<p>In this study, a modified Chua’s circuit driven by a dual-frequency excitation is employed as a representative system to investigate Filippov-type slow-fast dynamics. The two excitation terms function as slow variables. When the two excitation frequencies have a 3:2 ratio, novel bursting oscillations are observed. Using mathematical methods, the bifurcation analysis of the fast subsystem relative to the two slow variables is converted into an analysis relative to one slow variable and the excitation amplitude. In this bifurcation analysis, two-parameter bifurcation sets of Hopf and fold bifurcations are obtained. The necessary conditions for a codimension-2 bifurcation, namely branch point bifurcation, are also obtained. The research identifies crossing-sliding and grazing-sliding bifurcations in the slow-fast coupled system through numerical simulations. Drawing upon the bifurcation results, the generation mechanism of seven representative bursting oscillations is explained. It is demonstrated that, in a Filippov system, a hysteresis loop generated by the slow passage effect through a subcritical Hopf bifurcation serves to trigger bursting oscillations. The adding-sliding bifurcation can lead to adding-sliding motion and sliding motion before or after the bifurcation, causing unsynchronized oscillation in different state variable directions. Moreover, a branch point in the fast subsystem can cause two unstable equilibrium lines to break after colliding. The collision point then splits into two fold bifurcation points, altering the trajectory’s jumping behavior in the slow-fast coupled system.</p>

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Bursting oscillations in a modified Chua’s circuit of Filippov type with double-frequency excitation

  • Chun Zhang,
  • Zhixiang Wang,
  • Qinsheng Bi

摘要

In this study, a modified Chua’s circuit driven by a dual-frequency excitation is employed as a representative system to investigate Filippov-type slow-fast dynamics. The two excitation terms function as slow variables. When the two excitation frequencies have a 3:2 ratio, novel bursting oscillations are observed. Using mathematical methods, the bifurcation analysis of the fast subsystem relative to the two slow variables is converted into an analysis relative to one slow variable and the excitation amplitude. In this bifurcation analysis, two-parameter bifurcation sets of Hopf and fold bifurcations are obtained. The necessary conditions for a codimension-2 bifurcation, namely branch point bifurcation, are also obtained. The research identifies crossing-sliding and grazing-sliding bifurcations in the slow-fast coupled system through numerical simulations. Drawing upon the bifurcation results, the generation mechanism of seven representative bursting oscillations is explained. It is demonstrated that, in a Filippov system, a hysteresis loop generated by the slow passage effect through a subcritical Hopf bifurcation serves to trigger bursting oscillations. The adding-sliding bifurcation can lead to adding-sliding motion and sliding motion before or after the bifurcation, causing unsynchronized oscillation in different state variable directions. Moreover, a branch point in the fast subsystem can cause two unstable equilibrium lines to break after colliding. The collision point then splits into two fold bifurcation points, altering the trajectory’s jumping behavior in the slow-fast coupled system.