Dynamics of a general jerky equation

Abstract

Some classic nonlinear dynamical systems, such as Rössler's toroidal model, the Genesio model, and 19 Sprott's models, can be classified into seven distinct basic classes of jerky dynamics, labeled by $J{D}_1-J{D}_7$ . This paper is devoted to the dynamics of a general jerky equation which contains $J{D}_1-J{D}_7$ as parameters vary. It is shown that the system undergoes fold, Hopf, zero-Hopf, and Bogdanov–Takens bifurcations based on the center manifold theorem and normal form theory. Numerical simulations are also given to make the theoretical results visible and to detect more complicated dynamical behaviors, including degenerate Hopf bifurcation, fold bifurcation of cycle, and limit cycles. Especially, an apple-like attractive portrait is discovered near the zero-Hopf bifurcation point for the first time. Finally, according to the conclusions of the general jerky equation, exact conditions are summarized by two tables on how bifurcations will occur for $J{D}_1-J{D}_7$ , respectively.