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As a kind of dynamical system with a particular nonlinear structure, a multi-time scale nonlinear system is one of the essential directions of the current development of nonlinear dynamics theory. Multi-time scale nonlinear systems in practical applications are often complex forms of coupling of high-dimensional and high codimension characteristics, leading to various complex bursting oscillation behaviors and bifurcation characteristics in the system. For exploring the complex bursting dynamics caused by high codimension bifurcation, this paper considers the normal form of the vector field with triple zero bifurcation. Two kinds of codimension-2 bifurcation that may lead to complex bursting oscillations are discussed in the two-parameter plane. Based on the fast–slow analysis method, by introducing the slow variable W = A sin (ω t) , the evolution process of the motion trajectory of the system changing with W was investigated, and the dynamical mechanism of several types of bursting oscillations was revealed. Finally, by varying the frequency of the slow variable, a class of chaotic bursting phenomena caused by the period-doubling cascade is deduced. Developing related work has played a positive role in deeply understanding the nature of various complex bursting phenomena and strengthening the application of basic disciplines such as mechanics and mathematics in engineering practice. [ABSTRACT FROM AUTHOR]

Since the famous slow–fast dynamical system referred to as the Hodgkin–Huxley model was proposed to describe the threshold behaviors of neuronal axons, the study of various slow–fast dynamical behaviors and their generation mechanisms has remained a popular topic in modern nonlinear science. The primary purpose of this paper is to introduce a novel transition route induced by the comprehensive effect of special rest spike bistability and timescale difference rather than a common bifurcation via a modified Chua's circuit model with an external low-frequency excitation. In this paper, we attempt to explain the dynamical mechanism behind this novel transition route through quantitative calculations and qualitative analyses of the nonsmooth dynamics on the discontinuity boundary. Our work shows that the whole system responses may tend to be various and complicated when this transition route is triggered, exhibiting rich slow–fast dynamics behaviors even with a very slight change in excitation frequency, which is described well by using Poincaré maps in numerical simulations. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study point-cycle type bistability as well as induced periodic bursting oscillations by taking a modified Filippov-type Chua's circuit system with a low-frequency external excitation as an example. Two different kinds of bistable structures in the fast subsystem are obtained via conventional bifurcation analyses; meanwhile, nonconventional bifurcations are also employed to explain the nonsmooth structures in the bistability. In the following numerical investigations, dynamic evolutions of the full system are presented by regarding the excitation amplitude and frequency as analysis parameters. As a consequence, we can find that the classification method for periodic bursting oscillations in smooth systems is not completely applicable when nonconventional bifurcations such as the sliding bifurcations and persistence bifurcation are involved; in addition, it should be pointed out that the emergence of the bursting oscillation does not completely depend on bifurcations under the point-cycle bistable structure in this paper. It is predicted that there may be other unrevealed slow–fast transition mechanisms worthy of further study. [ABSTRACT FROM AUTHOR]