A simple multi-stable chaotic jerk system with two saddle-foci equilibrium points: analysis, synchronization via backstepping technique and MultiSim circuit design

Sambas, Aceng; Vaidyanathan, Sundarapandian; Moroz, Irene M.; Idowu, Babatunde A.; Mohamed, Mohamad Afendee; Mamat, Mustafa; Sanjaya, W. S. Mada

doi:10.11591/ijece.v11i4.pp2941-2952

Cited by 6 publications

(5 citation statements)

References 25 publications

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“…where D q is the fractional-order derivative by Caputo's definition, x i , i � 1, 2, 3 denotes the system state variables, σ, β are constants of the Chua circuit, and ϕ(x 1 ) is a nonlinear function defined in equation (24), with constant values a, b.…”

Section: Application Example 2: Chua Systemmentioning

confidence: 99%

“…On the other hand, the use of fractional calculus has been extensively studied in nonlinear systems (see, e.g., [14][15][16][17][18]) and also, there exist notable contributions related to the study of synchronization in fractional-order systems (see, e.g., [16,[19][20][21][22][23]). For example, there is work based on applying sliding modes to fractional-order models to achieve synchronization [24][25][26][27][28][29]. e modeling and analytical study of fractional-order systems is also a fruitful field, e.g., the use of the Razumikhin approximation for fractional-order systems with delay [30,31], the extrapolation of Lyapunov theory to fractional systems [32,33], and the existence and uniqueness of equilibrium points of the Mittag-Leffler criteria [34,35].…”

Section: Introductionmentioning

confidence: 99%

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Synchronization in Dynamically Coupled Fractional‐Order Chaotic Systems: Studying the Effects of Fractional Derivatives

et al. 2021

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This study presents the effectiveness of dynamic coupling as a synchronization strategy for fractional chaotic systems. Using an auxiliary system as a link between the oscillators, we investigate the onset of synchronization in the coupled systems and we analytically determine the regions where both systems achieve complete synchronization. In the analysis, the integration order is considered as a key parameter affecting the onset of full synchronization, considering the stability conditions for fractional systems. The local stability of the synchronous solution is studied using the linearized error dynamics. Moreover, some statistical metrics such as the average synchronization error and Pearson’s correlation are used to numerically identify the synchronous behavior. Two particular examples are considered, namely, the fractional-order Rössler and Chua systems. By using bifurcation diagrams, it is also shown that the integration order has a strong influence not only on the onset of full synchronization but also on the individual dynamic behavior of the uncoupled systems.

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Section: Application Example 2: Chua Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Synchronization in Dynamically Coupled Fractional‐Order Chaotic Systems: Studying the Effects of Fractional Derivatives

et al. 2021

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“…Ju H.Park, in his work [22] showcased a very effective back-stepping procedure to synchronize two same ordered chaotic systems so called in Triangular form. Synchronization of two same order chaotic systems belonging to the triangular feedback class has already been demonstrated [29][30][31]. Shihua Chen demonstrated the use of the back-stepping technique to construct a very successful generalized procedure for designing a scalar controller to achieve synchronization between two same ordered general class of chaotic systems in General Strict Feedback form in his paper [32].…”

Section: Introductionmentioning

confidence: 99%

Reduced Order Synchronization of Two Non-Identical Special Class of Strict Feedback Systems Via Back-Stepping Control Technique

Bora,

Sharma

2023

Journal of Computational and Nonlinear Dynamics

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This work offers a systematic technique to achieve reduced order synchronization (ROS) between two different order general classes of chaotic systems in a master-slave configuration. In this study, the dynamics of the master and slave systems are assumed to follow a special class of strict-feedback form, namely, the generalized triangular feedback form. The main objective is to design a suitable scalar controller using a Lyapunov theory-based back-stepping approach such that the mth order slave system gets synchronized with the nth order master system. Due to the difference in the order of the systems (m<n and m=(n−1)), it is only possible to achieve the synchronization between m numbers of states of the slave systems with (n−1) numbers of states of the master system, respectively. We cannot conclude on the stability of the nth state of the master system as there is no counterpart (state) available in the slave system to be synchronized with. Adding an additional (m+1)th state dynamics along with a nonlinear feedback controller (U1) to the slave system ensures that the nth state of the master system is synchronized with the (m+1)th state dynamics of the slave system. With the suggested technique proposed in this article, complete state-to-state synchronization can be achieved with only two controllers. The analytical results are successfully validated through numerical simulations presented in the end.

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“…Many of the jerk systems reported in the chaos literature involve jerk systems without any equilibrium point [20] or with unstable equilibrium points [17,18,[21][22][23][24], etc. In 2021, Vijayakumar et al [25] proposed a new chaotic jerk system with a stable equilibrium.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis, Synchronization and FPGA Implementation of a New 3-D Jerk System with a Stable Equilibrium

et al. 2023

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This research paper addresses the modelling of a new 3-D chaotic jerk system with a stable equilibrium. Such chaotic systems are known to exhibit hidden attractors. After the modelling of the new jerk system, a detailed bifurcation analysis has been performed for the new chaotic jerk system with a stable equilibrium. It is shown that the new jerk system has multistability with coexisting attractors. Next, we apply backstepping control for the synchronization design of a pair of new jerk systems with a stable equilibrium taken as the master-slave chaotic systems. Lyapunov stability theory is used to establish the synchronization results for the new jerk system with a stable equilibrium. Finally, we show that the FPGA design of the new jerk system with a stable equilibrium can be implemented using the FPGA Zybo Z7-20 development board. The design of the new jerk system consists of multipliers, adders and subtractors. It is observed that the experimental attractors are in good agreement with simulation results.

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A simple multi-stable chaotic jerk system with two saddle-foci equilibrium points: analysis, synchronization via backstepping technique and MultiSim circuit design

Cited by 6 publications

References 25 publications

Synchronization in Dynamically Coupled Fractional‐Order Chaotic Systems: Studying the Effects of Fractional Derivatives

Synchronization in Dynamically Coupled Fractional‐Order Chaotic Systems: Studying the Effects of Fractional Derivatives

Reduced Order Synchronization of Two Non-Identical Special Class of Strict Feedback Systems Via Back-Stepping Control Technique

Bifurcation Analysis, Synchronization and FPGA Implementation of a New 3-D Jerk System with a Stable Equilibrium

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