The mechanism of bursting oscillations with different codimensional bifurcations and nonlinear structures

Abstract

The main purpose of this article is to demonstrate that bursting oscillations can be observed not only in the slow–fast autonomous dynamical systems with multiple scales associated with time domain, but also in the non-autonomous dynamical systems with periodic excitations when an order gap exists between the exciting frequency and the natural frequency, implying multiple scales in frequency domain. Furthermore, we try to investigate the influence of different codimensional bifurcations between the quiescent states (QS) and repetitive spiking states (SP) and the nonlinear structures with different equilibrium branches on the bursting oscillations. By introducing an inductor as well as a periodically changed electrical current source in a traditional Chua’s circuit and taking suitable parameter values, a modified four-dimensional periodically excited oscillator with multiple scales in frequency domain is established. Bursting oscillations for two cases with nonlinear terms up to third and fifth order with codimension-1 and codimension-2 bifurcations have been explored, respectively. It is found that more equilibrium states may exist when higher order nonlinear terms are introduced in the vector field, which may cause multiple quiescent states, and accordingly, multiple forms of repetitive spiking oscillations in one bursting attractor, leading to more complicated bursting phenomena. Furthermore, instead of jumping from one stable equilibrium branch to settle down to another stable equilibrium branch when codimension-1 bifurcations (fold bifurcations) exist between QSs and SPs, codimension-2 bifurcation (fold-Hopf bifurcation) may cause QS approximately located on one stable equilibrium branch to jump to repetitive spiking oscillations surrounding another stable equilibrium branch of the generalized autonomous system.

1 Introduction

Many dynamical systems in natural and engineering problems involve two time scales [1, 2], which often behave in periodic states characterized by a combination of relatively large-amplitude and nearly harmonic small-amplitude oscillations, conventionally denoted by $N^K$ with N and K corresponding to large and small amplitude oscillations, respectively [3, 4]. Generally, we say the system is in quiescent state (QS) stage when all the variables are at rest or exhibit small-amplitude oscillations [5]. The coupling of two time scales may lead the systems to spiking state (SP), in which the variables may behave in large-amplitude oscillations [6, 7]d. Bursting phenomena can be observed when the variables alternate between QS and SP. Two important bifurcations associated with the bursters exist: bifurcation of a quiescent state to repetitive spiking and bifurcation of a spiking attractor to quiescence [8, 9].

At the beginning of the analysis for the dynamical systems with multiple time scales, a obvious method is introduced by eliminating the slow variables upon solving the related equations [10]. Perturbation method is also employed to find the asymptotic solutions by expanding all the variables in terms of small parameters [11]. Many other approaches have been developed to investigate the dynamics, which, however, cannot be used to explore the interaction between different time scales, until the so-called slow–fast analysis method is introduced [12–14]. By dividing the whole system into two subsystems, i.e., the fast subsystem and the slow subsystem [15], QSs and SPs as well as the bifurcations between them may be approximated by the equilibrium states as well as the related bifurcation forms in the fast subsystem, while the slow system can be used to understand the slow passage effect on the moderation of QS and SP [16, 17].

Based on the slow–fast analysis, a lot of results related to dynamics with two time scales are presented, in which several bursting oscillations such as the fold/fold [18] and fold/Hopf [19] bursters are obtained. However, most of the reports are focused on the autonomous systems, in which obvious slow and fast subsystems can be obtained [20], while for non-autonomous systems, such as periodically excited systems, when an order gap exists between the frequency of periodic excitation and natural frequency, no obvious slow and fast subsystem exists, while the effect of two time scales can also be observed [21], on the fact that the trajectories of systems are related to both the two frequencies [22], behaving in relaxation oscillations [23]. Here we define the frequency of the free vibration of autonomous nonlinear dynamical system as the natural frequency of the related nonlinear system. Unlike the linear system, the natural frequency of the nonlinear system is related to the dynamical behaviors, which may change with the parameters of the system. For example, when the trajectory settles down to a focus, the natural frequency can be determined by the imaginary parts of the pair of the conjugate eigenvalues of the focus, while when the system behaves in periodic movement, the natural frequency can be described by the frequency of the oscillation. Since no obvious slow and fast subsystem exists, the method of slow–fast analysis [24] can not be directly employed to approach the mechanism of the bursting. How to explore the characteristics of the multi-mode oscillations in non-autonomous system still remains an open problem. For the typical periodically excited dynamical system, when the exciting term changes on a much smaller time scale comparing with the original system, the whole exciting term can be regarded as a slow-varying parameter, leading to the so-called generalized autonomous and the transformed phase portrait can be employed to explore the influence of the exciting term on the evolution of the dynamical behavior [25].

Furthermore, up to now, only codimension-1 bifurcations [26, 27], such as fold bifurcation and Hopf bifurcations, exist between QS and SP and only one form of QS and SP [28, 29] involves in the most of bursting oscillations reported [30, 31]. How the bursting oscillations behave when higher codimensional bifurcations or multiple forms of QSs and SPs involve in the dynamics still need to be explored.

Here we introduce an inductor as well as a periodically changed electrical power in a traditional Chua’s circuit [32] and taking suitable parameter values so that not only an order gap exists between the excited frequency and the natural frequency, but also different codimensional bifurcations as well as multiple equilibrium branches may involve in generalized autonomous system corresponding to the oscillations, the evolution of the dynamics of the system is investigated and different types of bursting oscillations as well as the bifurcation mechanism will be presented.

2 Mathematical model

Based on the canonical three-dimensional Chua’s model [32], a modified version has been constructed with a controller composed of an inductor as well as a linear resistor, which may exhibit more complex dynamics [33]. By introducing a periodically changed electrical current source in the model with a nonlinear resistor, shown in Fig. 1, the related mathematical model can be expressed as

$$\begin{aligned} \begin{array}{rcl} \frac{\mathrm{d}V_1}{\mathrm{d}t}\!&{}=&{}\!\frac{1}{C_1}[G_1(V_2\!-\!V_1)\!-\!G_1\,g(V_1)\!+\!I_G \sin (\omega t)], \\ \frac{\mathrm{d}V_2}{\mathrm{d}t}&{}=&{}\frac{1}{C_2}[G_1(V_1-V_2)+i_{L1}+i_{L2}],\\ \frac{\mathrm{d}i_{L1}}{\mathrm{d}t}&{}=&{}-\frac{1}{L_1} (V_2-R_2\,i_{L2}),\\ \frac{\mathrm{d}i_{L2}}{\mathrm{d}t}&{}=&{}-\frac{1}{L_2} V_2, \end{array} \end{aligned}$$

(1)

where $G_1=1/R_1, G_2=1/R_2$, while $g(V_1)=\Delta _1\,V_1+\Delta _2\,V_1^3+\Delta _3\,V_1^5$ describes the relationship between the current and the voltage passing across the nonlinear resistor, in which $\Delta _1$ is a non-dimensional parameter, while $\Delta _2$ and $\Delta _3$ corresponds to the dimension $(v)^{-2}$ and $(v)^{-4}$, respectively.

Fig. 1

Circuit with periodic excitation

By introducing the transformations $V_1=E_{10}\,x$, $V_2=E_{20}\,y$, $i_{L1}=\frac{E_{20}}{R_1}\,u$, $i_{L2}=\frac{E_{20}}{R_1}\,v$, $t=R_1 C_2 \tau$ and taking $R_2=\frac{L_1}{R_1\,C_2}$,  (1) can be written in the non-dimensional form as

$$\begin{aligned} \begin{array}{llll} \frac{\mathrm{d}x}{\mathrm{d}\tau }&{}=&{}\alpha [ y - \delta f(x) ] + w, \\ \frac{\mathrm{d}y}{\mathrm{d}\tau }&{}=&{}\kappa x - y + u + v ,\\ \frac{\mathrm{d}u}{\mathrm{d}\tau }&{}=&{}- \gamma y + v,\\ \frac{\mathrm{d}v}{\mathrm{d}\tau }&{}=&{} -\beta y, \end{array} \end{aligned}$$

(2)

where $E_{10}$ and $E_{20}$ is used as the references and $\alpha =\frac{C_2\,E_{20}}{R_1\,E_{10}}$, $\beta =\frac{R_1^2\,C_2}{L_2}$, $\kappa =\frac{E_{20}}{E_{10}}$, $\delta =\frac{E_{20}}{E_{10}}(1+\Delta _1)$,  $\gamma =\frac{R_1^2\, C_2}{L_1}$,  $f(x)=a\,x+b\,x^3+c\,x^5$ with $a=1.0$, $b=\frac{\Delta _3\,E_{10}^2}{1+\Delta _1}$,  $c=\frac{\Delta _5\,E_{10}^4}{1+\Delta _1}$, $w=A\sin (\varOmega \tau )$ with $A=\frac{I_G\, R_1\,C_2}{C_1\,E_{10}}$, $\varOmega =\omega R_1 C_2$. System (2) with no external excitation may evolve from stable fixed point to periodic orbit and to chaos. Furthermore, super-chaotic movement can be observed for certain parameter values, corresponding to two positive Lyapunov exponents [34]. The external excitation may also cause the oscillator to evolve from periodic orbit to chaotic movements. However, when an order gap exists between $\varOmega$ and the natural frequency of the autonomous oscillator, the effect of two time scales may appear. Here we fix the parameter $\varOmega =0.005$ and other parameters at O(1.0). Obviously, the state variables $x,\,y,\,u,\,v$ may oscillate mainly according to the natural frequency, i.e., $O(\mathrm{d}x/\mathrm{d}\tau ,\,\mathrm{d}y/\mathrm{d}\tau ,\,\mathrm{d}u/\mathrm{d}\tau ,\,\mathrm{d}v/\mathrm{d}\tau )\approx O(1.0)\equiv T_1$, while the exciting term w oscillates periodically according to another much smaller scale, i.e., $O(\mathrm{d}w/\mathrm{d}\tau )\approx O(0.005)\equiv T_2$, leading to the coupling between two scales $T_1$ and $T_2$, which may cause the bursting oscillations in the system.

3 Bifurcation analysis of the generalized autonomous system

Since the state variables may oscillate mainly according to the natural frequency, for $\varOmega \ll \varOmega _N$, where $\varOmega _N$ represents the natural frequency, during an arbitrary period $T_N$, ($T_N=2 \pi /\varOmega _N$) i.e., $\tau \in [\tau _0,\tau _0+T_N]$, the exciting term w may change between $w_A=Acos(\varOmega \tau _0$ and $w_B=A cos [\varOmega (\tau _0+\tau _N)]$, implying $w_A\approx w_B$, which means w keeps almost a constant during any arbitrary period $T_N$. Therefore, (2) can be considered as a generalized autonomous system, in which w may be regarded as a slow-varying parameter though w may vary between $-A$ and $+A$. It can also be understood from the fact that the autonomous system (2) forms the fast subsystem, while $w=A\cos (\varOmega \tau )$ forms the slow subsystem.

Now we turn to the bifurcation analysis of the generalized autonomous system on regarding the periodic external excitation as a parameter w. The equilibrium point of (2) can be expressed in the form $EQ(x,y,u,v)=(X_0, 0, -\kappa X_0, 0)$, where $X_0$ satisfies

$$\begin{aligned} \alpha \delta (X_0+b X_0^3 +c X_0^5)+w=0, \end{aligned}$$

(3)

the stability of which can be determined by the associated characteristic equation, written in the form

$$\begin{aligned} \lambda ^4+h_1\,\lambda ^3+h_2\,\lambda ^2+h_3\,\lambda +h_4=0, \end{aligned}$$

(4)

where

$$\begin{aligned} \begin{array}{llll} &{}&{} h_1=1+\alpha \delta (1+3b X_0^2+5c X_0^4),\\ &{}&{} h_2=\beta +\gamma -\alpha \kappa +\alpha \delta (1+3b X_0^2+5c X_0^4),\\ &{}&{} h_3=\beta + \alpha \delta (1+3b X_0^2+5c X_0^4),\\ &{}&{} h_4=\alpha \beta \delta (1+3 b X_0^2+5c X_0^4). \end{array} \end{aligned}$$

(5)

Obviously, EQ is stable for $h_1>0$, $h_4>0$, $h_1h_2-h_3>0$ and $h_1h_2h_3-h_1^2h_4-h_3^2>0$, resulting in two possible bifurcation sets. One can be expressed in the form $h_4$=0, i.e.,

$$\begin{aligned} \left\{ \begin{array}{l} 1+3 b X_0^2+5c X_0^4=0,\\ 1+ b X_0^3+c X_0^5=0, \end{array} \right. \end{aligned}$$

(6)

on which fold bifurcation may occur, causing jumping phenomenon between different equilibrium points, while the other can be written in the form

$$\begin{aligned}&h_1h_2h_3-h_1^2h_4-h_3^2=0,\nonumber \\&\quad (h_1>0,\,h_4>0\,h_1h_2-h_3>0), \end{aligned}$$

(7)

on which Hopf bifurcation may take place, leading to periodic oscillation with the frequency $\varOmega _H=\frac{h_3}{h_1}$.

Furthermore, a codimension-2 bifurcation with a zero and a pair of pure imaginary eigenvalues $\pm \sqrt{\beta }\,I$ as well as negative eigenvalue $-1$ takes place on the set

$$\begin{aligned} \left\{ \begin{array}{l} a+3 b X_0^2+5c X_0^4=0,\\ a+ b X_0^3+c X_0^5=0,\\ \alpha \kappa -\gamma =0, \end{array} \right. \end{aligned}$$

(8)

leading to the occurrence of limit cycle away from the original fixed point.

As an example, the bifurcation sets on the plane $(\kappa , w)$ for the parameters fixed at $\alpha =8.0$, $\beta =0.5$, $\gamma =8.0$, $\delta =-0.15$, $b=-2.0$, $c=0.0$ are presented in Fig. 2a, which divide the parameter plane into eight regions corresponding to different dynamical behaviors. In Region (1) and (8), only one stable equilibrium point exists corresponding to $\textit{EB}_1$ and $\textit{EB}_3$ in Fig. 2b, respectively. On the fold bifurcation set ${F}_1$, two equilibrium branch $\textit{EB}_2$ and $\textit{EB}_3$ meet each other, which leads to a cusp point, together with a stable equilibrium point related to $\textit{EB}_1$. While on the fold bifurcation set ${F}_2$, $\textit{EB}_1$ and $\textit{EB}_2$ meet each other, leading to a cusp point, together with a stable equilibrium point related to $\textit{EB}_3$. On the Hopf bifurcation sets ${H}_1$, the stable focus related to $\textit{EB}_1$ loses its stability to form a limit cycle, while on ${H}_2$, the stable focus related to $\textit{EB}_3$ bifurcates to another limited cycle.

Fig. 2

Bifurcations for $\alpha =8.0$, $\beta =0.5$, $\gamma =8.0$, $\delta =-0.15$, $b=-2.0$, $c=0$. a Bifurcation sets on the $(\kappa ,\,w)$ plane b equilibrium branches with the variation of w

In region (2) and (5), a stable focus related to $\textit{EB}_1$ as well as a stale limit cycle bifurcated from the equilibrium branch $\textit{EB}_3$ can be observed, while in Region (4) and (6), a stable focus related to $\textit{EB}_3$ as well as a stale limit cycle bifurcated from the equilibrium branch $\textit{EB}_1$ exists. Two stable cycles bifurcate from equilibrium branches $\textit{EB}_1$ and $\textit{EB}_3$, respectively, in Region (3). However, in Region (7), two stable foci related to $\textit{EB}_1$ and $\textit{EB}_3$ as well as a saddle point can be obtained.

At the two intersection points ${B}_{+1}$ and ${B}_{-1}$ of the bifurcation sets ${F}_1$ and ${H}_1$ or ${F}_2$ and ${H}_2$, codimension two bifurcations of the equilibrium points corresponding to a zero as well as a pair of pure imaginary eigenvalues take place, which leads to different behaviors in the neighboring regions of ${B}_{+1}$ and ${B}_{-1}$, respectively, with small parameter perturbation.

However, at the intersection points ${H}_A$ and ${H}_B$ of ${H}_1$ or ${H}_2$, instead of a codimension two bifurcation, two codimension one bifurcations (super-Hopf bifurcations) corresponding to the equilibrium points located on $\textit{EB}_1$ and $\textit{EB}_3$, respectively, take place, leading to different limit cycles.

Remark 1

$\odot$ There also exist other forms of bifurcations for certain values of parameters, which we omitted here for simplicity, since these bifurcation conditions are so special that they may not really exist for the real systems. Furthermore, for certain fixed parameters, not all the above bifurcation sets can be observed, since the original stable fixed point may lose the stability via only one bifurcation form.

$\odot$ The bifurcation sets can be obtained by continuation method, using a properly continuation package for continuous-time system, such as MATCONT [35], which is a standard tool for numerical bifurcation analysis. Here we only list the conditions for the bifurcations, which may be used for the investigation of the bursting oscillations caused by the interaction between different time scales.

4 Evolution of bursting oscillations as well as the mechanism

Because of the order gap between the exciting frequency and the natural frequency, bursting oscillations, which always behave in the combination of small- and large-amplitude oscillations, can be observed. Note that the form of nonlinear function f(x) may influence the attractors of bursting oscillations. When higher-order nonlinear terms are introduced in the function, more equilibrium points of the generalized autonomous system can be observed, which may lead to more complicated bursting oscillations. Furthermore, the forms of the bifurcations between the quiescent states (QSs) and repetitive spiking states (SPs) may also affect the structures of bursting attractors. Here we fix the parameters at

$$\begin{aligned} \alpha =8,\quad \beta =0.5, \quad \gamma =8,\quad \varOmega =0.005, \end{aligned}$$

(9)

and consider the influence of the two factors, i.e., the nonlinear functions and the bifurcations between QSs and SPs, on the bursting oscillations.

We first introduce the conception of transformed phase portrait (TPP), in which the relationship between the variations of state variables and that of excited term will be presented, so that the bifurcation analysis above with the change of w can be employed to account for the mechanism of the bursting oscillations.

4.1 Transformed phase portrait (TPP)

Note that all the bifurcation analysis is investigated upon taking w as the bifurcation parameter, which may be employed to explore the mechanism of the bursting oscillations. Therefore, the relationship between the state variables and w is very important for the effect of two time scales on the dynamics, which can be described by the transformed phase portraits. The trajectory as well as w can be described as

$$\begin{aligned} \varPi _G= & {} [X(t),w]\nonumber \\= & {} \{[x(t),y(t),z(t), A\sin (\varOmega t)],\quad \forall t\in R\}, \end{aligned}$$

(10)

from which one may obtain the traditional phase portraits $\{[x(t),y(t),z(t)],\forall t\in R\}$, as well as its projections on subspaces such as $\{[x(t),y(t)],\forall t\in R\}$. Here we regard w as a generalized state variable and define $\varPi _G$ changing with four variables (x, y, z, w) as well as their projections on subspaces related to w, such as (x, w), (x, y, w), etc., as the transformed phase portraits (TPP), which may exhibit the relationships between the state variables and the slow variable w.

Fig. 3

Phase portraits for $\kappa =0.5$, $\delta =-0.15$, $b=-2.0$, $c=0$, $A=20.0$ a on the $(x,\,y)$ plane b on the $(y,\,v)$ plane

4.2 Bursting oscillations with codimension-1 bifurcation

For the parameters fixed at $\kappa =0.5$, $\delta =-0.15$, $b=-2.0$ and $c=0$, which implies that nonlinear terms only up to third order are considered in the function f(x), there are only codimension-1 bifurcations, i.e., fold bifurcations, occurring at the points ${FB}_{\pm }(x,w)=(\pm \sqrt{6}/{6}$, $\pm {2\sqrt{6}}/{15})$. The bifurcation points divide the x-axis into three parts. When $|x |> {\sqrt{6}}/{6}$, only one stable focus exists, while for $|x | < {\sqrt{6}}/{6}$, two stable foci as well as a unstable saddle point can be observed. At the bifurcation points $x= \pm {\sqrt{6}}/{6}$, one of the two stable foci may meet the saddle point to form a degenerate cusp point, which may lead to the jumping phenomena between different equilibrium points.

While for $\kappa =0.5$, $\delta =0.15$,  $b=-2.0$ and $c=\frac{4}{3}$, implying nonlinear terms up to fifth order are included in the function f(x), there still exist only codimension-1 bifurcations, occurring at the four points ${FB}_{\pm 1}(x,w)=(\pm 0.470,\pm 0.352)$, ${FB}_{\pm 2}(x,w)=(\pm 0.824,\pm 0.254)$, leading to the variation of the number of equilibrium points between one, three and five. At the bifurcation points, one stable focus may meet one unstable saddle point to form a degenerate cusp point, causing the jumping phenomena between different stable equilibrium points.

4.2.1 Bursting oscillations with third-order nonlinear term

Figure 3 gives the phase portrait of the bursting oscillations for $\kappa =0.5$, $\delta =-0.15$, $b=-2.0$, $c=0$, $A=20.0$, which implies that nonlinear terms only up to third order are considered in the function f(x), and the related time history of y is plotted in Fig. 4.

Fig. 4

Time history of y for $\kappa =0.5$, $\delta =-0.15$, $b=-2.0$, $c=0$, $A=20.0$

The trajectory, starting from the point $A_1$, moves almost strictly along the stable equilibrium branch $\textit{EB}_1$ to form the first stage of quiescent state (${QS}_1$), until it arrives at the bifurcation point ${FB}_1$. Jumping phenomenon occurs, which causes the trajectory to quickly jump to the point ${A}_3$. Then the trajectory may tend to another equilibrium branch $\textit{EB}_2$, resulting in repetitive spiking $({SP}_1)$ with large-amplitude oscillations, since the point ${A}_3$ is not exactly located on $\textit{EB}_2$. The amplitudes of the oscillations may gradually decrease, and finally the trajectory may settle down to stable $\textit{EB}_2$ to form ${QS}_2$. At the point ${A}_2$, the external excitation reaches its maximum $w=A=20.0$, which implies w may decrease with the evolution of time. Therefore, the trajectory may return almost strictly along $\textit{EB}_2$ until it arrives at the second fold bifurcation point ${FB}_2$. The jumping phenomenon between different equilibrium branches results in the repetitive spiking $({SP}_2)$ with large-amplitude oscillations. When the oscillations settle down to the equilibrium branch $\textit{EB}_1$ to the point ${A}_1$, one period of the bursting oscillations is finished.

The amplitudes as well as the frequency of the repetitive spiking oscillations can be approximated by the eigenvalues of equilibrium points located on the equilibrium branches. Here we take ${SP}_2$ as an example. The spiking oscillations ${SP}_2$ may take place between two points on x-axis, denoted by $\varPi _1$ and $\varPi _2$, with $\varPi _1$ for $x=0.932$ and $\varPi _2$ for $x=1.491$. The related eigenvalues can be approximated at

$$\begin{aligned} \begin{array}{lll} &{}&{}\lambda _1=-0.570,\quad \lambda _2=-0.059,\quad \lambda _{\pm }=-0.147\pm 2.740\,{I},\\ &{}&{}\lambda _1=-15.08,\quad \lambda _2=-.0591,\quad \lambda _{\pm }=-0.334\pm 2.863\,{I}, \end{array}\nonumber \\ \end{aligned}$$

(11)

at $\varPi _1$ and $\varPi _2$, respectively. Therefore, the frequency related to the spiking oscillations may vary from $\varOmega _{\varPi 1}\approx 2.740$ to $\varOmega _{\varPi 2}\approx 2.863$, which agrees well with the numerical results via $\varOmega _{NF}=\frac{2\pi }{T_{E}}$ varying from 2.764 to 2.881, obtained from the time history in Fig. 3.

The mean frequency of spiking oscillations can be computed at $\varOmega _{M}=\frac{1}{2}(\varOmega _{\varPi 1}+\varOmega _{\varPi 2})=2.802$, yielding the mean period at $T_{M}=\frac{2\pi }{\varOmega _{M}}\approx 2.242$, while the mean value for the real parts of the pair of the conjugate eigenvalues between $\varPi _1$ and $\varPi _2$ can be approximated at $Re_{M}\approx -0.2405$ and the y-value of the ${SP}_2$ oscillations may be approximated at $y=Y_{A}e^{-0.2405(\tau -\tau _0)}\cos [\varOmega _{M}(\tau -\tau _0)]$, where $Y_{A}$ is the y-value at the beginning of the repetitive spiking with non-dimensional time $\tau _0$, approximated at $Y_{A}=-0.0858$ and $\tau _0=1891.376$.

The curve $y=Y_{A}e^{-0.2405(\tau -\tau _0)}$, corresponding to $AP_{E2}$ in Fig. 4, can be used to approximate the amplitudes of the repetitive spiking oscillations, which agrees well with the extreme values of ${SP}_2$. Therefore, the amplitudes of ${SP}_2$ oscillations can be approximated at $A_i=|Y_{A}|e^{-0.5392\,i},\, (i=1,2,\ldots )$. For examples, $A_2$ and $A_3$ can be theoretically approximated at $A_2=0.051$ and $A_3=0.010$, respectively, which agrees well with the values of next two amplitudes of spiking oscillations in ${SP}_2$ via the numerical simulations (see Fig. 4).

To reveal the mechanism of the bursting oscillations, we plotted the overlap of the transformed phase portrait and the equilibrium branch on the $(x,w)=[x(\tau ),\,A\sin (\varOmega \tau )]$ plane in Fig. 5.

The trajectory, starting from the point ${A}_1$, moves almost strictly along the stable equilibrium branch $\textit{EB}_1$ to form ${QS}_1$ until it arrives at the point ${FB}_1$, at which fold bifurcation occurs, leading to the jumping phenomenon of the trajectory, which may try to approach another stable equilibrium branch $\textit{EB}_2$, yielding repetitive spiking oscillations $({SP}_1)$, since the stable $\textit{EB}_1$ is stopped at the point ${B}_2$ with $(w,x)=(0.3266,-0.4082)$, while $\textit{EB}_3$ is unstable branch associated with saddle point. The trajectory in spiking state $({SP}_2)$ may tend to the stable equilibrium branch $\textit{EB}_2$ via the gradual decrease of the amplitudes of the oscillations, which may finally settle down to $\textit{EB}_2$, leading the second stage of the quiescent state $({QS}_2)$. When the trajectory moves at the point ${A}_2$, the external excitation reaches its maximum with $w=A=20.0$, which may cause the return of the trajectory with further increase of the non-dimensional time $\tau$. The trajectory moves almost strictly along $\textit{EB}_2$, until another fold bifurcation occurs at the point ${A}_3$, which leads to the jumping phenomenon from $\textit{EB}_2$ to $\textit{EB}_1$, yielding repetitive spiking oscillations $({SP}_2)$. The spiking oscillations may settle down to stable $\textit{EB}_1$, until it reaches the starting point ${A}_1$, which finishes one period of the bursting oscillations.

Fig. 5

a Transformed phase portrait b Overlap of the TPP and equilibrium branches of the generalized autonomous system

Remark 2

$\odot$ There exists very small distance between the fold bifurcation points of the equilibrium branch and connecting points between QSs and SPs on the trajectory, which may be caused by inertia of the trajectory.

$\odot$ The repetitive spiking oscillations are caused by the transient procedures of the trajectory to settle down to the stable equilibrium branches of the generalized autonomous system, in which the fold bifurcation points can be considered as the initial points to approach the focus-type stable equilibrium branches.

4.2.2 Bursting oscillations with fifth-order nonlinear term

Figure 6 gives the phase portrait of the bursting oscillations for $\kappa =0.5$, $\delta =0.15$, $b=-2.0$, $c=\frac{4}{3}$ and $A=1.0$, which implies that nonlinear terms up to fifth order are considered in the function f(x), and the related time history of y is plotted in Fig. 7, where only the parts related to ${SP}_1$ and ${SP}_2$ are presented because of the symmetric structures for ${SP}_3$ and ${SP}_4$.

Fig. 6

Phase portraits for $\kappa =0.5$,  $\delta =0.15$, $b=-2.0$, $c=\frac{4}{3}$ and $A=1.0$ a on the $(x,\,y)$ plane b on the $(y,\,v)$ plane

Fig. 7

Time history of y for $\kappa =0.5$, $\delta =0.15$, $b=-2.0$, $c=\frac{4}{3}$ and $A=1.0$

The trajectory, starting from the point ${A}_1$, moves along the curve ${A}_1{FB}_{-2}$, which is almost a straight line with very small y-value, to form ${QS}_1$. When it moves to the point ${FB}_{-2}$, fold bifurcation occurs, leading to jumping phenomenon, which results in repetitive spiking oscillations $({SP}_1)$. The trajectory quickly passes across the point ${A}_2$ to ${A}_3$ and oscillates to settle down to the curve ${A}_4{FB}_{+1}$, which is almost a straight line with very small y-value. The trajectory moves along ${A}_4{FB}_{+1}$ to form ${QS}_2$ until it arrives at the point ${FB}_{+1}$, at which another fold bifurcation takes place. Jumping phenomenon appears, leading to the repetitive spiking oscillations ${SP}_2$, which causes the trajectory to quickly pass across ${A}_5$ to ${A}_6$ and then oscillates around the curve ${FB}_{+2}{B}_1$, an almost straight line with very small y-value. When the amplitudes of the oscillations settle down to zero, the trajectory moves along with the curve ${FB}_{+2}{B}_1$ to form ${QS}_3$. The trajectory returns at the point ${B}_1$ since the state variable x reaches its maximum and moves along ${FB}_{+2}{B}_1$. Fold bifurcation occurs at the point ${FB}_{+2}$, leading to the repetitive spiking oscillations ${SP}_3$. The trajectory quickly passes across ${B}_2$ to ${B}_3$ and oscillates to settle down to the curve ${B}_4{FB}_{-1}$. When the amplitudes of the oscillations decrease to zero, it may move along the curve ${B}_4{FB}_{-1}$, which is an almost straight line with very small y-value, to form ${QS}_4$. Fold bifurcation takes place at the point ${FB}_{-1}$, causing the trajectory to quickly pass across ${B}_5$ to ${B}_6$. The jumping phenomenon leads to the repetitive spiking ${SP}_4$ around the curve ${A}_1{FB}_{-2}$. The gradual decreasing of the amplitudes of the oscillations may lead to the trajectory to settle down to the curve. When the trajectory returns to the point ${A}_1$, one period of the bursting oscillations is finished.

The amplitudes as well as the frequency of the repetitive spiking oscillations can be approximated by the eigenvalues of equilibrium points located on the curves associated, which are almost straight lines with very small y-values by employing the same method for the analysis of the spiking oscillations in Fig. 3. Here we only list the results related to frequencies of the spiking oscillations. For the ${SP}_1$, the spiking oscillations may take place from the point ${FB}_{-2}$ to ${A}_4$, in which the x-value may vary from $-0.824$ to $-0.0242$, causing the pair of complex conjugate eigenvalues to vary from $-.4429\pm 2.050\,\texttt {I}$ to $-.0830 \pm 2.289\,\texttt {I}$. Therefore, the frequency related to ${SP}_1$ may increase gradually from 2.050 to 2.289, which agrees very well with the numerical simulations varying from $\frac{2\pi }{2.983}\approx 2.106$ to $\frac{2\pi }{2.741}\approx 2.292$ (see Fig. 7).

Remark 3

Two factors cause the difference between the two frequencies obtained via theoretical method and numerical simulations, respectively. Firstly, the starting point to compute the frequency from time history is taken at ${A}_2$, which is away from the bifurcation point ${FB}_{-2}$. Secondly, the absolute value of x for the ending point for numerical computation is taken a bit larger than the absolute value of x at the point ${A}_4$.

To reveal the mechanism of the bursting oscillations, we plotted the overlap of the transformed phase portrait and the equilibrium branches of the generalized autonomous system on the $(x,w)=(x(\tau ),\,Asin(\varOmega \tau ))$ plane in Fig. 8.

Fig. 8

a Transformed phase portrait on $(w,\,x)$ b Overlap of the phase portrait and the equilibrium branches

The trajectory, starting from the point ${A}_1$, moves almost strictly along the stable equilibrium branch $\textit{EB}_1$ to form ${QS}_1$ until it arrives at the point ${FB}_{-2}^{(2)}$, at which fold bifurcation occurs, leading to the jumping phenomenon of the trajectory, which may try to approach another stable equilibrium branch $\textit{EB}_3$, yielding repetitive spiking oscillations $({SP}_1)$. The trajectory in spiking state ${SP}_1$ may tend to the stable equilibrium branch $\textit{EB}_3$ via the gradual decrease of the amplitudes of the oscillations, and finally settles down to $\textit{EB}_3$, leading the second stage of the quiescent state $({QS}_2)$. Trajectory then moves almost strictly along $\textit{EB}_3$ until it reaches the point ${FB}_{+1}^{(2)}$, at which another fold bifurcation takes place. The trajectory may jump to another stable equilibrium branch $\textit{EB}_5$, yielding repetitive spiking oscillations $({SP}_2)$, which oscillate around $\textit{EB}_5$. When the trajectory oscillates to settle down to $\textit{EB}_5$, it moves almost strictly along the stable equilibrium branch $\textit{EB}_5$ to form ${QS}_3$. The trajectory may return along $\textit{EB}_5$ when it arrives at the point ${B}_1$, at which w reaches its maximum value $w=+1$. The system keeps in quiescent state ${QS}_3$, until the trajectory arrives at the point ${FB}_{+2}^{(2)}$, at which fold bifurcation occurs, leading to jumping phenomenon of the trajectory to approach $\textit{EB}_3$. Repetitive spiking oscillations ${SP}_3$ occur, where the trajectory oscillates around $\textit{EB}_3$. The amplitudes of the oscillations may gradually decrease and finally settle down to zero. Then the trajectory moves almost strictly along $\textit{EB}_3$ to form ${QS}_4$ until the trajectory arrives at the point ${FB}_{-1}^{(2)}$, at which fold bifurcation occurs. Jumping phenomenon can be observed, which may cause the trajectory to oscillate around the stable equilibrium branch $\textit{EB}_1$ via repetitive spiking oscillations $({SP}_4)$. The amplitudes of the oscillations may gradually settle down to zero, causing the trajectory to move almost strictly along the stable $\textit{EB}_1$. When the trajectory arrives at the starting point ${A}_1$, one period of the movement is finished.

Remark 4

$\odot$ From the overlap of the transformed phase portrait and the equilibrium branches in Fig. 8b, one may find that there exists a bit difference between the theoretical fold bifurcation points and the corresponding connecting points between QSs and SPs on the trajectory, which may be caused by the inertia of the trajectory since the nominal parameter w is not static, but changes slowly with the non-dimensional time $\tau$.

$\odot$ The spiking oscillations are caused by the transient procedures of the trajectory to approach the stable focus-type equilibrium branches, since the trajectory at the fold bifurcation points may jump from one equilibrium branch to another focus-type equilibrium branch. Therefore, the fold bifurcation points, located near the corresponding bifurcation points on one equilibrium branch, can be considered as the initial points to approach another equilibrium branch. Because of the focus-type of the equilibrium branch, the trajectory may cycle around the equilibrium branch to settle down to the branch, which may cause the spiking oscillations around the corresponding equilibrium branch.

4.3 Bursting oscillations with codimension-2 bifurcation

When the conditions in (8) are satisfied, codimension-2 bifurcation (fold-Hopf bifurcation) associated with a zero and a pair of pure imaginary eigenvalues may take place. For the parameters fixed in (9) and $\kappa =1.0$, $\delta =-0.15$, $b=-2.0$, $c=0$, which implies that nonlinear terms only up to third order are considered in the function f(x), there exist codimension-2 bifurcations, i.e., the combination of fold and Hopf bifurcations, occurring at the points ${FB}_{\pm }(x,w)=(\pm \frac{\sqrt{6}}{6},\pm \frac{2\sqrt{6}}{15})$. While for $\kappa =1.0$, $\delta =0.15$, $b=-2.0$, $c=\frac{4}{3}$, implying that nonlinear terms up to fifth order are included in the function f(x), codimension-2 bifurcations may occur at the four points ${FB}_{\pm 1}(x,w)=(\pm 0.470,\pm 0.352)$, ${FB}_{\pm 2}$ $(x,w)=(\pm 0.824,\pm 0.254)$. The codimension-2 bifurcations may cause different structures of bursting attractors, since there exist not only the jumping phenomena between different equilibrium branches, but also limit cycle oscillations with the frequency $\sqrt{\beta }$. In the following, we will investigate the bursting oscillations as well as the mechanism for the two cases with different orders of nonlinear terms in the function f(x).

4.3.1 Bursting oscillations with third-order nonlinear terms

Figure 9 gives the phase portrait of the bursting oscillations for $\kappa =1.0$, $\delta =-0.15$, $b=-2.0$, $c=0$ and $A=20.0$, which implies that nonlinear terms only up to third order are considered in the function f(x), and the related time history of y is plotted in Fig. 10, where only one period is presented.

Fig. 9

Phase portraits for $\kappa =1.0$, $\delta =-0.15$, $b=-2.0$, $c=0$ and $A=20.0$ a on the $(x,\,y)$ plane b on the $(y,\,v)$ plane

The phase portrait of the bursting oscillations with codimension-2 bifurcations connecting the quiescent states (QSs) and spiking states (SPs) is greatly different from that of the bursting attractors with codimension-1 bifurcations between QSs and SPs. The trajectory may oscillate around two cycles, which may be caused by the super-Hopf bifurcation part included in the codimension-2 bifurcations (see Fig. 8a), while the fold bifurcation part included in the codimension-2 bifurcation may lead to the jumping phenomenon between the two oscillations around the two cycles (see Fig. 9b), which can also be demonstrated by the time history of x plotted in Fig. 10.

Fig. 10

Time history of x for one period

It can be found that the structures of two SPs are symmetric to each other because of the symmetry of the system. When spiking oscillations occur from a QS, which is almost strictly located on one equilibrium branch via a codimension-2 bifurcation, the trajectory does not try to settle down to another equilibrium branch, but to oscillate to an large-amplitude cycle with the increase of the amplitudes of the oscillations. Therefore, the codimension-2 bifurcation causes not only the jumping phenomenon, but also limit cycle oscillations. When the amplitudes of the oscillations increase to an extent, the trajectory may oscillate to settle down to another QS with gradual decrease of the amplitudes of the oscillations.

To reveal the mechanism of the bursting oscillations, we plot the related transformed phase portrait in Fig. 11. The trajectory, starting from the point ${A}_1$, moves along ${A}_1{A}_2{A}_3$ to form ${QS}_1$. A codimension-2 bifurcation causes the trajectory to jump to the repetitive spiking oscillations $({SP}_1)$. The amplitudes of the oscillations increase with the evolution of the non-dimensional time $\tau$ until the trajectory reaches the potential limit cycle. Then the amplitudes of the oscillations may decrease to settle down to ${QS}_2$. Similar situations occur when the trajectory returns from the maximum value of w at ${B}_1$.

Fig. 11

a Transformed phase portrait b and c Locally enlarged TPP

Fig. 12

a Overlap of the transformed phase portrait and the equilibrium branches b Locally enlarged of the overlap

Fig. 13

a Overlap of the transformed phase portrait and the equilibrium branches for the bursting oscillations with fifth order nonlinear term b Locally enlarged of the overlap

More clear mechanism for the bursting oscillations can be observed in Fig. 12, in which the overlap of the transformed phase portrait and the equilibrium branches is plotted. The trajectory, starting form the point ${A}_1$, moves almost strictly along $\textit{EB}_1$ to form ${QS}_1$ until it reaches the point ${B}_{-1}^{(1)}$, at which a codimension-2 bifurcation occurs, not only leading to the jumping phenomenon to the equilibrium branch $\textit{EB}_3$, but also causing the trajectory to approach the limit cycle via repetitive spiking $({SP}_1)$. The amplitude of the limit cycle, caused by super-Hopf bifurcation part included in the codimension-2 bifurcation, may increase with the increase of w, which leads to the increase of amplitudes of the spiking oscillations. Further increase of w may cause the gradual decrease of the amplitude of the oscillations, because of the attraction of the stable focus-type equilibrium branch $\textit{EB}_3$. The trajectory may gradually settle down to $\textit{EB}_3$ to form ${QS}_2$ and returns when w reaches its maximum value at $w=20.0$. Then the trajectory moves almost strictly along the equilibrium branch $\textit{EB}_3$, until it arrives at another codimension-2 bifurcation point ${B}_{+1}^{(1)}$. The bifurcation not only causes the jumping phenomenon from $\textit{EB}_3$ to $\textit{EB}_1$, but also leads to the repetitive spiking oscillations around potential limit cycle to form ${SP}_2$. When the trajectory settles down to $\textit{EB}_1$ and moves to the starting point ${A}_1$, one period of the bursting oscillations is finished.

4.3.2 Bursting oscillations with fifth-order nonlinear terms

When nonlinear terms up to fifth order are included in the function, another two codimension-2 bifurcation points can be observed on the equilibrium branches of the generalized autonomous system. Figure 13 gives the overlap of the transformed phase portrait and the equilibrium branches for the bursting oscillations with $\kappa =1.0$, $\delta =0.15$, $b=-2.0$, $c=\frac{4}{3}$ and $A=10.0$, while the phase portrait as well as the time history is a bit similar to those of the bursting oscillations in Figs. 8 and 9.

The trajectory, starting form the point ${A}_1$ in Fig. 13a, moves almost strictly along the equilibrium branch $\textit{EB}_1$ to form ${QS}_1$, until it arrives at the point ${B}_{-2}^{(1)}$, which is very close to the point ${B}_{-2}^{(2)}$ on $\textit{EB}_1$, at which a codimension-2 bifurcation occurs, causing the trajectory to jump to the stable focus-type equilibrium branch $\textit{EB}_3$. Note that the x-value of the point on $\textit{EB}_3$ is between the two codimension-2 bifurcation points ${B}_{-1}^{(2)}$ and ${B}_{+1}^{(2)}$, and the trajectory may oscillate to the potential cycle around $\textit{EB}_3$, starting the stage of ${SP}_1$. When the trajectory moves near the bifurcation point ${B}_{+1}^{(2)}$, it turns to oscillate to the potential limit cycle around $\textit{EB}_5$. When the amplitudes of the oscillations reach the maximum value, they may gradually decrease to settle down to $\textit{EB}$ to start the second stage of quiescent state ${QS}_2$. Then the trajectory moves almost strictly along $\textit{EB}_5$ to the point ${B}_1$, at which w reaches its maximum value. The trajectory may return along $\textit{EB}_5$. Similar phenomenon can be observed for pross for trajectory to return to the starting point ${A}_1$.

Since two different limit cycles around $\textit{EB}_3$ may appear from the codimension-2 bifurcation points ${B}_{-2}^{(1)}$ and ${B}_{+2}^{(1)}$, respectively, the trajectory cannot settle down to $\textit{EB}_3$. Similarly, two limit cycles around $\textit{EB}_1$ and $\textit{EB}_5$, respectively, may bifurcate from ${B}_{-1}^{(1)}$ and ${B}_{+1}^{(1)}$, which may lead to the alternation between the oscillations around $\textit{EB}_1$ and $\textit{EB}_3$ or between the oscillations around $\textit{EB}_3$ and $\textit{EB}_5$.

5 Conclusions

When an order gap exists between the periodically exciting frequency and the natural frequency, bursting oscillations alternating between quiescent states (QSs) and repetitive spiking states (SPs) may be observed. For the case when exciting frequency is far smaller than the natural frequency, the whole exciting term can be regarded as a small-varying parameter, leading to a so-called generalized autonomous system, which can be considered as a fast subsystem. The equilibrium states as well as the related bifurcations of the fast subsystem determine the forms of QSs and SPs as well as the bifurcations between QSs and SPs. Therefore, when higher-order nonlinear terms are included in the vector fields, more equilibrium points as well as the related bifurcations may involve the structure of the bursting attractors, which may lead to more complicated bursting oscillations. Furthermore, bifurcations with high codimension may result in complicated transitions between QSs and SPs, which may also cause the complexity of the bursting attractors.