Synchronization and parameter estimations of an uncertain Rikitake system

Aguilar-Ibáñez, Carlos; Martínez-Guerra, Rafael; Aguilar-López, Ricardo; Mata-Machuca, Juan L.

doi:10.1016/j.physleta.2010.06.056

Cited by 14 publications

(5 citation statements)

References 10 publications

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“…The Rikitake system has also been considered to investigate synchronization. Aguilar-Ibañez et al [7] have used a master-slave scheme in conjunction with the adaptive control technique to address the synchronization and parameter estimation of an uncertain Rikitake system. Hamiltonian realizations of the Rikitake system were considered in Ref.…”

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confidence: 99%

A Parameter-Space Analysis of the Rikitake System

Silva¹,

Rech²

2013

Chinese Phys. Lett.

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We investigate analytically and numerically the dynamics of the Rikitake system. The Routh-Hurwitz criterion is used to study the stability of the equilibrium points of the differential equation system model, as functions of two parameters. The dynamics of the model are numerically studied using diagrams that associate colors to the largest Lyapunov exponent value, in two-dimensional parameter spaces. Additionally, phase-space plots and bifurcation diagrams are used to distinguish periodic and chaotic attractors.

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confidence: 99%

A Parameter-Space Analysis of the Rikitake System

Silva¹,

Rech²

2013

Chinese Phys. Lett.

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“…The problem of the unidirectional synchronization of chaotic systems consists of finding an appropriate control law such that when this is applied to a system with coupled inputs called "slave" or "response," such system follows the dynamics of an autonomous chaotic system called "master" or "drive" [1][2][3][4][5][6][7][8][9][10][11]. This proper control action is necessary because, without it, two identical autonomous chaotic systems could never be synchronized due to their high sensitivity to initial conditions [12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Exponential Synchronization of Chaotic Xian System Using Linear Feedback Control

et al. 2019

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In this paper, a new linear feedback controller for synchronization of two identical chaotic systems in a master-slave configuration is presented. This controller requires knowing a priori Lipschitz constant of the nonlinear function of the chaotic system on its attractor. The controller development is based on an algebraic Riccati equation. If the gain matrix and the matrices of Riccati equation are selected in such a way that a unique positive definite solution is obtained for this equation, then, with respect to previous works, a stronger result can be guaranteed here: the exponential convergence to zero of the synchronization error. Additionally, the nonideal case is also studied, that is, when unmodeled dynamics and/or disturbances are present in both master system and slave system. On this new condition, the synchronization error does not converge to zero anymore. However, it is still possible to guarantee the exponential convergence to a bounded zone. Numerical simulation confirms the satisfactory performance of the suggested approach.

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“…In the second case, a chaotic system with control inputs known as slave must follow the dynamics of an autonomous chaotic system known as master. Both systems should produce the same response in spite of the different initial conditions [39]. This problem can be posed as the stabilization of the difference between the states of slave system and master system (synchronization error).…”

Section: Introductionmentioning

confidence: 99%

Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control

Pérez-Cruz

2018

Complexity

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Standard adaptive control is the preferred approach for stabilization and synchronization of chaotic systems when the structure of such systems is a priori known but the parameters are unknown. However, in the presence of unmodeled dynamics and/or disturbance, this approach is not effective anymore due to the drift of the parameter estimations, which eventually causes the instability of the closed-loop system. In this paper, a robustifying term, which consists of a saturation function, is used to avoid this problem. The robustifying term is added to the adaptive control law. Consequently, the learning law is also modified. The boundedness of the states and the parameter estimations is rigorously and thoroughly proven by means of Lyapunov like analysis based on Barbalat’s lemma. On these new conditions, the convergence to zero cannot be achieved due to the presence of unmodeled dynamics and/or disturbance. However, it is still possible to guarantee the asymptotic convergence to a bounded zone around zero. The width of this zone can be adjusted by the designer. The performance of this robust approach is verified by numerical simulations. Although, for simplicity, this strategy is only applied to the stabilization and synchronization of Zhang system, the procedure can easily be generalized to a broad class of chaotic systems.

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Synchronization and parameter estimations of an uncertain Rikitake system

Cited by 14 publications

References 10 publications

A Parameter-Space Analysis of the Rikitake System

A Parameter-Space Analysis of the Rikitake System

Exponential Synchronization of Chaotic Xian System Using Linear Feedback Control

Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control

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