Analysis and synthesis of oscillator systems described by a perturbed double-well Duffing equation

• Published in: Nonlinear dynamics 2018-10, Vol.94 (1), p.57-85
• Main Authors: Georgiev, Zhivko D., Uzunov, Ivan M., Todorov, Todor G.
• Format: Article
• Language: English
• Subjects: Automotive Engineering; Classical Mechanics; Competitive advantage; Computer simulation; Control; Duffing equation; Dynamical Systems; Economic models; Elliptic functions; Engineering; Mechanical Engineering; Original Paper; Oscillators; Series expansion; Synthesis; Trajectories; Vibration
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-018-4345-4

This paper presents an investigation of limit cycles in oscillator systems described by a perturbed double-well Duffing equation. The analysis of limit cycles is made by the Melnikov theory. Expressing the solutions of the unperturbed Duffing equation by Jacobi elliptic functions allows us to calculate explicitly the Melnikov function, whereupon the final result is a function involving the complete elliptic integrals. The Melnikov function is analyzed with the aid of the Picard–Fuchs and Riccati equations. It has been proved that the considered oscillator system can have two small hyperbolic limit cycles located symmetrically with respect to the y -axis, or one large hyperbolic limit cycle, or two large hyperbolic limit cycles, or one large limit cycle of multiplicity 2. Moreover, we have obtained the conditions under which each of these limit cycles arises. The present work gives the conditions for the arising of limit cycles around the homoclinic trajectory. In this connection, an alternative approach is proposed for obtaining a series expansion of the Melnikov function near the homoclinic trajectory. This approach uses the series expansion of the complete elliptic integrals as the elliptic modulus tends to 1. It is shown that a jumping phenomenon may occur between limit cycles in the analyzed oscillator system. The conditions for the occurrence of this jumping phenomenon are given. A method for the synthesis of an oscillator system with a preliminary assigned limit cycle is also presented in the article. The obtained analytical results are illustrated and confirmed by numerical simulations.