Shock and Vibration

Volume 2018, Article ID 5207910, 11 pages

https://doi.org/10.1155/2018/5207910

## Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation

Correspondence should be addressed to Shuqian Cao; nc.ude.ujt@oacqs

Received 16 August 2017; Revised 23 January 2018; Accepted 4 February 2018; Published 14 March 2018

Academic Editor: Matteo Aureli

Copyright © 2018 Xindong Ma and Shuqian Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 by means of the transformed phase portrait. Finally, we can find that the time interval between two symmetric adjacent spikes of bursting oscillations exhibits dependency on the periodic excitation frequency.

#### 1. Introduction

Multiple time scales problems can be observed in many real systems, such as catalytic reactions in chemical systems [1, 2], electrophysiological experiments [3–6], and circuit systems [7, 8]. The coupling effect of these different fast-slow time scales generally results in a system that exhibits periodic motion characterized by a combination of relatively large amplitude and nearly harmonic small amplitude oscillations [9, 10], conventionally denoted by with and corresponding to the numbers of the large and small amplitude oscillations, respectively [11]. Variables that exhibit large amplitude oscillations generate a spiking state (SP) [12], whereas the systems may be in a quiescent state (QS) when all the variables exhibit small amplitude oscillations [13]. The dynamic behaviors connecting the spiking states and the quiescent states are called bursting oscillations. There are two types of bifurcations existing in one bursting oscillation, specifically one that transforms the system from spiking states to quiescent states and the other one that transforms the system from quiescent states to spiking states [14, 15]. Therefore, the bursting patterns not only depend on the patterns of the spiking states and quiescent states, but also depend on the patterns of the bifurcations that connect the two states [16].

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 [17]:where is a small parameter that satisfies and represents different time scales. Vector models the dynamics of a relatively fast changing process, while vector describes the relatively slowly changing quantity that modulates . Characterization of the dynamic behaviors at different time scales has long intrigued scholars. Rinzel [18] introduced the fast-slow analysis method to examine the Chay-Keizer model and characterize dynamic behaviors at different time scales and the mechanism of bursting oscillations. Wang et al. [19] examined the synchronization of bursting through the introduction of finite delays. Kiss et al. [20] studied the bursting oscillations in electrochemical systems caused by changes in the geometrical structure. Li et al. [21] examined the mechanism of bursting oscillations in a nonsmooth generalized Chua’s circuit with two time scales. However, most of the current work focused on the single slow variable, the mechanism of which is relatively simple, and some work is concentrated on the autonomous system. However, the mechanism of bursting oscillations in nonautonomous systems with slow-varying periodic excitation requires further exploration.

In the present work, we consider the Shimizu-Morioka system [22] with slow-varying excitation (SMSWSVE):where , , is the forcing amplitude, and is the forcing frequency, which satisfies . When the forcing frequency is much smaller than the natural frequency , we characterize the presented system as a typical fast-slow system involving two time scales. Inspired by Rinzel’s method, we present an analysis of bursting oscillation driven by slow-varying periodic excitation for the Shimizu-Morioka system. The purpose of this work is twofold. First, some results about bursting oscillations induced by the slow-varying periodic excitation are derived and how the excitation modulates the dynamics of bursting oscillations is drawn. Second, some important aspects of Rinzel’s method are highlighted in the forced Shimizu-Morioka system.

The rest of this paper is organized as follows. In Section 2, we summarize some of the results related to the stabilities and bifurcations of SMSWSVE, and based on this, we plot a two-parameter bifurcation set and obtain three SMSWSVE bifurcation diagrams following changes in parameter , which exhibit typical dynamic evolution behaviors of SMSWSVE. In Section 3, we investigate the generation of bursting oscillations based on three cases of different values of the parameter , of which four bursting patterns are obtained: symmetric “homoclinic/homoclinic” bursting oscillation for ; symmetric “fold/Hopf” bursting oscillation for ; symmetric “fold/fold” bursting oscillation; and symmetric “Hopf/Hopf” bursting oscillation via “fold/fold” hysteresis loop for . In Section 4, we focus on the effects of the excitation amplitude and frequency on the bursting oscillation. Finally, Section 5 concludes the paper.

#### 2. Stabilities and Bifurcations of the SMSWSVE

In order to analyze the influence of the slowly varying excitation, we can regard the external excitation as a slow-varying parameter . When the exciting frequency is much smaller than the natural frequency , during an arbitrary period , (), that is, , where is the initial timing point, the slow-varying parameter may change between and , thereby implying that , such that the parameter is kept almost constant during any arbitrary period . Therefore, (2) can be considered as a generalized autonomous system, the equilibrium points of which can also be considered as generalized equilibrium points. The generalized autonomous system in (2) forms the fast subsystem, whereas forms the slow subsystem.

The equilibrium point of (2) can be expressed in the form of with respect to the periodic external excitation as the parameter , where satisfiesthe stability of which can be determined by the corresponding characteristic equation, which is defined as Obviously, is stable for , , and . When the stability conditions are changed, different types of bifurcations may occur as follows.

*Fold Bifurcation*. For , the three equilibrium points may meet together to form one degenerate equilibrium point, thereby generating eigenvalues for the characteristic equation that can be expressed as , and , which imply the presence of fold bifurcation.

*Hopf Bifurcation*. defines the presence of a pair of pure imaginary eigenvalues, which implies the occurrence of supercritical Hopf bifurcation and thus results in periodic oscillation with a frequency of .

Here we fix the parameter and regard the parameters and as the control parameters. The two bifurcation sets of SMSWSVE are plotted in Figure 1, wherein the parameter plane is divided into eight regions. According to our numerical calculations, region 8 exhibits an unstable equilibrium point and two stable equilibria . When the parameters pass through the fold bifurcation curves LP1 and LP2 and enter regions 7 and 6, the numbers of the equilibria may decrease via fold bifurcation, thereby leaving two stable equilibria in region 7 and in region 6. As the parameters pass through the Hopf bifurcation curves supH1 and supH2 and enter regions 5 and 4, the two stable equilibria, specifically and , lose their stabilities via Hopf bifurcations and are left with two stable limit cycles, namely, in region 5 and in region 4. When the parameters pass through the fold bifurcation curves LP1 and LP2 and enter regions 3 and 2, the two stable limit cycles and are still observed, and the numbers of the equilibria may increase via fold bifurcation, specifically one stable equilibrium point and two unstable equilibrium points. As the parameters pass through the Hopf bifurcation curves supH1 and supH2 and enter region 1, the stable equilibrium point loses its stability, thereby producing one stable limit cycle via Hopf bifurcation. Therefore, three unstable equilibria and two stable limit cycles are observed in region 1.