PPubMed30 Mar 2018Cited 18×

Controlling and synchronizing a fractional-order chaotic system using stability theory of a time-varying fractional-order system Controlling and synchronizing a fractional-order chaotic system with time-varying

Yu Huang, Dongfeng Wang, Jinying Zhang, Feng Guo, Jun Ma

Abstract

Control and synchronization of fractional-order chaotic systems have attracted wide attention due to their numerous potential applications. To get suitable control method and parameters for fractional-order chaotic systems, the stability analysis of time-varying fractional-order systems should be discussed in the first place. Therefore, this paper analyzes the stability of the time-varying fractional-order systems and presents a stability theorem for the system with the order 0<α<1. This theorem is a sufficient condition which can discriminate the stability of time-varying systems conveniently. Feedback controllers are designed for control and synchronization of the fractional-order Lü chaotic system. The simulation results demonstrate the effectiveness of the proposed theorem.

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A new sufficient stability theorem for time‑varying fractional‑order systems enables effective control and synchronization of the fractional‑order Lü chaotic system.

What’s new

i.Proposed a sufficient stability theorem for time‑varying fractional‑order systems with order 0<α<1
ii.Designed a feedback controller for tracking control of the fractional‑order Lü chaotic system
iii.Designed a feedback controller for synchronization of the fractional‑order Lü chaotic system
iv.Validated the theorem and controllers through numerical simulations

How they did it

The authors derived a stability theorem based on eigenvalue analysis of a symmetric matrix H(t) for time‑varying fractional‑order systems. Using this theorem they formulated linear feedback controllers for the Lü chaotic system’s tracking and synchronization. Numerical simulations were then performed to verify stability and convergence.

What they found

Example 1: eigenvalues λ₁=‑1, λ₂=‑2 → system stable
Example 2 (a=0.25, b=0.5): eigenvalues ‑0.5 and ‑1.5 → stable; (a=1, b=0.5): eigenvalues 1 and ‑3 → theorem inconclusive
Control parameter k < ‑29.7 (e.g., k=‑35) guarantees uniform asymptotic stability of the controlled Lü system
Simulation shows state trajectories converge to the origin after control is applied at t=10 s

Caveats

✕Theorem provides only a sufficient (not necessary) condition for stability
✕Stability analysis demonstrated on only one chaotic system (Lü system)
✕Controller design relies on proper selection of the gain k, which may be system‑specific
✕Results are based solely on numerical simulations without experimental validation

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Controlling and synchronizing a fractional-order chaotic system using stability theory of a time-varying fractional-order system Controlling and synchronizing a fractional-order chaotic system with time-varying

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