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The coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the Shimizu-Morioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and supercritical Hopf bifurcation are observed in the system under a certain parameter condition. When the exciting frequency is much smaller than the natural frequency, we can regard the periodic excitation as a slow-varying parameter. Second, complicated dynamic behaviors are analyzed when the slow-varying parameter passes through different bifurcation points, of which the mechanisms of four different bursting patterns, namely, symmetric “homoclinic/homoclinic” bursting oscillation, symmetric “fold/Hopf” bursting oscillation, symmetric “fold/fold” bursting oscillation, and symmetric “Hopf/Hopf” bursting oscillation via “fold/fold” hysteresis loop, are revealed with different values of the parameter

Multiple time scales problems can be observed in many real systems, such as catalytic reactions in chemical systems [

For a typical system with different coupled time scales, its dynamic behaviors can be described by a singularly perturbed system with two time scales of the following form [

In the present work, we consider the Shimizu-Morioka system [

The rest of this paper is organized as follows. In Section

In order to analyze the influence of the slowly varying excitation, we can regard the external excitation

The equilibrium point of (

Here we fix the parameter

Two bifurcation sets of the system with respect to parameters

From the two bifurcation sets of SMSWSVE with respect to the parameters

To comprehensively investigate the influence of the slow-varying periodic excitation, three cases with different values of the parameter

Bifurcation diagrams for (a)

For

For

For

Since

The present section characterizes the dynamical behaviors of (

In case A for

Figure

Bursting oscillation for

When the parameter

Bursting oscillation is observed obviously in the time history in Figure

In case B for

When the amplitude

Bursting oscillation for

In order to reveal the mechanism of the bursting oscillation, we turn to the transformed portrait phase with the bifurcation diagram, as presented in Figure

Due to the slow passage effect [

When the parameter

The bursting oscillation can be classified as symmetric “fold/Hopf” bursting, since the repetitive spiking related to the limit cycle is generated by fold bifurcation and disappears by Hopf bifurcation.

In case C for

When the amplitude

Distribution of the equilibria for

From the phase portrait on the

Bursting oscillation for

In order to reveal the mechanism of the bursting oscillation, we turn to the transformed phase portrait on the

When the parameter

Bursting oscillation can be created since the system periodically switches between the two stable foci

According to the phase portrait on the

Bursting oscillation for

In order to explore the mechanism of the bursting oscillation in Figure

When the parameter

From Figure

We have investigated the dynamical mechanisms of bursting oscillations. In this section, we focus on the effects of excitation amplitude and excitation frequency on the bursting dynamics.

First, the effect of the excitation amplitude on the bursting dynamics is considered. Let us take the symmetric “fold/fold” bursting oscillation in case C as an example. According to the phase portrait related to

Second, the effect of the excitation frequency on the bursting dynamics is considered. The three bursting patterns are symmetrical in this article. Therefore, the time interval between two adjacent spikes of bursting in one oscillation period is half a period, which is the minimum time interval

Numerical results of the time interval between two adjacent spikes of bursting oscillations corresponding to Figure

The Shimizu-Morioka system may exhibit bursting oscillations with different waveforms under certain external forcing conditions. This article theoretically explores the mechanisms of different bursting patterns in the presence of two different frequency scales between the exciting frequency and the natural frequency. First, we regard the slow excitation

The relevant content of this article has been presented as a poster titled “Routes to Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation” at the 17th Asian Pacific Vibration Conference, 2017.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors are grateful to Dr. H. L. Guo for useful discussions. This work is supported by the National Key Basic Research Program of China (Grant no. 2015CB057400) and the National Natural Science Foundation of China (Grant no. 11672201).