Amplitude Modulation and Coexisting Attractors in a Slow-Fast Duffing System
Han Guo
*
Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, Jiangsu-212013, P.R. China.
*Author to whom correspondence should be addressed.
Abstract
This paper aims to investigate the influence of amplitude modulation and coexisting attractors on the formation mechanism of complex bursting oscillations in slow–fast coupled nonlinear systems. Taking a modified three-dimensional van der Pol–Duffing system as the research object, a parametric excitation with frequency far less than the natural frequency is introduced into the system, establishing a slow–fast system model with two-scale coupling characteristics in the frequency domain. Regarding the excitation term as a slow-varying parameter, the equilibrium points, limit cycles, and their bifurcation structures in the fast subsystem are systematically analyzed, revealing the coexistence conditions of different attractors and their impact on the dynamics of the full system.
When the modulation amplitude is relatively small, the slow evolution process of the system becomes notably extended, showing typical quasi-static relaxation–oscillation characteristics. As the modulation amplitude increases, these oscillations vanish rapidly, and the system transitions to a steady equilibrium or an aperiodic oscillatory state. Further investigation reveals that the coexistence of multiple attractors in the fast subsystem can result in diverse bursting behaviors of the full system: trajectories may remain within a single attraction basin or sequentially traverse several basins, thereby generating merged bursting oscillations. Moreover, due to the inertia effect, trajectories may directly cross parameter regions associated with specific attractors, leading to the disappearance of corresponding bifurcation phenomena. When the excitation frequency decreases to a sufficiently low value, previously vanished attractors may re-emerge. If the interval between two types of codimension-1 bifurcation points is extremely narrow, the system may also display compound bursting patterns analogous to those caused by codimension-2 bifurcations.
The research in this paper reveals the synergistic influence mechanism of coexisting attractors in slow-fast systems on relaxation oscillations and bursting dynamics under amplitude modulation conditions, enriching the multiscale dynamical theory of nonlinear oscillators and providing new theoretical references for the study of modulation-induced dynamical evolution in other complex nonlinear systems.
Keywords: Amplitude modulation, slow-fast system, coexisting attractors, bursting oscillations, Duffing system, nonlinear dynamics