> <mn>0.3655</mn> </mrow> </semantics></math> into the space <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>4</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The graph of the change in time of the Lyapunov exponent <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The graph of the change in time of the Lyapunov exponent <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>The graph of the change in time of the Lyapunov exponent <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The graph of the change in time of the Lyapunov exponent <math display="inline"><semantics> <mrow> <mi>L</mi> <msub> <mi>E</mi> <mn>4</mn> </msub> </mrow> </semantics></math>.</p> Full article ">
28 pages, 14466 KiB  
Article
Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems
by Jan Awrejcewicz, Anton V. Krysko, Nikolay P. Erofeev, Vitalyj Dobriyan, Marina A. Barulina and Vadim A. Krysko
Entropy 2018, 20(3), 175; https://doi.org/10.3390/e20030175 - 6 Mar 2018
Cited by 45 | Viewed by 5558
Abstract
The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the [...] Read more.
The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems)
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Figure 1

Figure 1
<p>Synchronization of perturbed and nonperturbed systems in the case of a logistic map (<math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> points to the largest Lyapunov exponent value).</p> Full article ">Figure 2
<p>Transformation of a sphere of initial states into a counterpart ellipsoid during the system evolution.</p> Full article ">Figure 3
<p>Single-layer feed forward neural network, which consists of input neurons, a layer of hidden neurons and one output neuron.</p> Full article ">Figure 4
<p>Transition function.</p> Full article ">Figure 5
<p>Nonlinear <b>c</b>haracteristics of the oscillation signal: (<b>a</b>) Time histories; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter.</p> Full article ">Figure 6
<p>Characteristics of the Hénon map: (<b>a</b>) Time history; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter; (<b>g</b>) Lyapunov exponents plane (Hénon map).</p> Full article ">Figure 6 Cont.
<p>Characteristics of the Hénon map: (<b>a</b>) Time history; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter; (<b>g</b>) Lyapunov exponents plane (Hénon map).</p> Full article ">Figure 7
<p>Signal characteristics: (<b>a</b>) Time history; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter; (<b>g</b>) Lyapunov exponents plane (generalized Hénon map).</p> Full article ">Figure 7 Cont.
<p>Signal characteristics: (<b>a</b>) Time history; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter; (<b>g</b>) Lyapunov exponents plane (generalized Hénon map).</p> Full article ">Figure 8
<p>Signal characteristics: (<b>a</b>) Time history; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter; (<b>g</b>) Lyapunov exponents plane (Rössler attractor).</p> Full article ">Figure 8 Cont.
<p>Signal characteristics: (<b>a</b>) Time history; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter; (<b>g</b>) Lyapunov exponents plane (Rössler attractor).</p> Full article ">Figure 9
<p>Signal characteristics: (<b>a</b>) Time history; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter; (<b>g</b>) Lyapunov exponents plane (Lorenz attractor).</p> Full article ">Figure 9 Cont.
<p>Signal characteristics: (<b>a</b>) Time history; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter; (<b>g</b>) Lyapunov exponents plane (Lorenz attractor).</p> Full article ">Figure 9 Cont.
<p>Signal characteristics: (<b>a</b>) Time history; (<b>b</b>) Time window; (<b>c</b>) Chaotic attractor; (<b>d</b>) Fourier frequency spectrum; (<b>e</b>) Wavelet spectrum; (<b>f</b>) Dependence of LLE on the control parameter; (<b>g</b>) Lyapunov exponents plane (Lorenz attractor).</p> Full article ">
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