Cluster oscillation and bifurcation of fractional-order Duffing system with two time scales
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RESEARCH PAPER
Cluster oscillation and bifurcation of fractional‑order Duffing system with two time scales Yanli Wang1 · Xianghong Li1,2 · Yongjun Shen2,3 Received: 29 November 2019 / Revised: 14 April 2020 / Accepted: 28 May 2020 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The dynamical behavior on fractional-order Duffing system with two time scales is investigated, and the point-cycle coupling type cluster oscillation is firstly observed herein. When taking the fractional order as bifurcation parameter, the dynamics of the autonomous Duffing system will become more complex than the corresponding integer-order one, and some typical phenomenon exist only in the fractional-order one. Different attractors exist in various parameter space, and Hopf bifurcation only happens while fractional order is bigger than 1 under certain parameter condition. Moreover, the bifurcation behavior of the autonomous system may regulate dynamical phenomenon of the periodic excited system. It results into the pointcycle coupling type cluster oscillation when the fractional order is bigger than 1. The related generation mechanism based on slow-fast analysis method is that the slow variation of periodic excitation makes the system periodically visit different attractors and critical points of different bifurcations of the autonomous system. Keywords Fractional-order Duffing system · Cluster oscillation · Slow-fast analysis method · Bifurcation
1 Introduction Fractional-order calculus is the extension of integer-order one, which has been put forward more than 300 years ago. Because it can explain the memory and nonlocality of the system in time and space, fractional-order calculus becomes one of the important tools for mathematical modeling of complex dynamics and physical processes. In the past three centuries, the study on fractional calculus mainly focused on the purely theoretical analysis. In recent decades, fractional calculus was widely used in various engineering fields such as physics [1], biology [2], environmental science [3] and economy [4], so it became the research hotspot of many fields [5–7]. At present, the research on fractional differential could be divided into two categories based on practice application. * Xianghong Li [email protected] 1
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
2
State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
3
Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
The first one is to introduce fractional order item into the integer order system [8–10]. Like the research on the analytical solution of integer-order system [11, 12], the analytical solution for fractional-order system is also interesting. For example, Shen et al. [9] obtained an approximate analytical solution for a linear singl
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