Abstract
A hybrid stability checking method is proposed to verify the establishment of synchronization between two hyperchaotic systems. During the design stage of a synchronization scheme for chaotic fractional-order systems, a problem is sometimes encountered. In order to ensure the stability of the error signal between two fractional-order systems, the arguments of all eigenvalues of the Jacobian matrix of the erroneous system should be within a region defined in Matignon’s theorem. Sometimes, the arguments depend on the state variables of the driving system, which makes it difficult to prove the stability. We propose a new and efficient hybrid method to verify the stability in this situation. The passivity-based control scheme for synchronization of two hyperchaotic fractional-order Chen-Lee systems is provided as an example. Theoretical analysis of the proposed method is validated by numerical simulation in time domain and examined in frequency domain via electronic circuits.
Highlights
Nonlinear systems may exhibit dynamical chaotic behavior
Time histories of the system states are first numerically evaluated for a sufficient long time, so that they can reflect the dynamical behavior of the chaotic fractional-order system
The Runge-Kutta method of order 4 was used to solve the differential equations in the systems (7) and (8), while the improved method [11] based on Caputo derivative was implemented to approximate the fractional differential equations in (15) and (16) with time step size Δt = 0.001 s
Summary
Nonlinear systems may exhibit dynamical chaotic behavior. The study of chaos synchronization has received increasing attention due to its predicted potentials in technological applications in recent years. Complete synchronization [20], synchronization and antisynchronization [21], controlling chaos with multiple time-delays [22], electronic circuit implementation [23, 24], fractional-order behavior [19, 25], and so forth have been studied for this system recently. Chaos synchronization in the unified chaotic system, the Rikitake attractor, and the hyperchaotic complex Chen system with unknown parameters has been achieved using the passive control technique [30,31,32]. It is not always that the stability can be determined by upper bounds of system variables This happens to the case when we apply passivity theory to the hyperchaotic Chen-Lee system. The synchronization between two hyperchaotic Chen-Lee systems with different initial conditions was established via the passive control technique first. The proposed controller was applied to fractional-order hyperchaotic Chen-Lee systems. Numerical simulation and the corresponding electronic circuits were included to show the feasibility and effectiveness of the proposed methods
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