Novel bursting patterns in a Van der pol-Duffing oscillator with slow varying external force

Mechanical Systems and Signal Processing 93 (2017) 164–174

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Novel bursting patterns in a Van der pol-Duffing oscillator with slow varying external force Yue Yu a,⇑, Min Zhao a, Zhengdi Zhang b a b

School of Science, Nantong University, Nantong 226007, PR China Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, PR China

a r t i c l e

i n f o

Article history: Received 6 October 2016 Received in revised form 2 January 2017 Accepted 27 January 2017

Keywords: Bursting oscillations Multi-stable system Codimension two bifurcations Time varying external force

a b s t r a c t In this paper, we investigate the emergence of bursting dynamics with complex waveforms and their relation to periodic behavior in typical Van der pol-Duffing equation with fifth order polynomial stiffness nonlinearity, when the external force changes slowly with the variation of time. We exploit bifurcation characteristics of the fast subsystem using the slowly changing periodic excitation as a bifurcation parameter to show how the bursting oscillations are created in this model. We also identify that some regimes of bursting patterns are related to codimension two bifurcation type over a wide range of parameters. A subsequent two-parameter continuation reveals a transition in the bursting behavior from fold/fold hysteresis cycle to sup-Hopf/sup-Hopf or limit point cycle/sub-Hopf bursting type. Furthermore, the effects of external forcing item on bursting oscillations are investigated. For instance, the time interval between two adjacent spikes of bursting oscillations is dependent on the forcing frequency. Some numerical simulations are included to illustrate the validity of our study. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction In the past several decades, the study of bursting oscillations have received great attention. Bursting oscillations are waveforms that consist of alternating small and large amplitude excursions. Such oscillations have been reported in both experiments and models of systems from chemistry, biochemistry, physics and neuroscience [1–5]. In fact, bursting oscillations are often observed to occur in signal transduction processes and associated with many biological phenomena, such as oscillatory enzyme responses [6], insulin production in the b-cell [7] and veratridine in rat injured sciatic nerves [8,9]. For a fast-slow dynamic, its dynamical behaviors can be described by a singularly perturbed system with two time scales of the following form [10–12],

x_ ¼ f ðx; uÞ ðFast SpikingÞ u_ ¼ egðx; uÞ ðSlow ModulationÞ

ð1Þ

where e  1 represents the ratio of time scales between spiking and modulation. Vector x 2 Rn models the dynamics of a relatively fast changing processes, while vector u 2 Rk describes the relatively slowly changing quantity that modulates x. A standard method of analysis of birsting oscillations, introduced by Rinzel et al. [13–15], is to set e ¼ 0 and consider the ⇑ Corresponding author. E-mail address: [email protected] (Y. Yu). http://dx.doi.org/10.1016/j.ymssp.2017.01.044 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

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fast and the slow subsystems separately, which is known as the dissection of neuronal bursting since it allows us to study the fast subsystem x and treat u as a vector of slowly changing bifurcation parameters. Van der Pol-Duffing equation, which is known to describe many important oscillating phenomena in nonlinear engineering systems, has become one of the commonest examples in nonlinear oscillation texts and research articles [16–20]. Many efforts have been made to find its approximate solutions or to construct simple maps that qualitatively describe the important features of its dynamics. In the classical Van der Pol-Duffing oscillator, it was shown that the oscillations occur when a stable equilibrium undergoes the singularity induced bifurcation, which in turn corresponds to the occurrence of supercritical Hopf bifurcations in the singularly perturbed models [21–24]. In our recent papers, using Rinzel’s method, we investigated dynamical behaviors of some multiple coupled systems with form of Eq. (1) and found some novel bursting patterns [25–28]. Here we will present an analysis of bursting oscillations driven by external forcing for Van der Pol-Duffing equation. Bursting oscillations to be studied in this paper are different from the usual fast-slow bursters since the trajectories are tumbling around the multiple equilibria and driven by external forcing. The purpose of this work is two fold. First, we study bursting oscillations induced by external forcing as well as how the external forcing modulates the bursting dynamics. Secondly, some important aspects of Rinzel’s method are highlighted in the multi-stable oscillator. In particular, we argue the particular constellation of a subcritical Hopf bifurcation together with a limit point cycle bifurcation. The rest of this paper is organized as follows. In the next section, the forced equation and its fast-slow analysis are presented. In Section 3, in order to investigate bursting oscillations, we consider the external force f cosðxtÞ as a slow variable and study its codimension one and two bifurcation on the controlled Van der Pol-Duffing equation as well as its influence. In Section 4, two strategies are developed to reveal the dynamical mechanisms of bursting oscillations. In Section 5, we focus on the effects of forcing amplitude and forcing frequency on bursting oscillations. Finally, Section 6 concludes the paper. 2. Mathematical model and its slow manifold In this paper, we consider the classical Van der Pol-Duffing equation with fifth order polynomial stiffness nonlinearity and periodic excitation, which is often written in the form

€x þ dðx2  1Þx_ þ a1 x  a2 x3 þ a3 x5 ¼ f cosðxtÞ;

ð2Þ

where d ðd > 0Þ can be regarded as dissipation or damping, a1 is linear stiffness parameter, a2 and a3 mean nonlinear stiffness, and parameters f ðf > 0Þ is the forcing amplitude and x ðx > 0Þ is the forcing frequency. This model exhibits many qualitatively different phenomena and is considered as one of the most intensely studied nonlinear systems and has served as a basic model in physics, electronics, biology, neurology and so on. Since we are interested in the case when fcosðxtÞ changes slowly, we assume 0 < x  1 here which implies the excited frequency x is far small from the natural frequency X of the oscillator. The effects of multiple time scales appear, which results in different types of bursting oscillations under different parameters. The oscillatory behaviors of bursting behave in periodic states characterized by a combination of relatively large amplitude (spiking process) and nearly harmonic small amplitude oscillations (quasi-stationary process), conventionally denoted by N k where N and K correspond to large and small amplitude oscillations respectively. The external excited oscillator can be considered as the coupling of two autonomous subsystems by regarding fcosðxtÞ as a generalized state variable q, the fast subsystem (FS) of which is presented as the following equation

€x þ dðx2  1Þx_ þ a1 x  a2 x3 þ a3 x5 ¼ q;

ð3Þ

while the slow manifold is written as q ¼ fcosðxtÞ. It can be checked that q_ ¼ f xsinðxtÞ, which forms the slow manifold for the far small value of x  1. In the following, two important bifurcation behaviors of the FS can be discussed. One corresponds to bifurcation from a quiescent state to repetitive spiking mode, while the other will induce the spiking attractor back to the quiescent state. 3. Bifurcation analysis for the fast subsystem Inspired by the dissection of neuronal bursting [29], we begin our study of the mechanism responsible for bursting oscillations by considering the slow variable q as a control parameter and studying its influence on the controlled Van der PolDuffing oscillator, which may help to understand bursting oscillations induced by the stability and bifurcation dynamics in the fast subsystem. 3.1. Analysis for Cusp bifurcation _ ¼ ðx0 ; 0Þ, where x0 is decided by the real Simple in form as it is, the equilibria of FS can be written in the form ðx; yÞ  ðx; xÞ roots of the following algebraic equation

a1 x þ a2 x3  a3 x5 þ q ¼ 0:

ð4Þ

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The number and stabilities of these equilibrium points are determined by the values of the parameters of FS. For the fixed parameters a2 ¼ 4 and a3 ¼ 4, double parameter bifurcation set of the FS related to Eq. (3) is analyzed and plotted in Fig. 1, where CP 1 ¼ ð0:52; 1:80Þ; CP 2 ¼ ð0:52; 1:80Þ are subcritical Cusp bifurcations and CP 3 ¼ ð0; 0Þ is supercritical Cusp bifurcation. The Cusp is a saddle-node bifurcation in which there is a degeneracy in the quadratic terms. This codimension two bifurcation involves the transition from one steady-state to three. With the influence of fifth nonlinear force stiffness, the Cusp in Eq. (3) is associated with bifurcations along which five equilibria exist in area I, II and III corresponding to different transition modes (see in Fig. 1). Note that the equilibrium points of the fast subsystem may change from five to one with the variation of the parameters. Small perturbation of the parameters may cause the degenerate equilibrium point to disappear or to split into two different types corresponding to fold bifurcations. Now we focus on the parameter area A ¼ fða1 ; qÞ j 0 < a1 < 1:8; q 2 R g, which can be classified as the upper subarea (transition mode I) AI ¼ fða1 ; qÞ j 1 < a1 < 1:8; q 2 R g, the middle subarea (transition mode II) AII ¼ fða1 ; qÞ j 0:8 < a1 < 1; q 2 Rg and the lower area (transition mode III) AIII ¼ fða1 ; qÞ j 0 < a1 < 0:8; q 2 Rg. Fix a1 at each subarea and slowly vary q from left to right, then some typical dynamics can be presented. We take the case as an example to illustrate the multiple stable dynamical structure, i.e., the case when we fix the parameters at q ¼ 0; a1 ¼ 2, a2 ¼ 3, and a3 ¼ 1. In this case, there are five equilibria, i.e., the trivial equilibrium of E0 ¼ ð0; 0Þ, two pffiffiffi sets of symmetric nontrivial equilibria of E1 ¼ ð1; 0Þ and E2 ¼ ð 2; 0Þ. The corresponding characteristic equations can be computed, which gives the following conclusions. E0 and E2 are sinks, while E1 are saddles, implying that the FS has multiple stable structure (coexistence of stable attractors). In particular, for the chosen parameters, E0 and E2 are stable foci. As q increases (decreases) from the initial state, the saddles E1 become gradually far from the sink E0 and approach the other sink Eþ2 (E2 ). When q increases (decreases) through the critical points, the equilibria E0 and Eþ1 (E1 ) collide and disappear by a fold bifurcation of equilibria, leaving Eþ2 ðE2 Þ, the only attractor of the system, on the phase plane. The FS then loses its multiple stability and becomes monostable. 3.2. Analysis for BT bifurcation In this subsection, we investigate the existence of the Bogdanov-Takens bifurcation (BT bifurcation), i.e., a typical type of codimension two bifurcation which is obtained by investigating the tangent points of the Hopf and saddle (or saddle-node) bifurcation curves. When a corresponding linearizable form of a system has a double-zero eigenvalue and geometric multiplicity one, the system is considered to exhibit BT bifurcation. This bifurcation is of particular importance for the theory of dynamical system because of providing the local existence of homoclinic cycle curves. Taking one untrivial equilibrium point E ¼ ðx0 ; 0Þ for an explanation and linearizing the FS at the equilibrium point E leads to the Jacobian matrix, expressed by



J¼

0 a1 þ 3a2 x20  5a3 x40

1 dðx20  1Þ



ð5Þ

The corresponding characteristic equation of Eq. (3) can be obtained from the equation of detðkI  JÞ ¼ 0 as

FðkÞ ¼ k2 þ dðx20  1Þk þ a1  3a2 x20 þ 5a3 x40 ¼ 0:

ð6Þ

2

CP 1.5

CP 2

1

transition mode I

α1

1

0.5

transition mode II transition mode III

0

CP 3 −0.5 −0.8 −0.6 −0.4 −0.2

0

ρ

0.2

0.4

0.6

0.8

Fig. 1. Cusp bifurcation diagram of FS with respect to the two parameters of a1 and q for the fixed parameters a2 ¼ 4 and a3 ¼ 4.

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It is easy to see that k ¼ 0 is a zero root of the characteristic Eq. (6) when the parameters of the oscillator satisfy the following equation

a1  3a2 x20 þ 5a3 x40 ¼ 0:

ð7Þ

The derivative of the characteristic equation FðkÞ with respect to k at the point of k ¼ 0 is given by

 dFðkÞ ¼ dðx20  1Þ ¼ 0: dk k¼0

ð8Þ

Furthermore, Eq. (6) has a double zero root, namely a zero eigenvalue with multiplicity two if Eqs. (7) and (8) hold simultaneously. For the characteristic equation of a linearized system has a zero eigenvalue with multiplicity two, a BT bifurcation occurs in the original nonlinear system, i.e., the FS has a BT bifurcation at the critical point of intersection associated with Eqs. (7) and (8). By a suitable blowup transformation, we can construct an approximation for the homoclinic orbits in the BT normal form and then transfer the normal form with the resulted homoclinic approximation to the original system. Then, a branch of saddle-homoclinic cycles can emanate from the given BT point. We note the FS has a certain symmetry: if ðxðtÞ; yðtÞÞ is a solution for given parameter values, then ðxðtÞ; yðtÞÞ is a solution when the parameter values become the opposite. Thus, two symmetric BT points can be found. Take an example for the fixed parameters a2 ¼ 4 and a3 ¼ 4, as shown in Fig. 3(a), one point BT 1 are found for a1 ¼ q ¼ 8 with normal form coefficients ð14; 2Þ, at which the fold bifurcation curve F 1 and Hopf bifurcation curve H1 interest. The procedure of homoclinic orbits that appear near singular points of BT bifurcation can be illustrated as follows. In a neighborhood of this codimension two point, the equilibrium grows out of nothing at first, and then bifurcates into a limit cycle via a Hopf bifurcation and disappears at last. In Fig. 3(a), we present the computed homoclinic orbit bifurcation ðHomo1 Þ to hyperbolic saddle Eþ1 in parameter space, while the corresponding homoclinic-to-saddle orbits in state space are illustrated in Fig. 3(b). 4. Bifurcation mechanism for bursting oscillations Codimension two bifurcation points can be the source of more complicated dynamics such as multistability, quasiperiodicity or chaos. When the parameters approach the critical values associated with codimension two bifurcations, bursting dynamic can be classified as well as the bifurcation mechanism of steady state which transits from the quiescence process to the repetitive spiking process. Based on bifurcation results above, we will take suitable parameter values to study the bursting phenomena, where the bifurcation diagram of FS can be projected onto phase space to understand the associated bursting mechanisms. Since different value of d will lead to a qualitative difference in the bifurcation behavior, we consider the case of d ¼ 0:05 in Section 4.1, while other cases of d ¼ 0:5 will be discussed in Sections 4.2 and 4.3 respectively. 4.1. Different symmetric hysteresis cycles around multiple equilibria We have given a classification of the transition modes near the Cusp bifurcations, i.e., the multiple stable case corresponding to transition mode I-III. A hysteresis cycle can be obtained by a slowly changing external forcing which periodically passes through fold bifurcation values. Consider the case of transition mode I (1 < a1 < 1:8), for a1 ¼ 1:2, as shown in Fig. 4(a), the trajectory slowly and periodically passes through the fold bifurcation points, and the system undergoes the transition from the stable focus E0 to E2 . Thus a hysteresis cycle with sharp wrap is created.

Fig. 2. Multiple stable symmetric structure (multiple symmetric equilibrium attractors) of the FS separated by saddles for the fixing parameters at a1 ¼ 2, a2 ¼ 3; a3 ¼ 1 and q ¼ 0.

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−3

8

(a)

F

(b) 1

6

−4

H

4

1

−5 2 1

y

α1

Homo −6

0 −2

−7

BT 1

−4

−8 −8

−7

−6

ρ

−5

−4

−3

−6 −2

−1

0

x

1

2

Fig. 3. Numerical simulations diagrams of FS for the fixed parameters a2 ¼ 4; a3 ¼ 4. (a) BT bifurcation simulation diagram with respect to two parameters of a1 and q. (b) Phase diagram of ‘‘growing” limit cycles near the BT bifurcation.

1

3

(a)

(b) 2

0.5 0

0 E−2

E

+2

y

y

1

E

0 E −2

E+2

−1 −0.5 −2 −1

−1.5

−1

−0.5

0

0.5

1

1.5

−3 −2

x

−1

0

1

2

x

Fig. 4. Bursting oscillations related to Eq. (2) formed by the switching among the multiple attractors for a2 ¼ a3 ¼ 4, f ¼ 2 and x ¼ 0:01. (a) Hysteresis cycles with four wraps at a1 ¼ 1:2; (b) hysteresis cycles with two wraps at a1 ¼ 0:5.

The dynamics of the hysteresis cycle can be explained as follows. The FS system is controlled by the excitation term of q and the trajectory tracks along the left attractor traces of equilibrium E2 which is corresponding to the quiescent state until it arrives the fold bifurcation. Because of the slow passage effect (hysteresis) and fold bifurcation, the fast state variable jumps to the middle stable branch of equilibrium E0 . The trajectory moves along with stable traces of equilibrium E0 corresponding to spiking state. Similarly, when the trajectory passes through the other symmetric fold point, the system jumps to the right stable attractor of Eþ2 thereby completing half the bursting loop. Bursting oscillations are generated since the system periodically switches between the stable branches around E0 and E2 . The cases of transition mode II and III are simple, since when the fold bifurcation happens, the FS system has no choice but to jump to the attractors of both sides around E2 for the system trajectories spin out of the control of the attractor of trivial equilibrium E0 of Eq. (2). Therefore, by a proper excitation item, a hysteresis cycle just around the equilibria E2 may be created. The associated hysteresis cycle is illustrated by Fig. 4(b) for a1 ¼ 0:5. To get a clear idea of the role of slow manifold, we plot the ‘‘transformed phase diagram”, that is, the evolution process of Eq. (2) with respect to the slow valuable q, and overlay it with the bifurcation diagram of FS to detect the associated bifurcation mechanisms. In Fig. 5 for the fixed parameters of for a1 ¼ 0:9; a2 ¼ a3 ¼ 4; x ¼ 0:01 and f ¼ 3, such hysteresis cycle on the phase space of ðq; xÞ is overlayed with the bifurcation diagram, at which the jumps at the fold bifurcation points F 1;2 between different stable branches are presented clearly.

4.2. Symmetric sup-Hopf bifurcation bursting patterns via hysteresis cycles According BT bifurcation theory, the equilibrium grows out of nothing at first, and then, it bifurcates into a limit cycle via a Hopf bifurcation. Among these processes, if the Hopf bifurcation is supercritical (sup-Hopf), there must be a stable limit

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1.5 1

F

2

0.5

x

0

−0.5

F

1

−1 −1.5

−3

−2

−1

0

1

2

3

ρ Fig. 5. The transformed phase diagram of hysteresis cycles of FS for a1 ¼ 0:9, a2 ¼ a3 ¼ 4, x ¼ 0:01 and f ¼ 3, where F 1;2 are fold bifurcation points.

cycle bifurcated from the equilibrium, which leads to another type of bursting behaviors. Oscillations of this type in a quiescent state lose their stability via sup-Hopf bifurcation and approach to the limit cycle attractor, which comes into the repetitive spiking mode. After that, the spiking shrinks to the equilibrium also via sup-Hopf bifurcation. Transitions between quiescent state and spiking state always occur via sup-H bifurcations. For the fixed parameters d ¼ 0:5; a1 ¼ 1:2, a2 ¼ 4 and a3 ¼ 4, the parameter bifurcation set with respect to slowly excited item q is plotted in Fig. 6, at which the different attractors bifurcated by symmetric fold bifurcations F 1;2;3;4 , and supercritical Hopf bifurcations supH1;2 . The solid point curve LC 1;2 correspond to the maximum and minimum amplitudes of the stable oscillatory solutions bifurcated from supH1;2 . When we fix the parameters at d ¼ 0:5; a1 ¼ 1:2, a2 ¼ 4; a3 ¼ 4; x ¼ 0:001 and f ¼ 4, the corresponding phase portraits as well as the time series are presented in Fig. 7. The trajectory slowly and periodically passes through the fold bifurcations, sup-Hopf bifurcations, and the system undergoes the transition between the two stable limit cycles around the nontrivial equilibriums on two sides via fold-fold hysteresis. In Fig. 8, the transformed phase diagram of this cycle bursting is overlayed with the bifurcation diagram to get a clear understanding of the dynamical mechanism, at which the jumps between different stable branches are presented clearly. As shown in Fig. 8, the FS system is controlled by the excitation term of q and the trajectory tracks along the left attractor traces of equilibrium E2 which is corresponding to the quiescent state until it arrives the sup-Hopf bifurcation. Because of the occurrence of stable cycles near the equilibrium of E2 , the trajectories begin to oscillate in small amplitudes. When the q arrives the fold bifurcation, the state variable jumps to the middle stable branch of equilibrium E0 leading the trajectory

2

stable limit cycle

supH 2

x

1

F3

0

F2 F1

LC 1

−1

LC 2

F4

supH 1

stable limit cycle −2 −4

−2

0

ρ

2

4

Fig. 6. The stability and bifurcation diagram of FS related to slow variable q for the fixed parameters d ¼ 0:5, a1 ¼ 1:2; a2 ¼ 4 and a3 ¼ 4.

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Y. Yu et al. / Mechanical Systems and Signal Processing 93 (2017) 164–174

2

2

(a)

(b)

1.5 1

1

0

y

y

0.5 0

−0.5 −1

−1

−1.5 −2

−1.5

−1

−0.5

0

x

0.5

1

−2 2000

1.5

4000

6000

x

8000

10000

12000

Fig. 7. Symmetric sup-Hopf/sup-Hopf bursting oscillations via hysteresis cycles related to Eq. (2) for d ¼ 0:5, a1 ¼ 1:2; a2 ¼ 4; a3 ¼ 4; x ¼ 0:001 and f ¼ 4. (a) Phase diagram. (b) Time series diagram.

2 stable limit cycle

supH

2

1

LC

F

2

4

x

F

3

0

F

2

LC

F

1

−1

1

supH stable limit cycle

−2 −4

−2

1

0

2

4

ρ Fig. 8. The transformed phase diagram of symmetric ‘‘sup-Hopf/sup-Hopf” bifurcation bursting corresponding to Fig. 7.

moving along with stable traces of equilibrium E0 . Similarly, when the trajectory passes through the other symmetric fold and sup-Hopf bifurcation points, the system jumps to the right stable limit cycle attractor around Eþ2 thereby completing half the bursting loop. Bursting oscillations are generated since the system periodically switches between the stable limit cycles around E2 . 4.3. Symmetric ‘‘limit point cycle/sub-Hopf” bifurcation bursting patterns via hysteresis cycles Bogdanov-Takens bifurcation can give rise to the appearance of a homoclinic cycle orbit for nearby parameter values. Based on the conclusion we have analyzed for the characteristic equation and BT bifurcations about the FS above, we can verify the occurrence about limit point cycle bifurcation (LPC), i.e., there is a bifurcation curve at which the fold bifurcation of limit cycles (fold-cycle bifurcation) takes place, where a stable and an unstable limit cycle collide, forming a nonhyperbolic cycle with multiplier of 1 near this bifurcation. For the fixed parameters d ¼ 0:5; a1 ¼ 0:9, a2 ¼ 4 and a3 ¼ 4, the parameter bifurcation set with respect to slowly excited item q is plotted in Fig. 9, at which the different attractors bifurcated by symmetric fold bifurcations F 1;2 , subcritical Hopf bifurcations subH1;2 and limit point cycle bifurcations LPC 1;2 . The solid point curve A1;2 correspond to the maximum and minimum amplitudes of the stable oscillatory solutions bifurcated from fold bifurcation of limit point cycle LPC 1;2 , while the hollow point curve LC 1;2 correspond to the unstable limit cycles amplitudes bifurcated from subH1;2 . Further numerical investigation shows that the two cycles LC 1;2 grow in size until they respectively collide with another limit cycle (homoclinic orbits to saddle E1 ) and combine into one limit cycle with large amplitude (LPC bifurcation). According to the associated characteristic equations, one can conclude that: E2 is stable for jqj > 0:9. The stability of E2 changes by subcritical Hopf bifurcation at the critical value subH1;2 ¼ 0:9, and the stable limit cycle bifurcated from limit point cycle

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A1

2 1.5

LPC 2 F2

1

subH 2

0.5

LC 2

x

0

LC 1

−0.5

F1

subH 1

−1 −1.5

LPC 1

E −2

A2

−2 −6

E +2

−4

−2

0

2

4

6

ρ Fig. 9. The stability and bifurcation diagram of FS related to slow variable q for the fixed parameters d ¼ 0:5, a1 ¼ 0:9; a2 ¼ 4 and a3 ¼ 4.

6

(a)

6

(b)

4 4 2

E

0

E

−2

y

y

2

+2

0 −2

−2

−4

−4

−6 −6

−2

−1

0

x

1

2

2400

2600

2800

3000

3200

3400

t

Fig. 10. Symmetric Limit point cycle/sub-Hopf bursting oscillations via hysteresis cycles related to Eq. (2) for d ¼ 0:5, a1 ¼ 0:9; a2 ¼ 4; a3 ¼ 4; x ¼ 0:01 and f ¼ 5. (a) Phase diagram. (b) Time series diagram.

bifurcation (LPC 1;2 ) is then created. Next we investigate the generation of bursting patterns related to limit point cycle/subHopf bifurcation behaviors, which exhibit a remarkable global bursting dynamical evolution behaviors (see in Fig. 10). When we fix the parameters at d ¼ 0:5; a1 ¼ 0:9, a2 ¼ 4; a3 ¼ 4; x ¼ 0:01 and f ¼ 5, the corresponding phase portraits as well as the time series are presented in Fig. 10. The trajectory slowly and periodically passes through the fold bifurcations of limit cycles, and the system undergoes the transition between the two stable wide range limit cycles via fold-fold hysteresis. In Fig. 11, the transformed phase diagram of this cycle bursting is overlayed with the bifurcation diagram to get a clear understanding of the dynamical mechanism, at which the jumps between different stable branches are presented clearly. The specific mechanism of such novel patterns can be explained as follows. As the multiple stable system (see in Fig. 2), the system has three stable equilibria and two unstable saddles. Meanwhile, with the variation of slow variable q, the stable branch approaches to the unstable saddle, where the fold bifurcation occurs, which starts the trajectory to oscillate with large amplitude, i.e., the trajectory evolves into the spiking process. Such dynamical behaviors will not disappear until the two coexisting limit point cycles occur induced by the LPC bifurcations, which leads to repetitive more complex spiking. With the slow variation of q, the stable wide range limit cycle terminates out of the spiking process and then decreases to settle down to the stable focus E2 , which forms the quiescent process.Similar situation takes place back to the initial stable branch of the attractor, which completing the rest half of the bursting loop. 5. Effects of excitation amplitude and excitation frequency on such bursting dynamics We have investigated the dynamical mechanisms of bursting oscillations. In this section, we focus on the effects of forcing amplitude and frequency on such bursting oscillations.

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2

LPC cycle attractor

1.5

E +2

1 0.5

x

0

−0.5 −1 −1.5

E−2 LPC cycle attractor

−2 −6

−4

−2

0

2

ρ

4

6

Fig. 11. The transformed phase diagram of symmetric ‘‘LPC/sub-Hopf” bifurcation bursting corresponding to Fig. 10.

First, the effect of forcing amplitude on bursting oscillations is considered. As described by the second strategy, bursting oscillations are created since external forcing passes through fold bifurcation values periodically. So, when the forcing amplitude f is below certain threshold value, bursting oscillations do not occur because the external force is not able to pass through the fold bifurcations, while if it is more than the threshold, they may be created.

8

(a) 4

symmetric burster d=0

0 −4 −8 750

800

850

900

950

1000

1050

1100

1150

1200

1250

1100

1150

1200

1250

1150

1200

1250

8

(b) break down to symmetry

4

d=1

y

0 −4 −8 750

800

850

900

950

1000

1050

8

(c)

merge into one large burster

4

d=1.5 0 −4 −8 750

800

850

900

950

1000

1050

1100

t Fig. 12. The evolution processes of break down to symmetry bursting with the transition of d, for d ¼ 0:5; a1 ¼ 0:9, a2 ¼ 4; a3 ¼ 4, and f ¼ 5. (a) Symmetric bursting for d ¼ 0. (b) Break down to symmetry for d ¼ 1. (c) Merge into one unique burster for d ¼ 1:5.

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We would like to point out that if the external force destroys the symmetry of bifurcations in Van der Pol-Duffing oscillator, then unequal time intervals between two adjacent spikes are obtained. For example, we consider the following equation:

€x þ dðx2  1Þx_ þ a1 x  a2 x3 þ a3 x5 ¼ f cosðxtÞ þ d;

ð9Þ

where d is a constant controller. When d ¼ 0, i.e., f cosðxtÞ þ d is able to pass through the critical bifurcation values symmetrically, according to the above analysis, system (2) may exhibit bursting oscillations (see Fig. 12(a)). When d > 0, FS is not symmetric with respect to d anymore, i.e., the control parameter of f cosðxtÞ þ d is no longer symmetric about the trivial equilibrium. Thus this type of asymmetric and folded equilibria may occur asymmetric bursting oscillations, where the burster has unequal time intervals (see in Fig. 12(b)). Further research can show that with the increasing of the translation amplitude of d, the two LPC attractors will interact with each other and form the unique spikes (see in Fig. 12(c)). Secondly, the effect of force frequency on bursting oscillations is considered. As a kind of coupling vibration system, the frequency of the Eq. (2) should be equal to the frequency of the two subsystems corresponding to the FS of Eq. (3) and the slow subsystem of q ¼ fcosðxtÞ respectively. Thus, the bursting dynamic of the whole fast-slow coupling system has the frequency of x. Furthermore, we can conclude the time intervals between two symmetric adjacent spikes of bursting oscillap . When x ¼ 0:005; 0:01 and 0.02, we have DT ¼ 200p; 100p and 50p, respectively. Take the waveforms in tions is DT ¼ x Fig. 10 as an example, when x ¼ 0:005; 0:01 and 0:02 respectively, the corresponding time intervals between two adjacent spikes are measured and shown in Fig. 13. Furthermore, we can discuss the thresholds of forcing frequency on bursting oscillations of symmetric ‘‘limit point cycle/ sub-Hopf” type. Because such bursting oscillations are reflected by two attractors (fold cycles in FS), we should regulate the period of bursting oscillation (periodic solution) to contain two fold cycle period, so that the bursting dynamics can reflect the dynamical properties of two fold cycles. On the other extreme, the bursting oscillations will disappear for the oscillator p
6 3

(a)

0

ΔT≈628

−3 −6 1.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4 x 10

4

6 3

y

(b)

0

ΔT≈314

−3 −6 1.2

1.21

1.22

1.23

1.24

1.25

1.26

1.27

1.28

1.29

1.3 x 10

6

4

(c)

3 0

ΔT≈157

−3 −6 1.2

1.21

1.22

1.23

t

1.24

1.25

1.26 x 10

4

Fig. 13. Numerical results of time interval between two adjacent spikes of bursting dynamics corresponding to Fig. 10 for the external force frequency x ¼ 0:005; 0:01; 0:02 respectively. (a) DT  628; (b) DT  314; (c) DT  157.

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Y. Yu et al. / Mechanical Systems and Signal Processing 93 (2017) 164–174

6. Conclusion Van der Pol-Duffing’s equation may exhibit bursting oscillations with different waveforms under certain external forcing conditions. A theoretical analysis of bursting oscillations induced by external forcing has been described to investigate the associated dynamical mechanisms. We first consider the external forcing as a manifold (slow control parameter) and study its influence on the controlled system. Then two strategies are developed to account for bursting oscillations: one is to overlay the projection of the FS system’s bifurcation diagram with bursting trajectory and the other is to introduce the local bifurcation characteristic of this multiple stability structure. We believe that the strategies are applicable to other dynamical systems whose bursting oscillations are induced by external forcing. Moreover, our analysis presents a quantitative investigation of bursting oscillations for Van der Pol-Duffing’s equation. 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