Shadowing in hidden attractors

Kamal, Neeraj Kumar; Varshney, Vaibhav; Shrimali, Manish Dev; Prasad, Awadhesh; Kuznetsov, Nikolay V.; Леонов, Г. А.

doi:10.1007/s11071-017-4022-z

Cited by 7 publications

(3 citation statements)

References 26 publications

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“…We deal with a hidden attractor when the mathematical model of a given system does not have a constant dynamic equilibrium point (stationary point). Previous literature reports on this subject concern both dynamic continuous systems (e.g., [ 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 ]), and discrete multidimensional dynamic systems (for example: [ 26 , 27 , 28 , 29 , 30 ]). Research on this issue is crucial because it can protect the system from dangerous, chaotic oscillations.…”

Section: Introductionmentioning

confidence: 99%

Hidden Attractors in Discrete Dynamical Systems

Berezowski

Lawnik

2021

Entropy

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Research using chaos theory allows for a better understanding of many phenomena modeled by means of dynamical systems. The appearance of chaos in a given process can lead to very negative effects, e.g., in the construction of bridges or in systems based on chemical reactors. This problem is important, especially when in a given dynamic process there are so-called hidden attractors. In the scientific literature, we can find many works that deal with this issue from both the theoretical and practical points of view. The vast majority of these works concern multidimensional continuous systems. Our work shows these attractors in discrete systems. They can occur in Newton’s recursion and in numerical integration.

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Section: Introductionmentioning

confidence: 99%

Hidden Attractors in Discrete Dynamical Systems

Berezowski

Lawnik

2021

Entropy

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“…Therefore, some traditional chaotic systems with a self-excited attractor are widely used in secret communication and have significant advantages [ 1 , 2 , 3 , 4 , 5 ]. Further, in recent years a hidden chaos attractor has been found, which makes the development of a high-dimensional nonlinear system an attractive challenge [ 6 , 7 , 8 , 9 ]. At present, most scholars primarily study the dynamic characteristics of hidden attractors.…”

Section: Introductionmentioning

confidence: 99%

A Novel Algorithm to Improve Digital Chaotic Sequence Complexity through CCEMD and PE

Fan

Xie

Ding

2018

Entropy

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In this paper, a three-dimensional chaotic system with a hidden attractor is introduced. The complex dynamic behaviors of the system are analyzed with a Poincaré cross section, and the equilibria and initial value sensitivity are analyzed by the method of numerical simulation. Further, we designed a new algorithm based on complementary ensemble empirical mode decomposition (CEEMD) and permutation entropy (PE) that can effectively enhance digital chaotic sequence complexity. In addition, an image encryption experiment was performed with post-processing of the chaotic binary sequences by the new algorithm. The experimental results show good performance of the chaotic binary sequence.Empirical mode decomposition (EMD) in digital signal processing has been extensively applied in nonlinear signal processing [15][16][17][18]. EMD was first proposed by Huang et al. [19][20][21]. It is an effective tool for analyzing nonlinear and non-stationary signals. The EMD method is closely related to the corresponding Hilbert transform method. Through the decomposition of nonlinear and non-stationary signals, a series of intrinsic mode functions (IMFs) are obtained, which makes each IMF a stable signal for narrowband [22]. The IMFs play a crucial role in the analysis of non-stationary or nonlinear signals. However, there are some problems with the EMD method, of which the main one is mode mixing. Complementary ensemble empirical mode decomposition (CEEMD) can effectively restrain the mode mixing of EMD at a certain level [23][24][25]. Based on the above considerations, we proposed a new algorithm which combines CEEMD with permutation entropy (PE) [26] to effectively improve the complexity of the digital chaotic sequence.The rest of this paper is organized as follows: Section 2 describes a hidden chaos attractor with no equilibria. The dynamic characteristics of a complex chaotic system are studied by means of numerical simulation and theoretical analysis. Section 3 proposes a new algorithm to improve the complexity of the digital chaotic sequence. Section 4 considers image encryption with post-processing of the chaotic binary sequences by the algorithm outlined in Section 3. The security of the encrypted image is analyzed through key sensitivity, information entropy, and histogram analysis. Section 5 summarizes the discussions of this paper. The Characteristic Analysis of a Chaotic SystemIn this section, a system can be expressed as the following set of differential equations:where a, b, c are real parameters. When a = 2, b = 0.35, c = 1 and the initial value is (−1.6, 0.82, 1.9), the system displays a single-scroll chaotic system [27]. Different projections of the chaotic attractor for this system are shown in Figure 1.

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“…Limit cycles of the Lorenz system driven by the Wiener process have been analyzed by P-bifurcation and D-bifurcation in [11], and the e®ect of time delay on the stochastic bifurcation in Van der Pol oscillator perturbed by white noise have been investigated in [12]. The e®ect of noise on the dynamical systems which have hidden attractors has been investigated in [13], and the analytical and numerical methods for the problem of localization of hidden attractors in Chua's circuit have been developed in [14,15].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Bifurcation in Generalized Chua’s Circuit Driven by Skew-Normal Distributed Noise

2018

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In this study, the stochastic phenomenological bifurcations (P-bifurcations) of generalized Chua’s circuit (GCC) driven by skew-normal distributed noise have been investigated by numerically obtaining the stationary distributions of the stochastic responses. The noise intensity and/or skewness parameters of skew-normal distributed noise have been chosen as the bifurcation parameters to change the structure of the stochastic attractor. While the number of breakpoints in the piecewise-linear characteristics of the GCC are fixed, it has been observed that the number of scrolls have been changed by tuning the noise intensity and the skewness parameter of the skew-normal distributed noise.

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Shadowing in hidden attractors

Cited by 7 publications

References 26 publications

Hidden Attractors in Discrete Dynamical Systems

Hidden Attractors in Discrete Dynamical Systems

A Novel Algorithm to Improve Digital Chaotic Sequence Complexity through CCEMD and PE

Stochastic Bifurcation in Generalized Chua’s Circuit Driven by Skew-Normal Distributed Noise

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