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Citation: Zhang, D.; Qian, Y. Bursting
Oscillations in General Coupled
Systems: A Review. Mathematics 2023,
11, 1690. https://doi.org/10.3390/
math11071690
Academic Editor: Rami
Ahmad El-Nabulsi
Received: 28 February 2023
Revised: 26 March 2023
Accepted: 30 March 2023
Published: 1 April 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
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4.0/).
mathematics
Review
Bursting Oscillations in General Coupled Systems: A Review
Danjin Zhang and Youhua Qian *
School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China; zhangdj@zjnu.edu.cn
*Correspondence: qyh2004@zjnu.edu.cn
Abstract:
In this paper, the bursting oscillation phenomenon in coupled systems with two time scales
is introduced. Firstly, several types of bifurcation are briefly introduced: fold bifurcation, Hopf
bifurcation, fold limit cycle bifurcation, homoclinic bifurcation, etc. The bursting oscillations of the
system with two excitation terms and the bifurcation delay in the bursting oscillations are considered.
Secondly, some simple bursting oscillations are introduced, such as fold/fold bursting, fold/supHopf
bursting, subHopf/subHopf bursting, fold/LPC bursting, Hopf/LPC bursting, fold/homoclinic
bursting, Hopf/homoclinic bursting, etc. At the same time, the system also has some complex
bursting oscillations, such as asymmetric bursting, delayed bursting, bursting with hysteresis loop,
etc. Finally, the practical applications of bursting oscillations, such as dynamic vibration absorbers
and nonlinear vibration energy harvesting technology, are introduced.
Keywords:
bursting oscillation; coupled system; bifurcation delay; fast–slow dynamic analysis
method; vibration energy harvesting
MSC: 34F10; 34F99
1. Introduction
As a type of system with special structures, coupled systems at different time scales [
1
]
have a wide-ranging background in practical engineering involving various fields of
science as well as engineering technology. At the same time, due to the coupling of different
time scales, the interaction of behaviors on different time scales will lead to many special
nonlinear phenomena. Its associated work has been highly attached by scholars all over
the world and has become one of the hot topics in nonlinear research. Coupling at different
time scales means that in the established mathematical model, state variables or some
combination of them can be divided into several different groups, among which there are
obvious differences in magnitude in the rate of change with time. In terms of dimensionless
mathematical models, coupled systems of different time scales can be divided into two
types. One is in the time domain; that is, there are vector field coupling of different orders
of magnitude. The other is in the frequency domain; that is, there is an order of magnitude
difference between the different frequencies of the coupled system. However, these two
types of systems are essentially the same [
2
]. The factors leading to the coupling of different
time scales can not only come from the fast–slow effect of the real time [
3
] but also from the
scale effect of geometry [
4
], as well as internal physical effects, system structure effects, and
so on. Thus, in the established mathematical model, state variables are divided into several
different groups, and the rate of change of state variables with time among each group is
significantly different in magnitude [
5
]. A two-time-scale coupled dynamic system is an
important part of a multi-time-scale coupled dynamic system. So far, a lot of research has
been conducted, and abundant results have been achieved on the fast–slow two-time-scale
coupled system, but due to the particularity of this kind of system, its related research is
still in the development stage, and many problems need to be further studied.
Many coupled systems at different time scales can produce complex behavioral modes,
and bursting oscillation is one of the typical dynamic behaviors. The fast–slow structure
Mathematics 2023,11, 1690. https://doi.org/10.3390/math11071690 https://www.mdpi.com/journal/mathematics
Mathematics 2023,11, 1690 2 of 16
plays an important role in the generation mechanism of the bursting oscillation. Kuehn [
1
]
defined the bursting oscillation as follows: if the system has a periodic orbit, and the time
series of the periodic orbit (at least one state variable or some coordinate scales) alternates
between fast oscillation and near-stable behavior, then the system is said to have a bursting
oscillation.
The study of the dynamic behavior of coupled systems at different time scales can be
traced back to the relaxation oscillation found in the Van der Pol oscillation equation [
6
], but
it did not receive much attention at that time. It was not until Nobel Prize winners Hodgkin
and Huxley established a two-time-scale neuron model (H–H model) and successfully
reproduced the firing behavior of neurons observed in the experiment that the coupled
problem at different time scales gradually attracted scholars’ attention [
7
]. Since then,
how to deal with the complex dynamic behavior and the inducing mechanism of coupled
systems at different time scales has been one of the topics puzzling scholars.
Due to the lack of effective analytical methods, most of the early relevant work focused
on approximate solutions, such as the quasi-static method [
8
] and the singular perturbation
method [
9
]. Although these methods could well approximate the exact solution of the
system, they could not explain the interaction between different scales. Therefore, geometric
singular perturbation theory [
10
] was proposed, whose main idea is to decompose the
system into two coupled fast and slow subsystems by means of invariant manifold theory
and obtain the dynamic behaviors and properties of the original system by analyzing the
dynamic behaviors and properties of the limiting fast and slow subsystems. However,
this method cannot deal with other types of attractors (such as limit cycles and chaotic)
bifurcations in coupled systems of different time scales.
It was not until Rinzel [
11
] proposed the fast–slow dynamic analysis method, which
revealed the induction mechanism of bursting oscillation in coupled systems with different
time scales, and pointed out that the essential cause of bursting oscillation is the mutual
transition between the system’s quiescent state and spiking state. The method provides
an effective tool for understanding the nature and induction mechanism of bursting oscil-
lations. The core idea of the fast–slow dynamic analysis method [
12
] is to take the slow
variable as the bifurcation parameter of a fast subsystem, use classical bifurcation theory to
analyze the bifurcation mechanism of the attractor of the fast subsystem changing with the
slow variable, and then analyze the interaction between different scales. With the change in
the slow variable, the fast subsystem has different motion modes, such as the equilibrium
point or the small amplitude limit cycle corresponding to the quiescent state and the large
amplitude limit cycle corresponding to the spiking state. When the slow variable changes
slowly and passes through the parameter domain of different vibration modes of the fast
subsystem, it will also cause the fast subsystem to transform back and forth between differ-
ent motion modes, resulting in the bursting oscillation of the coupled system. Therefore,
in the bursting phenomenon, with the slow variable as the bifurcation parameter, the fast
subsystem will generate two important bifurcation behaviors, namely the bifurcation from
the quiescent state to the spiking state and the bifurcation from the spiking state back to the
quiescent state (Figure 1). The introduction of the fast–slow dynamic analysis method can
explain the bifurcation connection between the spiking state and quiescent state and reveal
the generation mechanism of bursting oscillation. Based on this method, various oscillation
modes of bursting are obtained, such as fold/fold bursting, fold/Hopf bursting, etc.
Izhikevich improved Rinzel’s classification method based on the geometric shape
of the bursting and proposed a classification method based on the switching bifurcation
mode between the quiescent state and the spiking state. According to the difference in
bifurcation modes switching between quiescent state and spiking state, Izhikevich classified
all possible bursting oscillations in which only codimension one bifurcation occurred in
the fast subsystem [
13
,
14
], and pointed out six types of codimension one bifurcation in
the plane: fold bifurcation, saddle node bifurcation on the invariant torus, supercritical
Hopf bifurcation, subcritical Hopf bifurcation, fold limit cycle bifurcation and homoclinic
bifurcation. Different categories of bursting oscillations and their inducing mechanism are
Mathematics 2023,11, 1690 3 of 16
given. It is confirmed that the same category of bursting behavior is not only the same
bifurcation mechanism but also the same topology of bursting shape [15].
Mathematics 2023, 9, x FOR PEER REVIEW 3 of 17
Figure 1. Two important bifurcation behaviors: the bifurcation from the quiescent state to the spik-
ing state and the bifurcation from the spiking state back to the quiescent state.
Izhikevich improved Rinzel’s classification method based on the geometric shape of
the bursting and proposed a classification method based on the switching bifurcation
mode between the quiescent state and the spiking state. According to the difference in
bifurcation modes switching between quiescent state and spiking state, Izhikevich classi-
fied all possible bursting oscillations in which only codimension one bifurcation occurred
in the fast subsystem [13,14], and pointed out six types of codimension one bifurcation in
the plane: fold bifurcation, saddle node bifurcation on the invariant torus, supercritical
Hopf bifurcation, subcritical Hopf bifurcation, fold limit cycle bifurcation and homoclinic
bifurcation. Different categories of bursting oscillations and their inducing mechanism are
given. It is confirmed that the same category of bursting behavior is not only the same
bifurcation mechanism but also the same topology of bursting shape [15].
At present, Rinzel’s fast–slow dynamic analysis method and Izhikevich’s classifica-
tion criteria have become the classical tools for analyzing multi-time-scale coupled sys-
tems at home and abroad and have successfully solved a large number of problems of a
variety of bursting oscillation mechanisms of two time scale systems.
The remainder of this paper is organized as follows. In Section 2, the equilibrium and
bifurcation analysis, the bursting oscillations with two-frequency slow excitation system,
the bursting oscillations with the bifurcation delay, and the bursting oscillations of the
coupled system are discussed. In Section 3, learn more about the bursting oscillations from
four parts, namely the simple bursting, the asymmetric bursting, the delayed bursting,
and the bursting with a hysteresis loop. In Section 4, the practical application of the burst-
ing oscillations is introduced from vibration reduction and vibration energy harvesting.
In Section 5, we make conclusions and outlooks.
2. Dynamic Analysis
2.1. Bifurcation Analysis
2.1.1. Equilibrium Stability
Consider the -dimensional nonlinear system
.
(1)
Then the point satisfying is an equilibrium point of System (1). Sup-
pose that , the linear approximation system of System (1) is
Figure 1.
Two important bifurcation behaviors: the bifurcation from the quiescent state to the spiking
state and the bifurcation from the spiking state back to the quiescent state.
At present, Rinzel’s fast–slow dynamic analysis method and Izhikevich’s classification
criteria have become the classical tools for analyzing multi-time-scale coupled systems at
home and abroad and have successfully solved a large number of problems of a variety of
bursting oscillation mechanisms of two time scale systems.
The remainder of this paper is organized as follows. In Section 2, the equilibrium and
bifurcation analysis, the bursting oscillations with two-frequency slow excitation system,
the bursting oscillations with the bifurcation delay, and the bursting oscillations of the
coupled system are discussed. In Section 3, learn more about the bursting oscillations from
four parts, namely the simple bursting, the asymmetric bursting, the delayed bursting, and
the bursting with a hysteresis loop. In Section 4, the practical application of the bursting
oscillations is introduced from vibration reduction and vibration energy harvesting. In
Section 5, we make conclusions and outlooks.
2. Dynamic Analysis
2.1. Bifurcation Analysis
2.1.1. Equilibrium Stability
Consider the n-dimensional nonlinear system
.
x=f(x),x∈U⊂Rn,f:U⊂Rn→Rn. (1)
Then the point
x0
satisfying
f(x0)=
0 is an equilibrium point of System (1). Suppose
that x0=0, the linear approximation system of System (1) is
.
x=Ax, (2)
where
A=∂f(x)
∂x
x=0
is the Jacobian matrix of
f(x)
at
x=
0. The stability of System (1) at
the equilibrium point can be determined by the eigenvalues of the matrix
A
, as stated in
Theorem 1.
Theorem 1
([
16
])
.
If the eigenvalues of matrix
A
are
λi(i=1, 2, ·· · ,n)
, then the stability of
System (1) can be divided into the following three cases:
(1)
If
Re(λi)<
0
(i=1, 2, ·· · ,n)
, System (1) is asymptotically stable at the equilibrium point;
(2) If there exists at least one
λi
with
Re(λi)>
0, System (1) is unstable at the equilibrium point;
(3)
If there is no
λi
with
Re(λi)>
0, but there is at least one
λi
with
Re(λi)=
0, the stability of
System (1) at the equilibrium point needs to be determined by the higher order term.
Mathematics 2023,11, 1690 4 of 16
Let the characteristic equation of the matrix Abe
det(A−λI)=a0λn+a1λn−1+··· +an−1λ+an=0. (3)
We can judge the stability of the equilibrium point of System (1) by the stability of the
linear approximation in System (2).
Theorem 2 (Routh–Hurwitz criterion [16]).The sufficient and necessary condition for all roots
of Equation (3) to have negative real parts is that all the following determinants are of the same sign
∆0=a0,∆1=a1,∆2=
a1a0
a3a2
,∆3=
a1a00
a3a2a1
a5a4a3
,··· ,∆n=
a1a0··· 0
a3a2··· 0
.
.
..
.
.....
.
.
a2n−1a2n−2··· an
. (4)
In the elements of the determinant above, if subscript i of aigreater than n, let ai=0.
Therefore, when System (2) is asymptotically stable, the equilibrium point of System (1) is
asymptotically stable. When System (2) has eigenvalues of the positive real part, the equilibrium
point of System (1) is unstable.
2.1.2. Bifurcation of Smooth System
Bifurcation [
17
] refers to the sudden qualitative or topological changes in the behavior
of a dynamic system when its parameter values (bifurcation parameters) change slightly. In
the bifurcation theory of differential equations, the change in the number of singularities,
the change in the stability of singularities, and the change in the number of periodic
solutions when the parameter changes near a critical value are studied.
Bifurcation research can be divided into different bifurcation types according to differ-
ent emphases. Bifurcation can be divided into static bifurcation and dynamic bifurcation
according to different research objects. Static bifurcation refers to changes in the number
and stability of equilibrium points, such as fold bifurcation, etc. Dynamic bifurcation means
that the topological structure of the solution of a dynamic equation changes suddenly with
the change in parameters, such as Hopf bifurcation, homoclinic bifurcation, etc.
In addition, according to the different spatial regions where the bifurcation is located,
the bifurcation can be divided into local bifurcation and global bifurcation. Local bifur-
cation only considers the change in the topological structure of the trajectory near the
equilibrium point or the closed orbit, such as fold bifurcation, Hopf bifurcation, fold limit
cycle bifurcation, etc. Global bifurcation considers the behavior changes in the whole
vector field of the system in the bifurcation analysis, such as homoclinic bifurcation and
heteroclinic bifurcation. Sometimes the global structure of the vector field is affected by
local bifurcation.
Next, we briefly introduce several bifurcation types [18–21], as shown in Table 1.
Table 1. Several simple bifurcation types and their possible simple bursting types.
Bifurcation Type Possible Simple Bursting Type
Fold Bifurcation
Fold/fold bursting
Fold/supHopf bursting
Fold/LPC bursting
Fold/homoclinic bursting
Hopf Bifurcation
Fold/supHopf bursting
Delay subHopf/delay subHopf bursting
Hopf/LPC bursting
Hopf/homoclinic bursting
Fold limit cycle Bifurcation Fold/LPC bursting
Hopf/LPC bursting
Homoclinic Bifurcation Fold/homoclinic bursting
Hopf/homoclinic bursting
Mathematics 2023,11, 1690 5 of 16
Fold Bifurcation
Consider the one-dimensional dynamical system
.
x=f(x,α),x∈R,α∈R, (5)
where
f
is a smooth function. Suppose that
f(0, 0)=
0, if the following three conditions
are satisfied:
(1)
Bifurcation condition: λ=fx(0, 0)=0;
(2)
Non-degeneracy condition: fxx (0, 0)6=0;
(3)
Transversality condition: fα(0, 0)6=0.
Then System (5) will have a fold bifurcation at
α=
0, and the topology normal form
close to x=0 is .
x=α+sx2,s=±1. (6)
In Reference [
20
], the fold bifurcation diagram of System (6) is shown for
s=
1. With
the change in parameter
α
, two equilibrium points disappear after collision. When
α<
0,
there are two equilibrium points in the system, and the collision occurs when α=0. With
the increase in parameter α, the equilibrium point disappears.
Hopf Bifurcation
Consider the two-dimensional dynamical system
.
x=f(x,α),x=(x1,x2)T∈R2,α∈R, (7)
where
f
is a smooth function. Suppose that
f(0, 0)=
0, and there are a pair of complex
eigenvalues
λ1,2 =u(α)±iω(α)
in the domain of
x=
0. If the following conditions are
satisfied:
(1)
Bifurcation condition: when
α=
0, there exist a pair of pure virtual roots, namely
u(0)=0, ω(0)>0;
(2)
Non-degeneracy condition: the first Lyapunov coefficient is not zero, namely
l1(0)6=
0;
(3)
Transversality condition: uα(0)6=0.
Then System (7) will have a Hopf bifurcation at
α=
0, and the topology normal form
close to x=0 is .
x1
.
x2=β−1
1βx1
x2+sx2
1+x2
2x1
x2, (8)
where s=sgn(l1(0)) =±1.
When
l1(0)<
0, the Hopf bifurcation is supercritical, the equilibrium point becomes
unstable, and a stable limit cycle is generated. When
l1(0)>
0, the Hopf bifurcation is
subcritical, the equilibrium point becomes unstable, and the unstable limit cycle disappears.
Fold limit cycle Bifurcation
With the change in parameters, the phenomenon of instability, collision, and disap-
pearance of the limit cycle of the system is called fold limit cycle bifurcation.
By using the central manifold theorem and the Poincarémapping method, the bifurca-
tion problem of limit cycles can be transformed into the fixed-point problem of discrete
systems. In Reference [
20
], the fold limit cycle bifurcation diagram is shown. The trajectory
L0
is the limit cycle of the continuous system, and
Pα
is the Poincarémapping correspond-
ing to the limit cycle. At
α=
0, one of the characteristic roots of the fixed point of
Pα
is
u1=
1, and the other characteristic roots satisfy 0
<u2<
1. In this case, a fold bifurcation
occurs; that is, two limit cycles in the continuous system disappear due to collision.
Homoclinic Bifurcation
Consider the two-dimensional dynamical system (7). At
α=
0, there is a saddle
equilibrium point
x0=
0, eigenvalues
λ1(0)<
0
<λ2(0)
, and homoclinic orbit
Γ0
. If the
following conditions are satisfied:
(1)
σ0=λ1(0)+λ2(0)6=0;
Mathematics 2023,11, 1690 6 of 16
(2)
.
β(0)6=0, where β(α)is split function.
Then for any sufficiently small
|α|
, and existing domain
U0
of
Γ0∪x0
, the unique limit
cycle
Lβ
is bifurcated from it. When
σ0<
0 and
β>
0, the limit cycle is stable. When
σ0>
0
and β<0, the limit cycle is unstable.
In addition, there are many types of bifurcation, such as Pitchfork bifurcation [
22
],
Bogdanov–Takens bifurcation [
22
], heteroclinic bifurcation [
23
], Period-doubling bifurca-
tion [
24
], Bautin bifurcation [
25
], Cusp bifurcation [
25
], Neimark–Sacker bifurcation [
26
],
and so on.
2.2. Two-Frequency Slow Excitation Analysis
For the system with two excitation terms, we cannot directly use the traditional fast–
slow dynamic analysis method to analyze the bursting oscillation behavior of the system.
Consider the two-dimensional dynamical system
.
x=y
.
y=g(x,y)+f1cos(ω1t)+f2cos(ω2t). (9)
There are two excitation terms in the system. When one excitation frequency is
an integer multiple of the other excitation frequency, the two excitation terms can be
transformed into the function of one basic excitation term by using the De Moivre formula.
Then we analyze the bursting oscillation phenomenon of the system.
From the De Moivre formula, we can obtain the equation
(cosx +isinx)n=cos(nx)+isin(nx), (10)
where
i=√−1
. By balancing the real parts of both sides of Equation (10), we can obtain
the equation
cos(nx)=C0
ncosnx+C2
ncosn−2x(isinx)2+··· +C2m
ncosn−2mx(isinx)2m, (11)
where 2
m≤n
, and 2
m
is the maximum value not greater than
n
. Letting
cos(nx)=f∗
n(cosx)
,
we can obtain
f∗
n(x)=C0
nxn−C2
nxn−21−x2+··· +(−1)mC2m
nxn−2m1−x2m. (12)
Let
ω2=nω1
and
δ1=cos(ω1t)
, then
δ2=cos(ω2t)=f∗
n(δ1)
. In this case, System (9)
can become .
x=y
.
y=g(x,y)+f1δ1+f2f∗
n(δ1). (13)
Indeed, we can analyze the bursting oscillation phenomenon of System (9) by studying
System (13).
For example, Han et al. [
27
] obtained amplitude-modulated bursting by using the
multiple-frequency slow parametric modulation method, where the excitation frequencies
had great effects on amplitude-modulated bursting. Han et al. [
28
] studied pitchfork-
hysteresis bursting with additional slow parametric excitation. When the additional excita-
tion frequency is an integer multiple of the original excitation frequency, they found two
hysteresis modes. Zhang et al. [
29
] studied bursting oscillations of the system with two
slow-varying periodic excitation terms. Zhou et al. [
30
] studied the zero-crossing pinched
“Hopf/Hopf”-hysteresis bursting with the additional excitation term. Zhou et al. [
31
] stud-
ied the bursting oscillations of the smallest chemical reaction system with multi-frequency
slow parametric excitation terms and observed four novel bursting oscillations. Wang
et al. [
32
] studied a class of multi-timescale non-autonomous dynamical systems with
parametric and external excitation terms. They could observe novel multi-bifurcation
cascaded periodic, quasi-periodic, and chaotic bursting oscillations. Xiao et al. [
33
] studied
hysteresis, amplitude death, and oscillation death mechanisms of the Duffing–Van der Pol
Mathematics 2023,11, 1690 7 of 16
model with parametric and external excitation terms. Ma et al. [
34
] studied the complex
bursting oscillations of the Duffing–Van der Pol system with two slow-varying periodic
excitation terms and obtained four novel bursting oscillations.
When two excitation terms of the system have incommensurate frequencies, a frequency-
truncation fast–slow analysis method is used to truncate the incommensurate frequencies,
and the De Moivre formula is used to convert them into the function of one basic excitation
term, and the system’s bursting oscillation is analyzed. For example, Han et al. [
35
] proposed
the frequency-truncation fast–slow analysis method to study the bursting oscillations in
parametrically and externally excited systems with two slow incommensurate excitation
frequencies.
2.3. Bifurcation Delay
Bursting oscillation is a typical representative of the complex dynamic behavior of the
multiple time-scale systems, and bifurcation delay is a common phenomenon in bursting
oscillations. When the bifurcation parameter passes through the bifurcation point, the
stability changes, but the system state does not change immediately and remains in the
current state for a period of time, which is the phenomenon of bifurcation delay [
1
,
36
].
Bifurcation delay is manifested in a variety of bifurcations, such as Hopf bifurcation,
transcritical bifurcation, period-doubling bifurcation and so on. Two important factors
play a decisive role in the generation process of bifurcation-delayed bursting: one is the
parameter region of bifurcation-delayed termination, and the other is the attractor to which
the trajectory switches when the bifurcation-delayed termination occurs.
Many scholars have studied the bifurcation delay phenomenon of coupled systems. On
the basis of central manifold reduction and Neishtadt theory [
37
,
38
], Zheng and Wang [
39
]
proposed a method to determine the bifurcation delay exit point for the time-delay fast–slow
systems. Han et al. [
40
] studied the chaos bursting phenomenon of the parameter-driven
Lorentz system and obtained a new “delayed pitchfork/boundary crisis” chaos bursting
oscillation. Tasso and Theodore [
41
] studied the stability delay loss caused by the Hopf
bifurcation in reaction–diffusion equations with a slow-varying parameter. Zhang et al. [
42
]
studied the bursting oscillations of the Duffing system with multi-frequency parametric
excitation terms and revealed the complex “point-point” type bursting oscillation related
to delayed pitchfork bifurcation. Zheng et al. [
43
] studied the bursting oscillations of the
parametric excited three-dimensional chaotic system. When the slow-varying excitation
term periodically passed through the supercritical pitchfork bifurcation point, the system
had obvious delay behavior. Li et al. [
44
] studied a parameter-driven Rucklidge system
and discussed four bursting oscillations caused by delayed pitchfork bifurcation. Ma
et al. [
45
] studied a parameter-driven Van der Pol–Duffing system and obtained four mixed-
mode vibrations caused by the pitchfork bifurcation delay phenomenon. Li et al. [
46
]
studied a parameter-driven Shimizu–Morioka system and revealed some new bursting
oscillations caused by delayed transcritical bifurcation. Deng and Li [
47
] proposed a chaotic
memory circuit with external periodic disturbance, explored the mechanism of bursting
oscillation and delay effect caused by symmetric Hopf, and measured the delay time
of Hopf-induced bursting under different external excitation. The results show that the
relationship between delay time and external excitation frequency is given by a power law.
Ma et al. [
48
] studied the bursting oscillations caused by pitchfork bifurcation delay and
supHopf bifurcation delay based on a generalized parameter forced Van der Pol–Duffing
system. Zhang et al. [
49
] studied the mechanism of some special phenomena in the bursting
oscillations on an improved Van der Pol–Duffing system with periodic parameter excitation.
When the excitation amplitude or frequency increases to a certain extent, the delay caused
by the motion inertia becomes larger, which may cause the trajectory to pass through the
parameter region of the stable attractor by bifurcation.
Based on the continuous development of bifurcation delay, there are still many prob-
lems to be further studied. For example, the termination region of the bifurcation delay is
further calculated and analyzed in order to improve the type of bursting oscillations under
Mathematics 2023,11, 1690 8 of 16
the bifurcation delay. Further analysis is made on the bursting oscillation modes with
multiple bifurcation delay co-existing, and new types of bursting oscillations are sought.
The bifurcation delay phenomenon of bursting oscillations is analyzed mathematically, and
the mechanism of attractor transition induced by bifurcation delay is understood.
2.4. Coupled System
When using the fast–slow analysis method for the coupled system, it is necessary to
first divide the fast and slow subsystems of the system according to the actual situation and
then regard the slow variable as the bifurcation parameter. By analyzing the equilibrium
point and its stability of the fast subsystem, various equilibrium states and their correspond-
ing bifurcation behaviors for a change in the slow variable are given. The different motion
modes of the fast subsystem are discussed for transitions between different equilibrium
states. Based on the fast–slow analysis method, many scholars have performed analyses in
recent years on coupled systems with two time scales.
For the two-dimensional model of one fast and one slow, Chumakov et al. [
50
] estab-
lished a two-dimensional model of the metal catalytic oxidation process and analyzed the
bursting oscillations of the model. Kiss et al. [
51
] established a two-dimensional model of
point chemical oscillators caused by the geometric structure. Urvolgyi et al. [
52
] further
analyzed the bursting oscillations of the two-dimensional model. Yang et al. [
53
] and
Wermus et al. [54] studied the bursting oscillations of a laser system.
For the three-dimensional model of two fast and one slow, Bao et al. [
55
] proposed
a novel third-order autonomous memristive diode bridge-based oscillator and discussed
the bursting oscillations and their bifurcation mechanisms. Bao et al. [
56
] proposed a
fast–slow three-dimensional autonomous Morris–Lecar neuron model and studied the
bursting oscillation behaviors. Baldemir et al. [
57
] studied a three-dimensional reduced
IHC model and discussed the path connecting pseudo-plateau bursting and mixed-mode
oscillations. Barrio et al. [
58
] studied the Hindmarsh–Rose neuron model and the pancreatic
β
-cell model. Gou et al. [
59
] considered the vector field with the Hopf bifurcation at the
origin and observed several kinds of bursting attractors. Rakaric et al. [
60
] considered
the time-varying asymmetric potential method to study the bursting oscillations in the
system with low-frequency excitation. Zhao et al. [
61
] considered a hybrid Rayleigh–Van
der Pol–Duffing system driven by external and parametric slow-varying excitation terms
and discussed the complex bursting oscillations of the system.
For the four-dimensional model with two fast and two slow, Guckenheimer [
62
] stud-
ied the singular Hopf bifurcation condition with the two-dimensional slow subsystem and
obtained the bursting oscillations. Curtu [
63
] further analyzed the bursting oscillation be-
haviors of two fast and two slow four-dimensional systems. Domestic scholars, represented
by Lu et al., have carried out a lot of work on different neuron models in this field and
achieved a lot of achievements [64–66].
For the four-dimensional model with three fast and one slow, Ma et al. [
67
] studied the
bursting oscillations and its mechanism of a modified Van der Pol–Duffing circuit system with
slow-varying periodic excitation. Some compound bursting oscillations, namely “delayed
supHopf/fold cycle-subHopf/supHopf” bursting and “subHopf/supHopf” bursting via
“delayed supHopf/supHopf” hysteresis loop, are observed. Huang et al. [
68
] studied a
novel three-dimensional chaotic system and discussed different types of bursting oscillation
behaviors. Li et al. [
69
] studied the fast–slow Chay model and found out that bursting
oscillations exhibit period-adding bifurcations. Wu et al. [
70
] studied the Hindmarsh–Rose
model and discussed the fold/homoclinic bursting oscillation behaviors. Chen and Chen [
71
]
studied three types of aperiodic MMOs of a three-dimensional nonautonomous system with
slow-varying parametric excitation. Lin et al. [
72
] designed a simple autonomous three-
element-based memristive circuit and discussed the bifurcation mechanism of the symmetric
periodic fold/Hopf cycle-cycle bursting oscillation. Zhang et al. [
73
] studied the permanent
magnet synchronous motor system and discussed the influence of complex bursting oscillation
behaviors.
Mathematics 2023,11, 1690 9 of 16
In addition to the above types of models, many higher-dimensional models [
74
–
80
]
have been studied. At the same time, when there are time delay factors in the coupled
system, the system will produce a more abundant bursting oscillation phenomenon. For ex-
ample, Yu et al. [
81
–
83
] studied the influence of time-invariant delay on a non-autonomous
system with slow parametric excitation and discussed different types of bursting oscilla-
tions.
3. Analysis of Bursting Oscillations
Using Rinzel’s fast–slow dynamic analysis method, the bursting oscillations can be
classified in two ways. One is to classify bursting oscillations according to their geometric
structure, such as point/point bursting [
84
] and cycle/cycle bursting [
84
]. The other is
to classify according to the bifurcation mode of fast–slow conversion, such as fold/fold
bursting and fold/Hopf bursting. In this section, Izhikevich’s naming method for the
types of bursting oscillation is adopted [
14
]; that is, two bifurcations that make the system
enter the spiking state from the quiescent state and return to the quiescent state from the
spiking state are used to name the types of bursting oscillation. At present, the classification
method and the fast–slow dynamic analysis method have become the classical methods for
studying the bursting oscillation mechanism of multi-time-scale systems. Saggio et al. [
85
]
studied a model with two subsystems with different time scales.
3.1. Simple Bursting
In coupled systems, the common bifurcations are fold bifurcation, Hopf bifurcation
(including supercritical Hopf bifurcation and subcritical Hopf bifurcation), fold limit cycle
(that is LPC) bifurcation and homoclinic bifurcation. These bifurcations can form some
simple bursting oscillations, as shown in Table 1.
Fold/fold bursting
In Reference [
86
], the overlap of the equilibrium curve and transformed phase portrait
of fold/fold bursting oscillation is shown. The system enters the spiking state from the
quiescent state after two-fold bifurcations due to symmetry.
Fold/supHopf bursting
In Reference [
86
], the overlap of the equilibrium curve and transformed phase portrait
of fold/supHopf bursting oscillation is shown. When the fold bifurcation is passed, the
system enters the spiking state from the quiescent state. When the supHopf bifurcation is
passed, the system generates a limit cycle around which the orbit move.
Delay subHopf/delay subHopf bursting
In Reference [
86
], the overlap of the equilibrium curve and transformed phase portrait
of delay subHopf/delay subHopf bursting oscillation is shown. Due to symmetry, the
system enters the spiking state from the quiescent state after two subHopf bifurcations.
There is a delayed phenomenon when the subHopf bifurcation is encountered.
Fold/LPC bursting
In Reference [
87
], the overlap of the equilibrium curve and transformed phase portrait
of fold/LPC bursting oscillation is shown. When the fold bifurcation is passed, the system
enters the spiking state from the quiescent state. When passing through the LPC bifurcation,
the stable and unstable limit cycles collide and disappear, and the system enters the
quiescent state from the spiking state.
Hopf/LPC bursting
In Reference [
87
], the overlap of the equilibrium curve and transformed phase portrait
of Hopf/LPC bursting oscillation is shown. After passing the subcritical Hopf bifurcation
point, due to the influence of the slow-varying process, the system enters the spiking
state after a period of time. When passing through the LPC bifurcation, the stable and
unstable limit cycles collide and disappear, and the system enters the quiescent state from
the spiking state.
Mathematics 2023,11, 1690 10 of 16
Fold/Homoclinic bursting
In Reference [
87
], the overlap of the equilibrium curve and transformed phase portrait
of fold/homoclinic bursting oscillation is shown. When the fold bifurcation is passed, the
system enters the spiking state from the quiescent state. When passing through homoclinic
bifurcation, the stable limit cycle disappears, and the system enters the quiescent state.
Hopf/Homoclinic bursting
In Reference [
88
], the overlap of the equilibrium curve and transformed phase portrait
of Hopf/homoclinic bursting oscillation is shown. When passing homoclinic bifurcation,
the large amplitude limit cycle disappears, causing the system to move around the limit
cycle. Through the Hopf bifurcation, the system enters the quiescent state from the spiking
state.
3.2. Asymmetric Bursting
Many bursting oscillations generated in the system are symmetrical, but there are
also asymmetric cases. For instance, Huang et al. [
68
] studied a novel three-dimensional
chaotic system with multiple coexisting attractors. Four periodic bursting oscillations,
namely periodic asymmetric fold/fold bursting, periodic asymmetric fold/Hopf bursting,
periodic symmetric fold/fold bursting, and periodic symmetric fold/Hopf bursting, were
observed. Kpomahou et al. [
89
] studied the mixed Rayleigh–Liénard oscillator with asym-
metric double-well potential driven by parametric and external excitation terms. The model
had three types of asymmetric bursting oscillations, namely asymmetric fold/Hopf burst-
ing, asymmetric fold/fold bursting, and asymmetric Hopf/Hopf bursting via fold/fold
hysteresis loop.
3.3. Delayed Bursting
There is often a delay phenomenon in the bursting oscillation. Many scholars have
studied the delay phenomenon. For example, Wen et al. [
90
] studied the bursting oscillations
and their bifurcation mechanisms of a memristor-based Shimizu–Morioka system. Some
complex bursting oscillations with delayed, namely symmetric compound Fold/Fold-delayed
supHopf/supHopf bursting, symmetric delayed supHopf/delayed supHopf bursting, and
symmetric delayed supHopf-supHopf/supHopf bursting are revealed. Zhang et al. [
91
]
studied the bursting oscillations and their bifurcation mechanisms in a permanent magnet syn-
chronous motor system with external load perturbation. The system had the periodic delayed
subHopf/subHopf bursting with four scrolls, the signal period delayed subHopf/periodic
delayed subHopf bursting, and the periodic delayed symmetric subHopf/subHopf burst-
ing. Ma et al. [
92
] studied the bursting oscillations in a Lü system with orthogonal para-
metric and external excitation terms. Six novel bursting oscillations with delayed, namely
delayed supHopf/cascaded with PD and IPD/Fold/cascaded with PD and IPD/supHopf-
Fold bursting, delayed supHopf/supHopf/Fold-Fold bursting, delayed supHopf/supHopf-
Fold bursting, delayed supHopf/homoclinic connections/supHopf-Fold bursting, delayed
supHopf/homoclinic connections/supHopf bursting, and delayed supHopf/cascaded with
PD and IPD/supHopf-Fold bursting have been discovered.
3.4. Bursting with Hysteresis Loop
The hysteresis loop phenomenon will also appear in the bursting oscillation [
93
–
95
]. For
example, Lü et al. [
96
] studied the bursting oscillations of the pre-Bötzinger complex inspira-
tory neuron single-compartment model. Five types of bursting oscillations with hysteresis
loop, namely the Hopf/Hopf bursting via “fold/Hopf” hysteresis loop, the Hopf/homoclinic
bursting via “fold/homoclinic” hysteresis loop, the subHopf/homoclinic bursting via
“fold/homoclinic” hysteresis loop, the fold cycle/homoclinic bursting via “fold/homoclinic”
hysteresis loop, and the subHopf/subHopf bursting via “fold/homoclinic” hysteresis loop
were observed. Duan et al. [
97
] studied the bursting oscillations of the pre-Bötzinger com-
plex under a washout filter controller. Three bursting oscillations with hysteresis loop,
namely Hopf/fold limit cycle/homoclinic bursting via “fold/homoclinic” hysteresis loop,
Mathematics 2023,11, 1690 11 of 16
Hopf/Hopf bursting via “fold/homoclinic” hysteresis loop, and subHopf/subHopf burst-
ing via “fold/homoclinic” hysteresis loop were found. Ma et al. [
98
] studied the bursting
oscillations of the Rayleigh–Van der Pol–Duffing system. Six kinds of bursting oscillations
with hysteresis loop, namely compound fold/homoclinic-homoclinic/Hopf bursting via
“fold/Homoclinic” hysteresis loop, compound homoclinic/homoclinic bursting via “Homo-
clinic/Homoclinic” hysteresis loop, compound fold/homoclinic-Hopf/Hopf bursting via
“fold/Homoclinic” hysteresis loop, fold/homoclinic bursting via “fold/Homoclinic” hystere-
sis loop, fold/Hopf bursting via “fold/fold” hysteresis loop, and Hopf/Hopf bursting via
“fold/fold” hysteresis loop were studied.
4. The Application of Bursting
4.1. Vibration Reduction
A vibration system attached dynamic vibration absorber is one of the common vibra-
tion reduction methods. A dynamic vibration absorber is a device that is coupled with the
system to absorb the vibration energy of the main system and suppress the large vibration
of the system.
The vibration of a multi-time-scale coupled system usually has not only large vibration
but also high-frequency vibration. The damage effect of these vibrations on the system
cannot be ignored. Therefore, it is very necessary to study the vibration control of a multi-
time-scale coupled system deeply. For example, Wan et al. [
99
] studied the vibration control
of the two-time-scale coupled Duffing system under low frequency parametric excitation
by linear vibrator. It is found that the system will change from the single vibration mode
to the mixed vibration mode after the addition of dynamic vibration absorber, and the
vibration amplitude is significantly reduced, especially the high frequency vibration is
significantly inhibited.
4.2. Vibration Energy Harvesting
Vibration energy harvesting technology can reduce harmful vibration to protect equip-
ment. Nonlinear vibration energy harvesting technology has been widely applied in
engineering fields in the past decade. Yang et al. [
100
] summarized the research progress of
nonlinear vibration energy harvesting technology in the past ten years. Jiang et al. [
101
]
studied the vibration-based bistable Duffing energy harvester, the vibration-based tristable
energy harvester, and the vibration-based asymmetric bistable energy harvester. The
bursting oscillations are observed in the energy-harvesting systems. Jiang et al. [
102
]
discussed a novel bursting oscillation to collect energy and revealed the dynamical mecha-
nism of the bursting oscillation. Ma et al. [
103
] studied the Mathieu–Van der Pol–Duffing
energy harvester with parameter excitation. Five kinds of bursting oscillations, namely
the “delayed supHopf/supHopf” bursting, the “delayed pitchfork/pitchfork” bursting,
the “delayed Hopf-pitchfork/Hopf-pitchfork” bursting, the “delayed subHopf/supHopf”
bursting and the “delayed subHopf/fold-cycle” bursting were found. Qian and Chen [
104
]
analyzed the bursting oscillation behaviors of the bistable piezoelectric energy harvester.
Lin et al. [
105
] studied the bursting oscillations and energy harvesting efficiency of the
piezoelectric energy harvester in rotational motion with low-frequency excitation. Chen
and Chen [
106
] discussed energy harvesting in the bursting oscillations of the piezoelectric
buckled beam system. Chen et al. [
107
] studied the vibrational energy of a multistable
nonlinear mechanical oscillator using bursting energy harvesting. Lin et al. [
108
] studied
the bursting oscillations of the flow-induced vibration piezoelectric energy harvester with
magnets by low-frequency excitation. Ma et al. [
109
] investigated five novel compound
bursting oscillations of the parametrically amplified Mathieu–Duffing nonlinear energy
harvesters. Qian and Chen [
110
] studied the multi-stable series model. It was found that
the pentastable energy harvester has a richer multivalued response. Wu et al. [
111
] collected
the vibrational energy from the shape memory oscillator using bursting energy harvesting.
Mathematics 2023,11, 1690 12 of 16
5. Conclusions and Outlooks
5.1. Conclusions
In this paper, the bursting oscillations and their bifurcation mechanisms in the coupled
systems with two time scales are introduced. Firstly, the equilibrium stability analysis and
bifurcation analysis of the system are carried out, and several bifurcation types are briefly
introduced: fold bifurcation, Hopf bifurcation, fold limit cycle bifurcation, and homoclinic
bifurcation. For the system with two excitation terms, the two excitation terms are converted
into a function of one basic excitation term by using the De Moivre formula, and then
the bursting oscillations of the system are analyzed. Bifurcation delay often occurs in
bursting oscillations. Bifurcation delay is found in many kinds of bifurcations, such as Hopf
bifurcation, transcritical bifurcation, pitchfork bifurcation and period-doubling bifurcation.
The fast–slow systems are divided into several categories according to the dimensions of the
fast and slow subsystems, and the bursting oscillations are studied by using the fast–slow
dynamic analysis method. Then, using the naming method of Izhikevich to name the type of
bursting oscillation, the discovered bursting oscillation phenomenon is named. The system
has some simple bursting oscillations, such as fold/fold bursting, fold/supHopf bursting,
subHopf/subHopf bursting, fold/LPC bursting, Hopf/LPC bursting, fold/homoclinic
bursting, Hopf/homoclinic bursting, etc. At the same time, the system also has some
complex bursting oscillations, such as asymmetric bursting, delayed bursting, bursting
with hysteresis loop, etc. Finally, the bursting oscillation phenomenon of the system has
very important practical applications, such as dynamic vibration absorber and nonlinear
vibration energy harvesting technology.
5.2. Outlooks
Some achievements have been made in the study of coupled systems at different time
scales. However, many questions still need further study: Firstly, the study of bursting
oscillations in high-dimensional systems. At present, the research on nonlinear coupled
systems mainly focuses on low-dimensional systems, and the research results of the burst-
ing oscillations in high-dimensional systems are few. With the development of nonlinear
dynamics, the dynamics research of high-dimensional nonlinear systems needs further
research. Secondly, the study of the bursting oscillations under high codimension bifurca-
tion. At present, most of the research is on the bursting oscillations of codimension one
in fast subsystems. The high codimension bifurcation model is the focus and difficulty of
nonlinear multi-time-scale dynamics. The bursting oscillations under high codimension
bifurcation need further study. Thirdly, the bursting oscillations of the nonlinear coupled
systems with special structures are studied, such as non-smooth, time delay, etc. The
bifurcation analysis of the fast subsystems of unconventional systems is more complex
and may involve non-smooth bifurcations. Therefore, the nonlinear coupled systems with
special structures also need to be discussed in depth. Fourthly, the bifurcation delay phe-
nomenon of the bursting oscillation is studied. The bifurcation delay phenomenon still
has many problems to be further studied. For example, the bursting oscillation mode with
multiple bifurcation delays co-existing is further analyzed, and new bursting oscillation
types are searched for. The bifurcation delay phenomenon of the bursting oscillations is
analyzed mathematically, and the mechanism of attractor transition induced by bifurcation
delay is understood. Fifthly, the study of classification standard of the bursting oscillations.
According to Izhikevich’s method, different bursting can be classified according to the
bifurcation form of their fast and slow transition. However, the classification work is not
comprehensive, so it is necessary to further study the classification criteria of the bursting
oscillation and consider its essential structure.
Author Contributions:
Conceptualization, Y.Q.; methodology, Y.Q.; software, D.Z.; validation, D.Z.;
writing—original draft preparation, D.Z.; writing—review and editing, Y.Q. and D.Z. All authors
have read and agreed to the published version of the manuscript.
Mathematics 2023,11, 1690 13 of 16
Funding:
The reported study was funded by the National Natural Science Foundation of China
(NNSFC) for the research project Nos. 12172333, 11572288 by Youhua Qian, the Natural Science
Foundation of Zhejiang through grant No. LY20A020003 by Youhua Qian.
Data Availability Statement:
All data, models, and code generated or used during this study are
included within the article.
Acknowledgments:
The authors gratefully acknowledge the reviewers for thoroughly examining
our manuscript and providing useful comments to guide our revision.
Conflicts of Interest: The authors declare that there are no conflict of interest regarding the publica-
tion of this paper.
References
1. Kuehn, C. Multiple Time Scale Dynamics; Springer: Berlin/Heidelberg, Germany, 2015.
2. Zhang, W.; Hu, H.Y. New Advances in the Theory and Application of Nonlinear Dynamics; Science Press: Beijing, China, 2009.
3.
Gendelman, O.V.; Bar, T. Bifurcations of self-excitation regimes in a van der Pol oscillator with a nonlinear energy sink. Phys. D
2010,239, 220–229. [CrossRef]
4.
Dupont, P.; Haddad, C.; Ardissone, J.P.; Debieve, J.F. Space and time organisation of a shock wave/turbulent boundary layer
interaction. Aerosp. Sci. Technol. 2005,9, 561–572. [CrossRef]
5.
Teka, W.; Tabak, J.; Bertram, R. The relationship between two fast/slow analysis techniques for bursting oscillations. Chaos
2012
,
22, 043117. [CrossRef]
6. Van der Pol, B. On relaxation oscillations. Philos. Mag. 1926,7, 978–992. [CrossRef]
7.
Hodgdin, A.L.; Huxley, A.F. A quntaitative description of membrane current and its application to conduction and excitation in
nerve. J. Physiol. 1952,116, 424–448.
8.
Zmurchok, C.; Small, T.; Ward, M.J.; Edelstein-Keshet, L. Application of quasi-steadystate methods to nonlinear models of
intracellular transport by molecular motors. Bull. Math. Biol. 2017,79, 1923–1978. [CrossRef]
9. Johnson, R.S. Singular Perturbation Theory; Springer: Berlin/Heidelberg, Germany, 2005.
10.
Fenichel, N. Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ.
1979
,31, 53–98. [CrossRef]
11. Rinzel, J. Bursting oscillations in an excitable membrane model. Ordinary Part. Differ. Equ. 1985,1151, 304–316.
12.
Perc, M.; Marhl, M. Different types of bursting calcium oscillations in non-excitable cells. Chaos Solitons Fract.
2003
,18, 759–773.
[CrossRef]
13. Izhikevich, E.M. Synchronization of elliptic bursters. SIAM J. Appl. Math. 2000,60, 503–535. [CrossRef]
14. Izhikevich, E.M. Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 2000,10, 1171–1266. [CrossRef]
15.
Izhikevich, E.M.; Desai, N.S.; Walcott, E.C.; Hoppensteadt, F.C. Bursts as a unit of neural information: Selective communication
via resonance. Trends Neurosci. 2003,26, 161–167. [CrossRef] [PubMed]
16. Hu, H.Y. Applied Nonlinear Dynamics; Aviation Industry Press: Beijing, China, 2000.
17.
Wu, L. Fast-slow Behavior Analysis of Two Classes of Nonlinear Oscillator with Periodic Excitation. Master’s Thesis, Jiangsu
University, Jiangsu, China, 2016.
18.
Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields; Springer: New York, NY,
USA, 1983.
19. Wiggins, S. Global Bifurcations and Chaos; Springer: New York, NY, USA, 1988.
20. Kuznetsov, Y.A. Elements of Applied Bifurcation Theory; Springer: New York, NY, USA, 1995.
21.
Shilnikov, L.P.; Shilnikov, A.L.; Turaev, D.V.; Chua, L.O. Methods of Qualitative Theory in Nonlinear Dynamics; World Scientific:
Singapore, 1998.
22.
Liu, S.H.; Shao, T.T.; Wu, H.D.; Zhang, D.H.; Han, Q.Z.; Ferrara, M. Dynamic Behaviors and Mechanisms of Permanent Magnet
Synchronous Motor with Excitation. Math. Probl. Eng. 2021,2021, 5535856. [CrossRef]
23.
Yu, Y.; Zhang, C.; Chen, Z.Y.; Lim, C.W. Relaxation and mixed mode oscillations in a shape memory alloy oscillator driven by
parametric and external excitations. Chaos Solitons Fractals 2020,140, 110145. [CrossRef]
24.
Chtouki, A.; Lakrad, F.; Belhaq, M. Quasi-periodic bursters and chaotic dynamics in a shallow arch subject to a fast–slow
parametric excitation. Nonlinear Dyn. 2020,99, 283–298. [CrossRef]
25.
Zhang, Y.T.; Cao, Q.J.; Huang, W.H. Bursting oscillations in an isolation system with quasi-zero stiffness. Mech. Syst. Signal
Process. 2021,161, 107916. [CrossRef]
26.
Xiong, X.W.; Li, Y.Q.; Zheng, D.M. Dynamic mechanism of multiple bursting patterns in a whole-cell multiscale model with
calcium oscillations. J. Biol. Dyn. 2021,15, 308–326. [CrossRef]
27.
Han, X.J.; Wei, M.K.; Bi, Q.S.; Kurths, J. Obtaining amplitude-modulated bursting by multiple-frequency slow parametric
modulation. Phys. Rev. E 2018,97, 012202. [CrossRef]
28.
Han, X.J.; Zhang, Y.; Bi, Q.S.; Kurths, J. Two novel bursting patterns in the Duffing system with multiple-frequency slow
parametric excitations. Chaos 2018,28, 043111. [CrossRef]
Mathematics 2023,11, 1690 14 of 16
29.
Zhang, X.F.; Zheng, J.K.; Wu, G.Q.; Bi, Q.S. Mixed mode oscillations as well as the bifurcation mechanism in a Duffing’s oscillator
with two external periodic excitations. Sci. China Technol. Sci. 2019,62, 1816–1824. [CrossRef]
30.
Zhou, C.Y.; Li, Z.J.; Xie, F.; Ma, M.L.; Zhang, Y. Bursting oscillations in Sprott B system with multi-frequency slow excitations:
Two novel “Hopf/Hopf”-hysteresis-induced bursting and complex AMB rhythms. Nonlinear Dyn.
2019
,97, 2799–2811. [CrossRef]
31.
Zhou, C.Y.; Xie, F.; Li, Z.J. Complex bursting patterns and fast-slow analysis in a smallest chemical reaction system with two slow
parametric excitations. Chaos Solitons Fractals 2020,137, 109859. [CrossRef]
32.
Wang, M.J.; Li, J.H.; Zhang, X.N.; Iu, H.H.C.; Fernando, T.; Li, Z.J.; Zeng, Y.C. Multi-bifurcation cascaded bursting oscillations
and mechanism in a novel 3D non-autonomous circuit system with parametric and external excitation. Nonlinear Dyn.
2021
,105,
3699–3714. [CrossRef]
33.
Xiao, J.Y.; Chen, Z.Y.; Bi, Q.S.; Zou, Y.; Guan, S.G. Distinctive roles of hysteresis, amplitude death and oscillation death in
generating fast-slow phenomena in parametrically and externally excited systems. Chaos Solitons Fractals
2021
,150, 111189.
[CrossRef]
34.
Ma, X.D.; Bi, Q.S.; Wang, L.F. Complex bursting dynamics in the cubic-quintic Duffing-van der Pol system with two external
periodic excitations. Meccanica 2022,57, 1747–1766. [CrossRef]
35.
Han, X.J.; Liu, Y.; Bi, Q.S.; Kurths, J. Frequency-truncation fast-slow analysis for parametrically and externally excited systems
with two slow incommensurate excitation frequencies. Commun. Nonlinear Sci. Numer. Simul. 2019,72, 16–25. [CrossRef]
36.
Shishkova, M.A. Study of a system of differential equations with a small parameter at the highest derivatives. Dokl Akad Nauk
SSSR 1973,209, 576–579.
37. Neishtadt, A.I. On delayed stability loss under dynamic bifurcations i. Differ. Equ. 1987,23, 2060–2067.
38. Neishtadt, A.I. On delayed stability loss under dynamic bifurcations ii. Differ. Equ. 1988,24, 226–233.
39.
Zheng, Y.G.; Wang, Z.H. Delayed Hopf bifurcation in time-delayed slow-fast systems. Sci. China Technol. Sci.
2010
,53, 656–663.
[CrossRef]
40.
Han, X.J.; Yu, Y.; Zhang, C. A novel route to chaotic bursting in the parametrically driven Lorenz system. Nonlinear Dyn.
2017
,88,
2889–2897. [CrossRef]
41.
Tasso, J.K.; Theodore, V. Delayed loss of stability due to the slow passage through Hopf bifurcations in reaction-diffusion
equations. Chaos 2018,28, 091103.
42.
Zhang, Y.; Han, X.J.; Bi, Q.S. Series-mode pitchfork-hysteresis bursting oscillations and their dynamical mechanisms. Chin. J.
Theor. Appl. Mech. 2019,51, 228–236.
43.
Zheng, J.K.; Zhang, X.F.; Bi, Q.S. Bursting oscillations as well as the delayed pitchfork bifurcation behaviors in a class of chaotic
system. Chin. J. Theor. Appl. Mech. 2019,51, 540–549.
44.
Li, Z.J.; Fang, S.Y.; Ma, M.L.; Wang, M.J. Bursting Oscillations and Experimental Verification of a Rucklidge System. Int. J. Bifurc.
Chaos Appl. Sci. Eng. 2021,31, 2130023. [CrossRef]
45.
Ma, X.D.; Yu, Y.; Wang, L.F. Complex mixed-mode vibration types triggered by the pitchfork bifurcation delay in a driven van der
Pol-Duffing oscillator. Appl. Math. Comput. 2021,411, 126522. [CrossRef]
46.
Li, Z.J.; Li, Y.; Ma, M.L.; Wang, M.J. Delayed Transcritical Bifurcation Induced Mixed Bursting in a Modified SM System with
Asymmetrically Distributed Equilibria. Braz. J. Phys. 2021,51, 840–849. [CrossRef]
47.
Deng, Y.; Li, Y.X. Symmetrical Hopf-induced bursting and hyperchaos control in memristor-based circuit. Chaos
2021
,31, 043103.
[CrossRef]
48.
Ma, X.D.; Wang, L.F.; Bi, Q.S. Bursting behaviors induced by the bifurcation delay in a generalized parametrically forced van der
Pol-Duffing system. Indian J. Phys. 2022,96, 4269–4282. [CrossRef]
49.
Zhang, X.F.; Zhang, B.; Han, X.J.; Bi, Q.S. On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation
in a modified Van der Pol-Duffing system with slow-varying periodic excitation. Nonlinear Dyn.
2022
,108, 2097–2114. [CrossRef]
50.
Chumakov, G.A.; Chumakova, N.A. Relaxation oscillationsin a kineticmodel of catalytic hydrogen oxidationinvolving a chase on
canards. Chem. J. 2003,91, 151–158.
51.
Kiss, I.Z.; Pelster, L.N.; Wickramasinghe, M.; Yablonsky, G.S. Frequency of negative differential resistance electrochemical
oscillators: Theory and experiments. Phys. Chem. Chem. Phys. 2009,11, 5720–5728. [CrossRef] [PubMed]
52.
Urvolgyi, M.; Gaspar, V.; Nagy, T.; Kiss, I.Z. Quantitative dynamical relationships for the effect of rotation rate on frequency and
waveform of electrochemical oscillations. Chem. Eng. Sci. 2012,83, 56–65. [CrossRef]
53.
Yang, S.; Ponomarev, E.A.; Bao, X.Y. Experimental study on relaxation oscillation in a detuned FM harmonic mode-locked
Er-doped fiber laser. Opt. Commun. 2005,245, 371–376. [CrossRef]
54.
Wermus, D.; Shaffner, T.; Siemers, T.J.; Thompson, J.R. Interplay of classical noise sourcesin the relaxation oscillations of an
Nd:YAG laser. Opt. Commun. 2011,284, 3051–3059. [CrossRef]
55.
Bao, B.C.; Wu, P.Y.; Bao, H.; Wu, H.G.; Zhang, X.; Chen, M. Symmetric periodic bursting behavior and bifurcation mechanism in a
third-order memristive diode bridge-based oscillator. Chaos Solitons Fractals 2018,109, 146–153. [CrossRef]
56.
Bao, B.C.; Yang, Q.F.; Zhu, L.; Bao, H.; Xu, Q.; Yu, Y.J.; Chen, M. Chaotic Bursting Dynamics and Coexisting Multistable Firing
Patterns in 3D Autonomous Morris-Lecar Model and Microcontroller-Based Validations. Int. J. Bifurc. Chaos
2019
,29, 1950134.
[CrossRef]
57.
Baldemir, H.; Avitabile, D.; Tsaneva-Atanasova, K. Pseudo-plateau bursting and mixed-mode oscillations in a model of developing
inner hair cells. Commun. Nonlinear Sci. Numer. Simul. 2020,80, 104979. [CrossRef]
Mathematics 2023,11, 1690 15 of 16
58.
Barrio, R.; Ibáñez, S.; Pérez, L.; Serrano, S. Spike-adding structure in fold/hom bursters. Commun. Nonlinear Sci. Numer. Simul.
2020,83, 105100. [CrossRef]
59.
Gou, J.T.; Zhang, X.F.; Xia, Y.B.; Bi, Q.S. All Possible Bursting Attractors in the Neighborhood of Hopf Bifurcation Point Under
Periodic Excitation. Int. J. Bifurc. Chaos 2022,32, 2250079. [CrossRef]
60.
Rakaric, Z.; Lukesevic, L.R. On the phenomenon of bifurcation space symmetrization as mechanism for bursting oscillations
generation. Arch. Appl. Mech. 2023,93, 761–771. [CrossRef]
61.
Zhao, F.; Ma, X.D.; Cao, S.Q. Periodic bursting oscillations in a hybrid Rayleigh-Van der Pol-Duffing oscillator. Nonlinear Dyn.
2023,111, 2263–2279. [CrossRef]
62.
Guckenheimer, J. Singular Hopf bifurcation in systems with two slow variables. SIAM J. Appl. Dyn. Syst.
2008
,7, 1355–1377.
[CrossRef]
63.
Curtu, R. Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitor-y neural network. Phys. D
2010
,239,
504–514. [CrossRef]
64.
Duan, L.X.; Lu, Q.S.; Wang, Q.Y. Two-parameter bifurcation analysis of fring activities in the Chay neuronal model. Neurocomputing
2008,72, 341–351. [CrossRef]
65.
Wang, H.X.; Wang, Q.Y.; Lu, Q.S. Bursting oscillations, bifurcation and synchro- nization in neuronal systems. Chaos Solitons
Fractals 2011,44, 667–675. [CrossRef]
66.
Zhang, F.; Lu, Q.S.; Su, J.Z. Transition in complex calcium bursting induced by IP3 degradation. Chaos Solitons Fractals
2009
,41,
2285–2290. [CrossRef]
67.
Ma, X.D.; Cao, S.Q.; Guo, H.L. Routes to Bursting Oscillations in a Modified Van Der Pol-Duffing Oscillator with Slow-Varying
Periodic Excitation. J. Vib. Control 2018,24, 4960–4970.
68.
Huang, L.; Wu, G.Q.; Zhang, Z.D.; Bi, Q.S. Fast-Slow Dynamics and Bifurcation Mechanism in a Novel Chaotic System. Int. J.
Bifurc. Chaos 2019,29, 1930028. [CrossRef]
69.
Li, Y.Y.; Gu, H.G.; Ding, X.L. Bifurcations of Enhanced Neuronal Bursting Activities Induced by the Negative Current Mediated
by Inhibitory Autapse. Nonlinear Dyn. 2019,97, 2091–2105. [CrossRef]
70.
Wu, F.Q.; Gu, H.G.; Li, Y.Y. Inhibitory electromagnetic induction current induces enhancement instead of reduction of neural
bursting activities. Commun. Nonlinear Sci. Numer. Simul. 2019,79, 104924. [CrossRef]
71.
Chen, Z.Y.; Chen, F.Q. Complex aperiodic mixed mode oscillations induced by crisis and transient chaos in a nonlinear system
with slow parametric excitation. Nonlinear Dyn. 2020,100, 659–677. [CrossRef]
72.
Lin, Y.; Liu, W.B.; Bao, H.; Shen, Q. Bifurcation mechanism of periodic bursting in a simple three-element-based memristive
circuit with fast-slow effect. Chaos Solitons Fractals 2020,131, 109524. [CrossRef]
73.
Zhang, S.H.; Wang, C.; Zhang, H.L. Bursting oscillation analysis and synergetic control of permanent magnet synchronous motor.
Acta Phys. Sin. 2020,69, 210501. [CrossRef]
74.
Chen, Z.Y.; Chen, F.Q. Mixed mode oscillations induced by bi-stability and fractal basins in the FGP plate under slow parametric
and resonant external excitations. Chaos Solitons Fractals 2020,137, 109814. [CrossRef]
75.
Nyman, K.H.M.; Ashwin, P.; Ditlevsen, P.D. Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems.
Nonlinearity 2020,33, 2853–2904. [CrossRef]
76.
Duan, L.X.; Liang, T.T.; Zhao, Y.Q.; Xi, H.G. Multi-time scale dynamics of mixed depolarization block bursting. Nonlinear Dyn.
2021,103, 1043–1053. [CrossRef]
77.
Lu, W.; Zhang, Y.H.; Qian, Y.; Pandey, V.; Qu, Z.L.; Zhang, Z.Y. Bursting and complex oscillatory patterns in a gene regulatory
network model. Chaos Solitons Fractals 2021,152, 111348. [CrossRef]
78.
Zhang, M.R.; Bi, Q.S. On occurrence of bursting oscillations in a dynamical system with a double Hopf bifurcation and slow-
varying parametric excitations. Int. J. Non-Linear Mech. 2021,128, 103629. [CrossRef]
79.
Zhang, M.R.; Zhang, X.F.; Bi, Q.S. Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with
Double Hopf Bifurcations. Int. J. Bifurc. Chaos 2021,31, 2130022. [CrossRef]
80.
Huang, J.J.; Bi, Q.S. Bursting oscillations with multiple modes in a vector field with triple Hopf bifurcation at origin. J. Sound Vib.
2023,545, 117422. [CrossRef]
81.
Yu, Y.; Han, X.J.; Zhang, C.; Bi, Q.S. Mixed-mode oscillations in a nonlinear time delay oscillator with time varying parameters.
Commun. Nonlinear Sci. Numer. Simul. 2017,47, 23–34. [CrossRef]
82.
Yu, Y.; Zhang, C.; Han, X.J. Routes to bursting in active control system with multiple time delays. Nonlinear Dyn.
2017
,88,
2241–2254. [CrossRef]
83.
Yu, Y.; Wang, N.; Chen, Z.Y. Amplitude modulation control method for bursting dynamics under time-delayed feedback. J. Vib.
Control 2022,28, 3550–3558. [CrossRef]
84.
Ma, X.D.; Han, X.J.; Jiang, W.A.; Bi, Q.S. Two bursting patterns induced by system solutions approaching infinity in a modified
Rayleigh–Duffing oscillator. Pramana J. Phys. 2020,94, 159. [CrossRef]
85.
Saggio, M.L.; Spiegler, A.; Bernard, C.; Jirsa, V.K. Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the
Ultra-slow Transitions of Classes. J. Math. Neurosci. 2017,7, 1–47. [CrossRef]
86.
Qian, Y.H.; Zhang, D.J.; Lin, B.W. Bursting Oscillation and Its Mechanism of a Generalized Duffing–Van der Pol System with
Periodic Excitation. Complexity 2021,2021, 5556021. [CrossRef]
Mathematics 2023,11, 1690 16 of 16
87.
Xue, M.; Ge, Y.W.; Zhang, Z.D.; Bi, Q.S. Bursting oscillations as well as the classification in the field with codimension-3
fold-fold-Hopf bifurcation. Chin. J. Theor. Appl. Mech. 2021,53, 1423–1438.
88. Zhang, Z.D.; Li, Y.Y.; Bi, Q.S. Routes to bursting in a periodically driven oscillator. Phys. Lett. A 2013,377, 975–980. [CrossRef]
89.
Kpomahou, Y.J.F.; Adechinan, J.A.; Ngounou, A.M.; Yamadjako, A.E. Bursting, mixed-mode oscillations and homoclinic
bifurcation in a parametrically and self-excited mixed Rayleigh–Liénard oscillator with asymmetric double well potential.
Pramana-J. Phys. 2022,96, 176. [CrossRef]
90.
Wen, Z.H.; Li, Z.J.; Li, X. Bursting oscillations and bifurcation mechanism in memristor-based Shimizu-Morioka system with two
time scales. Chaos Solitons Fractals 2019,128, 58–70. [CrossRef]
91.
Zhang, S.H.; Zhang, H.L.; Wang, C.; Ma, P. Bursting oscillations and bifurcation mechanism in a permanent magnet synchronous
motor system with external load perturbation. Chaos Solitons Fractals 2020,141, 110355. [CrossRef]
92.
Ma, M.L.; Xiong, L.W.; Shu, Z.Y.; Fang, Y.J. New Bursting Patterns in the Lü System Driven by Orthogonal Parametric Excitation
and External Excitation. Int. J. Bifurc. Chaos 2021,31, 2150197. [CrossRef]
93.
Meng, P.; Dong, J.W.; Ji, Q.B. Bursting analysis for the pre-Bötzinger complex with periodic forcing. AIP Adv.
2019
,9, 065010.
[CrossRef]
94.
Ma, X.D.; Xia, D.X.; Jiang, W.A.; Liu, M.; Bi, Q.S. Compound bursting behaviors in a forced Mathieu-van der Pol-Duffing system.
Chaos Solitons Fractals 2021,147, 110967. [CrossRef]
95.
Zhang, E.; Yu, L.; Yang, Z.Q. New topological classification of bursting in multi-time-scale Chay–Cook model. Eur. Phys. J. Spec.
Top. 2022,231, 2277–2288. [CrossRef]
96.
Lü, Z.S.; Chen, L.N.; Duan, L.X. Bifurcation analysis of mixed bursting in the pre-Bötzinger complex. Appl. Math. Model.
2019
,67,
234–251. [CrossRef]
97.
Duan, L.X.; Liang, W.J.; Ji, W.C.; Xi, H.G. Bifurcation Patterns of Bursting within pre-Bötzinger Complex and Their Control. Int. J.
Bifurc. Chaos 2020,30, 2050192. [CrossRef]
98.
Ma, X.D.; Bi, Q.S.; Wang, L.F. Complex Periodic Bursting Structures in the Rayleigh-van der Pol-Duffing Oscillator. J. Nonlinear
Sci. 2022,32, 25. [CrossRef]
99.
Wan, H.L.; Li, X.H.; Shen, Y.J.; Wang, Y.L. Study on vibration reduction of dynamic vibration absorber for two-scale Duffing
system. Chin. J. Theor. Appl. Mech. 2022,54, 1–11.
100.
Yang, T.; Zhou, S.X.; Cao, Q.J.; Zhang, W.M.; Chen, L.Q. Some advances in nonlinear vibration energy harvesting technology.
Chin. J. Theor. Appl. Mech. 2021,53, 2894–2909.
101.
Jiang, W.A.; Han, X.J.; Chen, L.Q.; Bi, Q.S. Bursting vibration-based energy harvesting. Nonlinear Dyn.
2020
,100, 3043–3060.
[CrossRef]
102.
Jiang, W.A.; Ma, X.D.; Liu, M.; Wang, Y.; Chen, L.Q.; Bi, Q.S. Exploiting Bursting Oscillations to Improve Energy Capture from
Slowly Changing Excitation. J. Vib. Eng. Technol. 2021,9, 1923–1939. [CrossRef]
103.
Ma, X.Q.; Jiang, W.A.; Zhang, X.F.; Han, X.J.; Bi, Q.S. Complex bursting dynamics of a Mathieu-van der Pol-Duffing energy
harvester. Phys. Scr. 2021,96, 015213. [CrossRef]
104.
Qian, Y.H.; Chen, Y.N. Bursting oscillations and energy harvesting efficiency analysis of bistable piezoelectric energy harvester.
Chin. J. Theor. Appl. Mech. 2022,54, 3157–3168.
105.
Lin, B.W.; Wang, Y.H.; Qian, Y.H. Bursting oscillation phenomenon and efficiency analysis of a piezoelectric energy harvester in
rotational motion by low-frequency excitation. Eur. Phys. J. Plus 2022,137, 459.
106.
Chen, Z.Y.; Chen, F.Q. Bifurcation behaviors and bursting regimes of a piezoelectric buckled beam harvester under fast-slow
excitation. Nonlinear Dyn. 2023,111, 4121–4139. [CrossRef]
107.
Chen, Z.Y.; Jiang, W.A.; Chen, L.Q.; Bi, Q.S. Bursting analysis of multi-stable nonlinear mechanical oscillator and its application in
energy harvesting. Eur. Phys. J. Spec. Top. 2022,231, 2223–2236. [CrossRef]
108.
Lin, B.W.; Wang, Y.H.; Qian, Y.H. Bursting oscillation and its mechanism of the flow-induced vibration piezoelectric energy
harvester with magnets by low-frequency excitation. Eur. Phys. J. Spec. Top. 2022,231, 2237–2248. [CrossRef]
109.
Ma, X.D.; Zhang, X.F.; Yu, Y.; Bi, Q.S. Compound Bursting Behaviors in the Parametrically Amplified Mathieu-Duffing Nonlinear
System. J. Vib. Eng. Technol. 2022,10, 95–110. [CrossRef]
110.
Qian, Y.H.; Chen, Y.N. Research on multi-valued response and bursting oscillation of series multi-stable piezoelectric energy
harvester. Eur. Phys. J. Plus 2022,137, 588. [CrossRef]
111.
Wu, T.Y.; Jiang, W.A.; Bi, Q.S. Bursting Energy Harvesting of Shape Memory Oscillator. J. Vib. Eng. Technol.
2022
,10, 985–994.
[CrossRef]
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