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.
From the two bifurcation sets of SMSWSVE with respect to the parameters and , we can find that when the parameter is taken at different values, the two types of curves corresponding to fold bifurcation and Hopf bifurcation exhibit different positions. For example, when the parameter is smaller than 0667, no Hopf bifurcation is observed. When the parameter is greater than 0667 but smaller than 1.420, the supH1 curve is observed on the left side of the LP1 curve and the supH2 curve is observed on the right side of the LP2 curve, thereby indicating that a sufficiently large amplitude of the parameter generates an increase (decrease) in the parameter from point 0, wherein it first passes through the Hopf bifurcation points and then passes through the fold bifurcation points. When the parameter is greater than 1.420, the supH1 curve is observed on the right side of the LP1 curve and the supH2 curve is observed on the left side of the LP2 curve, thereby indicating that the absolute value of the Hopf bifurcation points is greater than the absolute value of the fold bifurcation points.
To comprehensively investigate the influence of the slow-varying periodic excitation, three cases with different values of the parameter , that is, case A: , case B: , and case C: , are considered, the bifurcations of which are presented in Figure 2, respectively.
(a)
(b)
(c)
For in Figure 2(a), twofold bifurcation points are observed at LP1 and LP2 . And two homoclinic bifurcation points are observed at Hom1 and Hom2 , at which the two stable limit cycles can be switched.
For in Figure 2(b), twofold bifurcation points are observed at LP1 and LP2 , whereas two supercritical Hopf bifurcation points are observed at supH1 and supH2 , at which the two limit cycles are produced via Hopf bifurcation.
For in Figure 2(c), twofold bifurcation points are observed at LP1 and LP2 , at which the jumping phenomenon may occur. In addition, two supercritical Hopf bifurcation points are observed at supH1 and supH2 , at which the two limit cycles are produced via Hopf bifurcation.
Since , the change interval of the parameter depends on its amplitude . Therefore, the influence of the amplitude on the system can be examined to characterize the dynamic behaviors of SMSWSVE when passes through different bifurcation points.
3. Dynamical Mechanisms of the Bursting Oscillations
The present section characterizes the dynamical behaviors of (2) for the three typical cases. In this article, SMSWSVE is a typical fast-slow system involving two time scales, which may generate bursting oscillations in the system. In the following analysis, the excitation frequency is fixed at .
3.1. Symmetric “Homoclinic/Homoclinic” Bursting in Case A
In case A for , no stable equilibrium points are observed. However, one stable limit cycle is observed for , one stable limit cycle is observed for , and two stable limit cycles are observed for simultaneously. The stable limit cycle disappears via homoclinic bifurcation of the saddle point at Hom2 , and the limit cycle disappears via homoclinic bifurcation of the saddle point at Hom1 . The two stable limit cycles can be switched via two homoclinic bifurcation points, namely, and .
Figure 3 presents the bursting oscillation for , wherein the trajectory moves around two stable limit cycles, namely, and , as presented in Figure 3(a), which are switched by the two homoclinic bifurcation points. In order to reveal the mechanism of this bursting oscillation, we turn to the help of the transformed phase portrait on the plane, as presented in Figure 3(b). The trajectory, starting from the point , moves along the attraction domain of the stable limit cycle , and the movement exhibits a repetitive spiking state with large amplitude. As the parameter decreases to −0.029, where the homoclinic bifurcation point is located, disappears via homoclinic bifurcation. In addition, the trajectory jumps into the vector field of stable limit cycle . Parameter continues to decrease until it reaches its minimum value of , after which it gradually increases.
(a)
(b)
(c)
When the parameter increases to 0.029, where the homoclinic bifurcation point is located, disappears via saddle homoclinic bifurcation, and the trajectory jumps into the vector field of the stable limit cycle . Parameter continues to increase until it reaches its maximum value of , after which it gradually decreases. The return of the trajectory to starting point finished one periodic movement.
Bursting oscillation is observed obviously in the time history in Figure 3(c), wherein the up-state and the down-state are switched via the two homoclinic bifurcation points that form the bursting oscillation with the symmetric structure. Therefore, the bursting can be classified as the symmetric “homoclinic/homoclinic” type.
3.2. Symmetric “Fold/Hopf” Bursting in Case B
In case B for , twofold bifurcation points and two Hopf bifurcation points are observed on the plane, which result in the transition between the repetitive spiking and quiescent states.
When the amplitude is taken at , the slow variable exhibits changes between −0.430 and 0.430, that is, , thereby implying the presence of the fold bifurcation points at and the Hopf bifurcation points at . From the phase portrait on the plane plotted in Figure 4(a), the attractors oscillate around two equilibrium points, as denoted by . In addition, a typical bursting oscillation is observed in the time history in Figure 4(b). The trajectory jumps between two spiking states , which are symmetric to each other.
(a)
(b)
(c)
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 4(c). The trajectory starts from the point with , where is taken at its positive extreme value. At this point, the attractor is categorized as the stable limit cycle , thereby allowing the trajectory to move within the vector field of , and repetitive spiking movement with large amplitude is observed. Parameter then decreases, as indicated by the arrow. When the parameter decreases to , where the Hopf bifurcation point is observed, thereby resulting in the disappearance of stable limit cycle and the appearance of stable equilibrium point .
Due to the slow passage effect [23], the trajectory will not move within the vector field of immediately, it still moves in the vector field of for a short time, and then it turns into the vector field of . The repetitive spiking state movement is terminated, and the quiescent state movement is created. Parameter then continues to decrease to , where the fold bifurcation point is located, thereby generating a trajectory jump into the vector field of and resulting in the termination of the quiescent state movement and the creation of repetitive spiking state movement. As the parameter decreases to , it reaches its negative extreme value, after which the parameter exhibits a gradual increase.
When the parameter increases to , where the other Hopf bifurcation point is observed, it results in the disappearance of the stable limit cycle and the appearance of the stable equilibrium point . The trajectory still moves in the vector filed of for a short time according to the slow passage effect, after which it turns into the vector field of . The Hopf bifurcation point terminates the repetitive spiking state movement and exhibits the quiescent state movement. When the parameter increases to , where the fold bifurcation point is observed, it results in the disappearance of the quiescent state movement and the appearance of the repetitive spiking state movement. In addition, this increase results in the loss of ’s stability and the generation of the stable limit cycle . Parameter then continues to increase to . The return of the trajectory back to the starting point finished one periodic movement.
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.
3.3. Bursting Oscillation in Case C
In case C for , twofold bifurcation points and two Hopf bifurcation points are observed on the plane, resulting in the transition between the repetitive spiking and quiescent states. Two different types of bursting oscillations are observed when the amplitude is taken at different values.
3.3.1. Symmetric “Fold/Fold” Bursting Oscillation
When the amplitude is taken at , the slow variable changes between −1.0 and 1.0, thereby implying the presence of only twofold bifurcation points at . Under this condition, three equilibria are present: two stable foci denoted by and one unstable saddle point denoted by , as presented in Figure 5.
From the phase portrait on the plane, as presented in Figure 6(a), the trajectory oscillates around two equilibrium points, namely, . A typical bursting oscillation is observed in the time history diagram in Figure 6(b). In addition, the trajectory jumps between two spiking states, specifically , which are symmetrical to each other.
(a)
(b)
(c)
In order to reveal the mechanism of the bursting oscillation, we turn to the transformed phase portrait on the plane with equilibrium branches, as presented in Figure 6(c). The trajectory starts from the point with , where the parameter is taken at its positive extreme value. The trajectory moves within the vector field of the stable focus that is associated with the quiescent state. When the parameter decreases to , where the fold bifurcation point is observed, thereby resulting in a trajectory jump to the vector field of the other stable focus and ending the quiescent state. The trajectory exhibits high-frequency oscillation with a gradual decrease in the amplitude, which corresponds to the spiking state. As the parameter decreases to its negative value, specifically , it may generate a gradual increase that corresponds to the quiescent state.
When the parameter increases to , where the fold bifurcation point is observed, it results in a trajectory jump to the vector field of and terminating the quiescent state. In addition, the trajectory is characterized as high-frequency oscillation with a gradual decreasing in the amplitude, which corresponds to the spiking state. As the parameter increases to , the trajectory returned back to its starting point , thereby finishing one periodic movement.
Bursting oscillation can be created since the system periodically switches between the two stable foci and by twofold bifurcations and the two parts are symmetrical. So the bursting pattern can be classified as symmetric “fold-fold” type.
3.3.2. Symmetric “SupHopf/SupHopf” Bursting Oscillation via “Fold/Fold” Hysteresis Loop
continues to increase to the range of , for example, , thereby implying that the parameter not only passes through the fold bifurcation points and at , but also passes through the supercritical Hopf bifurcation points and at , where the Hopf bifurcation occurs.
According to the phase portrait on the plane in Figure 7(a), the trajectory oscillates around two stable foci, namely, . In addition, a typical bursting oscillation is observed in the time history diagram in Figure 7(b), and the trajectory jumps between two spiking states, specifically , both of which are symmetrical to each other. However, a great difference is observed between Figure 7(a) and Figure 6(a) because two limit cycles are observed in Figure 7(a).
(a)
(b)
(c)
In order to explore the mechanism of the bursting oscillation in Figure 7, we turn to the overlap of the transformed phase portrait on the plane with the system bifurcation diagram, as presented in Figure 7(c). We assume that the trajectory, starting from the point with , moves within the vector field of the limit cycle , corresponding to the spiking state. When the parameter decreases to , where the Hopf bifurcation point is observed, thereby resulting in the disappearance of limit cycle and the presence of the stable focus . In addition, the Hopf bifurcation point resulted in the termination of the spiking state and the appearance of the quiescent state. As the parameter decreases to , where the fold bifurcation point is observed, the trajectory jumps into the vector field of the stable focus , and the fold bifurcation point also results in the termination of the quiescent state and the appearance of the spiking state. When the parameter decreases to , loses its stability via Hopf bifurcation, and a stable limit cycle is observed. As the parameter decreases to an negative extreme value, specifically , it exhibits a gradual increase.
When the parameter increases to , disappears and is observed due to Hopf bifurcation. In addition, this increase results in the termination of the spiking state and the appearance of the quiescent state. As the parameter increases to , where the fold bifurcation point is observed, which results in the trajectory jumps to the vector field of , thereby terminating the quiescent state and generating the presence of the spiking state. When the parameter exhibits an increase to , where the Hopf bifurcation point is observed, loses its stability and the stable limit cycle is produced via Hopf bifurcation. As the parameter increases to , the trajectory returns back to its starting point , thereby finishing one periodic movement.
From Figure 7(c), a hysteresis loop produced by the equilibrium branches between twofold bifurcation points, specifically , is observed. As a result, this bursting is classified as symmetric “Hopf/Hopf” type via the “fold/fold” hysteresis loop.
4. Effects of the Excitation Amplitude and Frequency on the Bursting Dynamics
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 in Figures 6 and 7, bursting oscillation is created when the slow excitation passes through twofold bifurcation values and periodically. Since , if the excitation satisfies , the bursting oscillations are not created because the slow excitation cannot pass through the twofold bifurcation values, while if , they may be created.
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 between two adjacent spikes, where is the period of one movement. Take the waveforms in Figure 7 as an example, when , , and , respectively, the corresponding time intervals between two adjacent spikes are measured, as presented in Figure 8, of which the numerical results agreed well with the analytical ones.
(a)
(b)
(c)
5. Conclusions
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 as a slow-varying parameter . Second, according to the two bifurcation sets of , three cases are considered: namely, , , and . Third, we investigate different bursting oscillations of different cases of : for , one type of bursting oscillation, namely, symmetric “homoclinic/homoclinic” bursting, is created when slow-varying passes through two homoclinic bifurcation points; for , symmetric “fold/Hopf” is created when slow-varying passes through twofold bifurcation points and two Hopf bifurcation points; for , two types of bursting oscillations, namely, symmetric “fold/fold” bursting when slow-varying passes through only two fold bifurcation points, and symmetric “Hopf/Hopf” bursting via “fold/fold” hysteresis loop is created when slow-varying passes through twofold bifurcation points and two Hopf bifurcation points. In a word, these four bursting oscillations are created due to the coexistence of different attractors. Fourth, we introduce two analysis strategies, specifically the transformed phase portrait and the overlap of the phase portrait related to with the bifurcation diagram, to explore the bursting oscillation mechanism in terms of the bifurcations and equilibria structures. Last of all, we find that the bursting oscillation can be easily controlled by the amplitude and frequency of the slow excitation.
Disclosure
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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
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).