[Paper Review] Synchronization of a class of master-slave non-autonomous chaotic systems with parameter mismatch via sinusoidal feedback control

Jianping Cai, Mihua Ma+1 more|ArXiv.org|Aug 22, 2009

Chaos control and synchronization37 references3 citations

TL;DR

This paper proposes a sinusoidal state error feedback control scheme to achieve synchronization in n-dimensional non-autonomous chaotic master-slave systems with parameter mismatch in external harmonic excitations. By deriving algebraic synchronization criteria via the Lyapunov direct method and Gerschgorin disc theorem, it analytically estimates synchronization error bounds, showing that increasing coupling strength or reducing mismatch minimizes error. The method is validated on a 3D gyrostat system, demonstrating robustness against 12% amplitude or phase mismatch with small synchronization errors.

ABSTRACT

In this paper we investigate a master-slave synchronization scheme of two n-dimensional non-autonomous chaotic systems coupled by sinusoidal state error feedback control, where parameter mismatch exists between the external harmonic excitation of master system and that of slave one. A concept of synchronization with error bound is introduced due to parameter mismatch, and then the bounds of synchronization error are estimated analytically. Some synchronization criteria are firstly obtained in the form of matrix inequalities by the Lyapunov direct method, and then simplified into some algebraic inequalities by the Gerschgorin disc theorem. The relationship between the estimated synchronization error bound and system parameters reveals that the synchronization error can be controlled as small as possible by increasing the coupling strength or decreasing the magnitude of mismatch. A three-dimensional gyrostat system is chosen as an example to verify the effectiveness of these criteria, and the estimated synchronization error bounds are compared with the numerical error bounds. Both the theoretical and numerical results show that the present sinusoidal state error feedback control is effective for the synchronization. Numerical examples verify that the present control is robust against amplitude or phase mismatch.

Motivation & Objective

• To address the challenge of synchronizing non-autonomous chaotic systems with parameter mismatch in external harmonic excitations.
• To develop a synchronization scheme that remains effective despite amplitude, frequency, or phase mismatches.
• To derive analytically estimable synchronization error bounds using Lyapunov methods and matrix inequality techniques.
• To simplify complex matrix inequalities into practical algebraic criteria for easier controller design and parameter tuning.
• To validate the effectiveness and robustness of the proposed control method through numerical simulations on a 3D gyrostat system.

Proposed method

• Introduces a master-slave synchronization framework using sinusoidal state error feedback control for n-dimensional non-autonomous chaotic systems.
• Applies the Lyapunov direct method to derive matrix inequality-based synchronization criteria under parameter mismatch.
• Utilizes the Gerschgorin disc theorem to transform matrix inequalities into simpler algebraic inequalities for analytical tractability.
• Estimates synchronization error bounds analytically by analyzing the relationship between system parameters and coupling strength.
• Derives explicit algebraic criteria that link coupling coefficients, mismatch magnitudes, and the upper bound of synchronization error.
• Validates the theoretical results through numerical simulations on a 3D gyrostat system, comparing estimated and numerical error bounds.

Experimental results

Research questions

• RQ1 Can sinusoidal state error feedback control achieve synchronization in non-autonomous chaotic systems with parameter mismatch in external harmonic excitations?
• RQ2 How do coupling strength and mismatch magnitude affect the synchronization error bound in such systems?
• RQ3 Can the derived synchronization criteria be simplified into algebraic inequalities that reveal clear relationships with system parameters?
• RQ4 How robust is the proposed control scheme against amplitude or phase mismatches in the external excitation?
• RQ5 To what extent do the analytically estimated error bounds match the actual numerical synchronization errors in practical systems?

Key findings

• The synchronization error bound can be made arbitrarily small by increasing the coupling strength or reducing the magnitude of parameter mismatch.
• For a 12% amplitude mismatch, the synchronization error is approximately 2.4%, demonstrating robustness.
• For a 12% phase mismatch, the synchronization error is approximately 2.7%, confirming robustness against phase differences.
• The estimated error bound H is inversely proportional to the coupling coefficient k, with H ≈ 0.5428/k - 7.245 for a phase mismatch of 0.1 radians.
• The derived algebraic criteria (e.g., H < 5.428/|Δφ| + 7.244/k) provide a practical design guide for selecting coupling coefficients.
• Numerical results confirm that the estimated error bounds closely match the actual numerical synchronization errors, validating the theoretical analysis.