Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems

Matouk, A. E.; Abdelhameed, T. N.; Almutairi, Dalal Khalid; Abdelkawy, Mohamed; Herzallah, Mohamed A. E.

doi:10.3390/math11030591

Cited by 9 publications

(3 citation statements)

References 31 publications

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“…Therefore, hidden attractors can be detected in some continuous chaotic or hyperchaotic systems with no equilibrium point or only with a stable equilibrium point. Researchers have widely researched hidden attractors and obtained many meaningful results [11][12][13][14][15][16]. In the mentioned results, the existence condition of the hidden attractor, the coexistence and transition of various hidden attractors, and localization of hidden attractors have been investigated, which enriched the research results of nonlinear dynamics.…”

Section: Introductionmentioning

confidence: 98%

Hidden Dynamics of a New Jerk-like System with a Smooth Memristor and Applications in Image Encryption

Wang,

Zhuang,

et al. 2023

Mathematics

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Considering the dynamic characteristics of memristors, a new Jerk-like system without an equilibrium point is addressed based on a Jerk-like system, and the hidden dynamics are investigated. When changing system parameter b and fixing other parameters, the proposed system shows various hidden attractors, such as a hidden chaotic attractor (b = 5), a hidden period-1 attractor (b = 3.2), and a hidden period-2 attractor (b = 4). Furthermore, bifurcation analysis suggests that not only parameter b, but also the initial conditions of the system, have an effect on the hidden dynamics of the discussed system. The coexistence of various hidden attractors is explored and different coexistences of hidden attractors can be found for suitable system parameters. Offset boosting of different hidden attractors is discussed. It is observed that offset boosting can occur for hidden chaotic attractor, period-1 attractor, and period-2 attractor, but not for period-3 attractor and period-4 attractor. The antimonotonicity of the proposed system is debated and a full Feigenbaum remerging tree can be detected when system parameters a or b change within a certain range. On account of the complicated dynamics of the proposed system, an image encryption scheme is designed, and its encryption effectiveness is analyzed via simulation and comparison.

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Section: Introductionmentioning

confidence: 98%

Hidden Dynamics of a New Jerk-like System with a Smooth Memristor and Applications in Image Encryption

Wang,

Zhuang,

et al. 2023

Mathematics

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show abstract

“…Several papers have been published on modeling chaotic and hyperchaotic systems, with widespread application across diverse fields, including electrical circuits, mathematics, and physics [1][2][3][4][5][6]. It has emerged as a prominent trend in recent years.…”

Section: Introductionmentioning

confidence: 99%

Cap like trajectories in 5D chaotic tangent hyperbolic memristive model: fractional calculus and encryption

Qureshi,

Alam Khan,

Raza

et al. 2024

Phys. Scr.

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This research aims to investigate the influence of model parameters and fractional order on a novel mathematical model with tangent hyperbolic memristor. This investigation conducted by applying Lyapunov exponents and bifurcation analysis. We utilize the Lyapunov exponent theory to understand and characterize these chaotic behaviors under fractional indices. The Lyapunov exponent, bifurcation, and phase diagrams have been depicted to explore the intricate dynamics of the chaotic system governed by the chaotic equation. This investigation reveals that the system exhibits a wealth of complex behaviors influenced by the system parameters and the order of the fractional derivative. A novel approach termed Atangana-Baleanu-Caputo (ABC) fractional derivative (FD) to generate phase portraits and gain insights into the system's behavior. The random numbers generated by the chaotic system are employed to distort the image through an amalgamated image encryption (AIE) technique. Subsequently, the integrity of the scrambled image has been assessed using various image security evaluation methods to reinforce the notion that combining the chaotic system and image can constitute a valuable encryption key. Finally, the chaotic model circuit realization uses active and passive components, and the outcomes are compared with the numerical simulations.

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“…The use of symmetrical chaotic models is acceptable because it is difficult to forecast a wide range of real-world occurrences. Numerous novel methods for assessing chaotic systems have appeared in recent years [12][13][14][15]. Two of these methods, asymptotic stability and Lyapunov exponents, shed light on how the parameters of the model affect the dynamics of the chaotic model.…”

Section: Introductionmentioning

confidence: 99%

A Comparative Numerical Study of the Symmetry Chaotic Jerk System with a Hyperbolic Sine Function via Two Different Methods

Alzahrani,

Abdoon,

Elbadri

et al. 2023

Symmetry

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This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have been compared to Runge–Kutta’s fourth order (RK4). It was discovered that the suggested techniques gave results comparable to the RK4 method. Our primary goal is to develop effective methods for addressing symmetrical, chaotic systems. Using ABC-FD and NILM presents innovative approaches for comprehending and effectively handling intricate dynamics. The findings of this study have significant significance for addressing the occurrence of chaotic behavior in diverse scientific and engineering contexts. This research significantly contributes to fractional calculus and its various applications. The application of ABC-FD, which can identify chaotic behavior, makes our work stand out. This novel approach contributes to advancing research in nonlinear dynamics and fractional calculus. The present study not only offers a resolution to the problem of symmetric chaotic jerk systems but also presents a framework that may be applied to tackle analogous challenges in several domains. The techniques outlined in this paper facilitate the development and computational analysis of prospective fractional models, thereby contributing to the progress of scientific and engineering disciplines.

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Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems

Cited by 9 publications

References 31 publications

Hidden Dynamics of a New Jerk-like System with a Smooth Memristor and Applications in Image Encryption

Hidden Dynamics of a New Jerk-like System with a Smooth Memristor and Applications in Image Encryption

Cap like trajectories in 5D chaotic tangent hyperbolic memristive model: fractional calculus and encryption

A Comparative Numerical Study of the Symmetry Chaotic Jerk System with a Hyperbolic Sine Function via Two Different Methods

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