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Commun Nonlinear Sci Numer Simulat 19 (2014) 1175–1184
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Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier.com/locate/cnsns
Bursting mechanism in a time-delayed oscillator with slowly
varying external forcing
Yue Yu a,b, Hongji Tang b, Xiujing Han a, Qinsheng Bi a,⇑
a
b
Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, PR China
Faculty of Science, Nantong University, Nantong 226007, PR China
a r t i c l e
i n f o
Article history:
Received 15 March 2013
Received in revised form 10 August 2013
Accepted 11 August 2013
Available online 30 August 2013
Keywords:
Bursting oscillations
Time delay
Dufﬁng oscillator
External forcing
Bifurcation mechanism
a b s t r a c t
This paper investigates the generation of complex bursting patterns in the Dufﬁng oscillator with time-delayed feedback. We present the bursting patterns, including symmetric
fold–fold bursting and symmetric Hopf–Hopf bursting when periodic forcing changes
slowly. We make an analysis of the system bifurcations and dynamics as a function of
the delayed feedback and the periodic forcing. We calculate the conditions of fold bifurcation and Hopf bifurcation as well as its stability related to external forcing and delay. We
also identify two regimes of bursting depending on the magnitude of the delay itself and
the strength of time delayed coupling in the model. Our results show that the dynamics
of bursters in delayed system are quite different from those in systems without any delay.
In particular, delay can be used as a tuning parameter to modulate dynamics of bursting
corresponding to the different type. Furthermore, we use transformed phase space analysis
to explore the evolution details of the delayed bursting behavior. Also some numerical simulations are included to illustrate the validity of our study.
Ó 2013 Published by Elsevier B.V.
1. Introduction
The studies of bursting or mixed-mode oscillations have received great attention recently in modeling realistic neuronal
networks, physics, mechanics and engineering systems and so on [1–5]. Bursting oscillations are waveforms that consist of
alternating small and large amplitude excursions. Mathematically, the generation of bursting oscillations is often associated
with fast and slow subsystems simultaneously [6,7]. Traditionally bursting oscillations can be created by the system switching between the coexisting attractors of the fast subsystem corresponding to the slow current. We call a system in downstate when variables exhibit small amplitude. Then the effect of two time scales may lead the system to up-state state, in
which variables behave in large amplitude.
Most of the related papers focused on analyzing dynamics of bursting oscillations. Lewis and Rinzel [8] and Izhikevich [9]
ﬁrst gave a complete classiﬁcation of different types of bursters. Their fast-slow dynamics analysis has become a well-accepted approach to study bursting. In addition, the geometrical bifurcation analysis is also used to investigate the generation
or transition of bursting modes in the single neuron and bifurcation mechanism for synchronized bursting of two-cell systems [10–12]. Simo and Woafo [13] presented the bursting oscillations in a system consisting of a double-well magnetically
coupled electrical oscillator. Shorten and Wall [14] studied bursting in a Hodgkin–Huxley type of neuronal model and its
transition to other models. Bertram et al. [15] studied bursting behavior in the Chay-Cook model with two-parameter analysis. In Refs. [16,17], bifurcation mechanism of bursting has also been studied by the unfolding and normal form theory.
⇑ Corresponding author. Tel.: +86 511 88791110; fax: +86 511 88797815.
E-mail address: [email protected] (Q. Bi).
1007-5704/$ - see front matter Ó 2013 Published by Elsevier B.V.
http://dx.doi.org/10.1016/j.cnsns.2013.08.010
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Y. Yu et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 1175–1184
These results attempt to propose methods to uncover the bifurcation mechanism of transition between down-state and upstate.
So far, the analysis and classiﬁcation of bursting are mainly concentrated on systems without any delay. However, it is
very useful that systems involve delays which can be in the description of the real world more explicitly [18,19]. Thus,
time-delay has become an important tool for describing and controlling various nonlinear phenomena in numerous and diverse ﬁelds [20–25]. Although much is known about bursting phenomena and delayed systems, little is known about the
relationship between them, as well as how time delay could affect bursting oscillations, the mechanism of which still need
to be investigated.
Here we consider the well-known Dufﬁng oscillator [26–28] with negative linear stiffness and linear time delay driven by
a slowly periodically varying external forcing, expressed in the form of
€x þ x_ x þ x3 þ Axs ¼ k sinðxtÞ;
ð1Þ
where k > 0 is the forcing amplitude, 0 < x 1 is the forcing frequency, xs ¼ xðt sÞ is a time delay. A means gain coefﬁcient about the delay. If A > 0, it means a positive delayed feedback, and negative feedback if A < 0.
Because two time scales evolve in the vector ﬁeld, bursting oscillations in different forms, can be observed. The dynamic
behaviors and mechanism of the system are explained by local divergence. The paper is organized as follows: In Section 2, an
analysis of the bifurcations and dynamics is obtained as a function of the linear delayed state feedback and the external periodic forcing. In Sections 3 and 4, bursting phenomena (symmetric fold/fold bursting, symmetric sup-Hopf/sup-Hopf bursting) and the effects of some parameters including time delay on different bursting are discussed. Investigations of occurrence
and mechanism of certain bursting dynamics are also presented. Finally, Section 5 concludes the paper.
2. Bifurcation analysis
System (1) can be reduced to the non-delay system with external excitation at s ¼ 0. For x 1, the external excitation
varies more slowly than the rest variables in the system. Periodic bursting and chaotic bursting, i.e. a series of spikes which
are periodically or chaotically interspersed with the spiking period and quiescence behavior in multiscale systems have been
studied more. Here, we concentrate on bursters under the condition of s–0. In this case, the properties of bursting are inﬂuenced by the delay. At ﬁrst, we shall analyze the bifurcation behavior of the equation of (1).
For x 1, considering k sinðxtÞ as a control parameter d, given by the following delay-differential equation (DDE):
€x þ x_ x þ x3 þ Axs ¼ d
ð2Þ
System (2) can easily be converted into (3)
x_ ¼ y
y_ ¼ €x ¼ y þ x x3 Axs þ d
ð3Þ
Denoting the equilibrium points as ðx0 ; y0 Þ, obviously y0 ¼ 0, x0 is decided by the equation
x x3 Ax þ d ¼ 0; i:e: x3 þ ð1 AÞx þ d ¼ 0
2
ð4Þ
3
The root discriminant of (4) is D ¼ 81d 12ð1 AÞ , which gives the results: for D < 0, there are three equilibrium points;
for D ¼ 0, there are two equilibrium points; otherwise D > 0, there is one and only equilibrium point in DDE. Particularly
taking A ¼ 0, d ¼ 0, three equilibrium points are E0 ¼ ð0; 0Þ and E ¼ ð1; 0Þ. For the chosen values of the parameters, E0
is a saddle and E are stable focuses, implying the non-delayed system is bi-stable. To get a clear idea of the distribution
of equilibrium points of DDE, we plot the curve of equilibrium points related to equation of (4) with respect to d, for ﬁxed
values A ¼ 0:5 and A ¼ 0:5 in Fig. 1, while the equilibrium points for other values can be derived accordingly. One should be
Fig. 1. Curve of equilibrium points in DDE related to Eq. (4) with respect to d, solid points mean fold bifurcation points, and hollow points mean equilibrium
points when d ¼ 1. (a) A ¼ 0:5. (b) A ¼ 0:5.
Y. Yu et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 1175–1184
1177
noticed on the curve of equilibrium points when d reaches its boundaries of two sides, the corresponding equilibriums are
labeled in hollow points, which will be referred later.
Fold bifurcation. The condition D ¼ 81d2 12ð1 AÞ3 ¼ 0 corresponds to the critical transition of the number of equilibrium points, implying fold bifurcation occurs. Therefore, the critical condition for fold bifurcation can be expressed in the
form
FB : 81d2 12ð1 AÞ3 ¼ 0;
ð5Þ
Fig. 2(a) illustrates the fold bifurcation curve on the plane of ðd; AÞ, which is independent of time lag s. For example, at
A ¼ 0:5, we can obtain the fold bifurcation points F1ð0:5; 0:136Þ and F2ð0:5; 0:136Þ, while at A ¼ 0:5, we can obtain the
fold bifurcation points FF1ð0:5; 0:707Þ and FF2ð0:5; 0:707Þ. As shown in Fig. 1, when d increases from 1 to 1, the equilibriums appear or disappear by a fold bifurcation of equilibrium points on the plane of ðd; xÞ.
Then shifting the coordinates so that the stationary solution is at the origin and linearization of system (2) results in
€x þ x_ x þ Axs ¼ 0
ð6Þ
The characteristic equation, obtained by substituting xðtÞ ¼ expðktÞ into (6), is
DðkÞ ¼ k2 þ k þ Aeks 1 ¼ 0
ð7Þ
Steady state bifurcation. Obviously, if A ¼ 1, (7) has a zero root. Thus, one expects a steady state bifurcation to occur and
is independent of s. To ensure that the root k ¼ 0 is a simple root of (7), we differentiate (7) with respect to k and obtain
dD
¼ Aseks þ 2k þ 1
dk
ð8Þ
It is easy to see that dDð0Þ=dk ¼ 0 only if s ¼ A1 ¼ 1 s . So excluding this value s , k ¼ 0 is only a simple root of (7).
In order to determine the crossing direction of characteristic root through the k ¼ 0, we calculate:
dk
1
j
¼
> 0ðor < 0Þ; if
dA A¼1 1 s
s < s or ðs > s Þ
ð9Þ
Therefore, excluding the special case s ¼ s , the trivial equilibrium point of DDE undergoes a steady state bifurcation at
A ¼ 1.
Hopf bifurcation and its direction. When the eigenvalues associated with equilibrium points pass across the imaginary
axis, Hopf bifurcation may occur. Hopf bifurcation curves on the plane of ðA; sÞ are obtained by substituting k ¼ iw into (7),
where w is real, yield
w2 þ wi þ Aewsi 1 ¼ 0
ð10Þ
Separating the real and imaginary parts to yield
(
w2 þ 1 ¼ A cosðwsÞ
w ¼ A sinðwsÞ
ð11Þ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Eliminating s from (11), we
have ﬃw4 þ 3w2 þ 1 A2 ¼ 0. By simple calculations, we obtain w2 ¼ 12 ð3 5 þ 4A2 Þ: For
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
2
w2 > 0, only w2 ¼ 12 ð3 þ 5 þ 4A Þ can be reserved and yields A > 1. Moreover, noting that cosðwsÞ > 0, w2 þ 1 > 0, we
obtain A > 1 and the following prescription for calculation of the bifurcation values about the time delay,
s¼
i
1h
ð1Þ w
2np þ sin
; where n ¼ 0; 1; 2 . . .
w
A
ð12Þ
Fig. 2. (a) Curves of fold bifurcation in DDE related to Eq. (5) on the plane of ðA; dÞ. (b) Curve of Hopf bifurcation in DDE related to Eq. (10) on the plane of
ðA; sÞ.
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Differentiating the characteristic Eq. (7) with respect to
dðRekÞ
ds !
Ae k
¼ Re
sAeks þ 2k þ 1 ks
s and A, for k ¼ iw, we have
3w2
>0
ð13Þ
dðRekÞ
eks
w2 ð1 þ s þ sw2 Þ
¼
Re
¼
>0
2
k
s
dA k¼iw
2k þ 1 Ase
Að1 sw2 Þ þ ð2w þ swÞ2
k¼iw
ð14Þ
k¼iw
k¼iw
¼
ð1 s
2
w2 Þ
þ ð2w þ swÞ2
Formula (13) and (14) imply the DDE obeys the sufﬁcient condition of the Hopf bifurcation theorem and undergoes a sequence of Hopf bifurcation on the critical stability boundaries which is given by Eq. (12). Fig. 2(b) illustrates a Hopf bifurcation curve deﬁned by Eq. (10) for n ¼ 0, relevant for the dynamics near the stationary solution on the plane of ðA; sÞ.
Furthermore, from analysis above, the region r is the stable region and there is a stable focus (seen in Fig. 3(a)). When s
increases till crossing the critical value deﬁned by equation of (10) to unstable region s, a pair of eigenvalues will cross
the imaginary axis and system (2) occurs Hopf bifurcation, i.e., a family of periodic solutions bifurcate from the equilibrium
point (seen in Fig. 3(b)).
3. Point–point burster in the delayed system
In this section, we ﬁrst analyze the bursting phenomena in DDE at A < 1. With the variation of slow variable quantity, the
behavior of bursting oscillations is much more complex than usual limit cycle, as the occurrence of them usually is associated with the bifurcation of the equilibrium points nearby. The fold/fold bursting is ﬁrstly deﬁned in Ref. [29]. In the following, the burster shown in Fig. 5 is point–point hysteresis loop of focus–focus type as the corresponding up-state and downstate are both stable focuses. The transitions between the up-state and down-state occur via fold bifurcations. Moreover, we
will explain the evolution of bursting oscillations of this type with the perturbations of some parameters including the delay.
3.1. Symmetric fold/fold bursting
In all this paper, we have limited the study to different two dimensional subspaces of the initial conditions. In this way,
we take initial conditions corresponding to linear functions of time, i.e. xðtÞ ¼ x0 þ l0 t, where ðx0 ; y0 Þ ¼ ð1; 1Þ and l0 ¼ 2, in the
initial interval ½s; 0. For A < 1, we ﬁx A ¼ 0:5, k ¼ 1, s ¼ 0:6, x ¼ 0:005, we can investigate the dynamical behaviors in DDE.
For x 1, d changes slowly. Based on the analysis above, when choosing appropriate value of k related for A, there is a
transition via fold bifurcation with the change of d periodically. The phase portrait as well as the time series is presented in
Fig. 4, from which one may ﬁnd the trajectories of the burster with symmetric structure oscillate between two parts associated with the two equilibrium points via fold bifurcation.
And then the concept of transformed phase space should be used to analyze oscillating behaviors of the system. Traditionally, phase portraits can be used to describe the relationship between different variables with the variation of time.
For example, the trajectory of an oscillator can be written in the form of ðx; yÞ ¼ ðxðtÞ; yðtÞÞ. However, the bursting phenomena above can be derived from the generalized autonomous system of (2), implying that the slow variable term d ¼ k sinðxtÞ
is regarded as one independent variable term. When d changes periodically with the variation of time t, the projection of (2)
may be expressed in the form of ðd; xÞ (or ðd; yÞ). On this generalized phase space, mechanism of bursting can be explored.
As shown in Fig. 5, we overlay the transformed portrait with the curve of equilibrium points in Fig 1(a). From the transformed phase space, the associated bifurcation mechanism can be described explicitly. When d changes from 1, the trajectory
starting at equilibrium point E1, moves strictly along the equilibrium curve which is in the attraction domain of the up
attractor, until fold bifurcation point F1. At that point, the trajectory of the system falls into the attraction domain of the
low attractor. So, the trajectory jumps to the low piece of equilibrium curve towards E2. For slow passage effect, the fast variable x spends much passage converging to the equilibrium curve. This procedure lasts until the trajectory arrives at point E2.
Fig. 3. Phase diagrams of DDE related to Eq. (1) for k ¼ 0. (a) A ¼ 1:5,
s ¼ 0:4. (b) A ¼ 1:5, s ¼ 0:75.
Y. Yu et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 1175–1184
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Fig. 4. Diagrams related to Eq. (1) for A ¼ 0:5, other parameters: ðs; k; xÞ ¼ ð0:6; 1; 0:005Þ. (a) Portrait phase. (b) Time series. (c) The enlargement of (b).
Fig. 5. Transformed phase diagram related to Eq. (1) on the plane of ðd; xÞ corresponding to Fig. 5, where the equilibrium curve of DDE shown in Fig. 1(a) is
superimposed.
When d changes from 1 to 1, similar dynamics take place for the symmetric structure of DDE. The trajectory tracks back
onto the attraction domain of the up attractor and bursting behavior occurs at fold point F2.
Sum to all, the system (2) depends on the slow variable d which controls transitions of the system. When the slow variable d moves up slowly, the trajectory tracks along the curve of the equilibrium points in down-state. However, if the
increasing rate of fast variables cannot be slow enough to agree with d, the solution starts to oscillate in large amplitude.
If the delayed oscillator does not have a limit cycle, we refer to such a burster as ‘symmetric fold/fold bursting’ of point–point
type.
3.2. Effects of gain coefﬁcient and time delay on this type of burster
We have investigated the dynamical mechanism of the fold bursting above. Obviously, bursting oscillations are created
since slow variable passes through values of the fold bifurcation. Only when forcing amplitude is greater than value of fold
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Fig. 6. Diagrams related to Eq. (1) for A ¼ 0:5, other parameters: ðs; k; xÞ ¼ ð0:6; 1; 0:005Þ. (a) Portrait phase. (b) Transformed phase diagram on the plane
of ðd; xÞ, where the equilibrium curve of DDE shown in Fig. 1(b) is also superimposed.
Fig. 7. Diagrams related to Eq. (1) for A ¼ 0:5, other parameters: ðs; k; xÞ ¼ ð6; 1; 0:005Þ. (a) Portrait phase. (b) Transformed phase diagram on the plane of
ðd; xÞ. (c) Time series. (d) The enlargement of (c).
bifurcation, this burster can appear. In the following, we focus on the effects of gain coefﬁcient and time delay on this
burster.
First, the effect of gain coefﬁcient is considered. Here, we plot the burster at A ¼ 0:5 in Fig. 6 where other parameters are
the same as A ¼ 0:5. We can contrast their equilibrium curves and fold bifurcation points in Fig. 1. By comparison, we would
like to point out that gain coefﬁcient A affects the amplitude of equilibrium curve and the critical values of fold bifurcation.
Thus the waveforms at A ¼ 0:5, have larger amplitude and keep much more time in down-state than ones at A ¼ 0:5.
Second, we turn to the effect of the time delay on this bursting. Basing on the analysis in second section, we can ﬁnd fold
bifurcation occurs which is independent of s. This indicates that such bursters will not lose its stability with varying of s and
a considerable large delay could still stabilize the bursting solution. Fig. 7 shows graphs of the burster ﬁxed at s ¼ 6, A ¼ 0:5,
k ¼ 1, where the interval of delay s ¼ 6 is much more than the original one of s ¼ 0:6. Bursting trajectories with symmetric
structure can also be presented, where the numerical results agree well with the analytical ones above. In particular, the
oscillations in down-state along the equilibrium curve is only controlled by the slow variable of d. So it can be seen in
Fig. 7(b), even if s increases so much, the bursting solution still has similar form of structure and nearly plots identical trajectory in down-state in comparison with Fig. 5. We would like to point out that the time delay will not affect the occurrence
of the fold bursting but it can be used to regulate its amplitude in up-state. This ‘twisted’ bursting generated by the delay are
shown in Fig. 7(d). The frequency of the fold bursting will not be affected by the delay and is still equal to forcing frequency.
Y. Yu et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 1175–1184
1181
Fig. 8. Diagrams related to Eq. (1) for A ¼ 1:5, other parameters: ðs; k; xÞ ¼ ð0:9; 4; 0:005Þ. (a) Phase portrait. (b) Time series. (c) The enlargement of (b).
4. Cycle-point burster in the delayed system
Further increase of the parameter of A > 1, may lead to disappearance of the fold bifurcation in equilibrium points. Meanwhile, if we modulate the delay, the supercritical Hopf bifurcation occurs and causes another type of bursting. Oscillations of
this type in down-state (rest) lose stability via Hopf bifurcation and approach to the limit cycle attractor which is called upstate (spiking). After that, the spiking shrinks to a point also via Hopf bifurcation. Transitions between the up-state and the
down-state occur via supercritical Hopf bifurcations. In the following, we illustrate the bursting mechanism of this type.
4.1. Symmetric Hopf/Hopf bursting
For A ¼ 1:5, s ¼ 0:8, x ¼ 0:005, k ¼ 4, another different type of bursting is plotted in Fig. 8. As shown in Fig. 8, the burster
is the connection of two parts of oscillations between equilibrium points with symmetric funnel structure via Hopf bifurcation. We can observe the critical value of Hopf bifurcation from Fig. 2(b) in the delay-differential equation.
The transformed phase portrait is presented in Fig. 9 to investigate the mechanism. When d changes from 1, the trajectory
starting at equilibrium point Eþ , moves almost along with the equilibrium curve which is in down-state, until it arrives at
point H2 . At that point, supercritical Andronov–Hopf bifurcation occurs, leading the trajectory to vibrate approximately
according to the limit cycle attractor, which belongs to up-state. Then the up-state loses stability at H1 also via supercritical
Andronov–Hopf bifurcation, i.e., the limit cycle attractor corresponding to up-state shrinks to the equilibrium point of E .
Similar situation takes place with the increase of d from 1 to 1.
Let us continue to elaborate. The mechanism of this symmetric Hopf bursting can be explained as follows: the delayed
system is controlled by the slowly periodic excitation term of d. The trajectory tracks along the curve of equilibrium points
which is corresponding to the down-state. Due to slow passage effect, the fast variable y (or x) diverges from down-state and
starts to oscillate with large amplitude via Hopf bifurcation caused by the delay. Then system goes into up-state. With the
increase of time, the slow variable d continues changing and passes through the other Hopf bifurcation point in the symmetric position and opposite direction. This provokes that the fast variable x (or y) shrinks to the related equilibrium point thereby completing half the bursting loop. The vector ﬁeld of DDE is invariant under the coordinate transformation of
ðx; yÞ ! ðx; yÞ, which leads to a symmetric transition behavior, ﬁnishing the whole bursting loop. In this case, the DDE
undergoes the switches between periodic limit cycle attractors and shrinking to one point via sup-Hopf bifurcations, we refer
to such a burster as ‘symmetric sup-Hopf/sup-Hopf’ bursting of Cycle-point type.
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Y. Yu et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 1175–1184
Fig. 9. Transformed phase diagram related to Eq. (1) on the plane of ðd; yÞ corresponding to Fig. 9, where H1 , H2 are Hopf bifurcation points and Eþ , E are
equilibrium points corresponding to d ¼ 4.
Fig. 10. Diagrams related to Eq. (1) for A ¼ 1:5, other parameters: ðs; k; xÞ ¼ ð0:9; 40; 0:005Þ. (a) Phase portrait. (b) Transformed phase diagram on the plane
of ðd; yÞ corresponding to (a).
Fig. 11. Phase portraits related to Eq. (1) for A ¼ 1:5, k ¼ 4, x ¼ 0:005. (a)
s ¼ 0:2. (b) s ¼ 0:6. (c) s ¼ 0:95. (d) s ¼ 1:2.
Y. Yu et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 1175–1184
1183
4.2. Effects of forcing amplitude and time delay on this type of burster
We have investigated the dynamical mechanism of Hopf/Hopf bursting above. Obviously, bursting oscillations are created
since system (2) occurs Hopf bifurcation on the space of ðA; sÞ. So only when A > 1 and s satisfy the critical condition of Hopf
bifurcation, this burster appears. Then we focus on the effects of forcing amplitude and time delay on this burster.
First, the effect of forcing amplitude is considered. As described by the previous section, this type of burster is mainly concerned with the occurrence of super-Hopf bifurcation for some ﬁxed value of d. This indicates that such burster will not lose
its stability with an increasing of k, when A and s satisfy the condition of Hopf bifurcation. A moderate large forcing amplitude could still stabilize the bursting solution. Here, we plot the bursters at k ¼ 40 in Fig. 10(a), where other parameters are
the same with k ¼ 4. We can compare their transformed phases (between Fig. 10(b) and Fig. 9), which show their dynamical
behaviors have similar structures. And the forcing amplitude can be large enough to affect the position of the equilibrium
points on both sides. So waveforms at k ¼ 40 have larger amplitude in the horizontal and vertical directions.
Second, we turn to the effect of time delay on this bursting. Basing on the analysis in the previous section, we can ﬁnd
Hopf bifurcation is concerned with time delay. If the value of the delay s cannot pass through the critical value of Hopf bifurcation, trajectory of the burster cannot be found either. By the numerical simulation in Fig. 11(a) and (b), we can see that
when s ¼ 0:2 or s ¼ 0:6, the solution does not have obvious bursting trajectories. The examples show roughly the time delay
can be used as a control parameter to change the occurrence of bursters of this type in delayed system. Consistently with the
intuition, when A > 1, ﬁxing other parameters, there is a domain of the values of s away from zero such that the bursting
solution is stable (in Fig. 11(c)), when s is in this domain and it is unstable for s leading to chaotic bursting (in
Fig. 11(d)), when s is above this domain. Numerical evidence shows that for such parameters of A > 1, k and s, the delayed
system is still symmetric with the bursting dynamics.
5. Conclusions
Dufﬁng oscillator with a linear delayed feedback and slowly varying external periodic force may exhibit different bursting
oscillations for the order gap existing between the frequency of the excitation and the natural frequency. We adopt the analytical approach by describing the external forcing as slow variable and combining this bifurcation analysis of the generalized autonomous system. Two types of bursters are obtained and their mechanism is discussed, which shows that
bifurcation forms play important role to the structures of these bursters. The bursting oscillations to be studied here are different from the usual. They are inﬂuenced by both external forcing and time delay.
When A < 1, the symmetric fold/fold burster related to the two symmetric equilibriums appears and is caused by the fold
bifurcation, which leads the transition between up-state and down-state. When A > 1, the symmetric Hopf/Hopf burster can
be observed by modulating the delay, in which two limit cycles caused by the Hopf bifurcation related to the two equilibriums interact with each other. Furthermore, we deal with the effect of some parameters including time delay on dynamics
of different bursters. The occurrence of the fold bifurcation bursting does not depend on the magnitude of the time delay. But
the Hopf bifurcation bursting depends critically on the magnitude of the delay. The time delay could be used as a tuning
parameter to change the dynamical behaviors of different bursters in different ways. Simulation results verify our results
and indicate methods used here are reliable and rather simple analysis techniques for bursting phenomena in delay-differential equations. It should be interesting to study bursting dynamics in systems with larger number of delays, or possibly
with an additive or multiplicative noise that might play some important role, and we will discuss those problems in the next
work.
Acknowledgements
The authors thank the editor and anonymous reviewers for their valuable comments and suggestions that helped to improve the presentation of the paper. The authors are supported by the National Natural Science Foundation of China (Grant
Nos. 21276115 and 11202085) and the Rearch Foundation for Advanced Talents of Jiangsu University (Grant Nos. 11JDG065
and 11JDG075).
References
[1] Rinzel J, Ermentrout GB. In: Koch C, Segev I, editors. Method in neuronal modeling. Cambridge: THE MIT Press; 1989.
[2] Zhen S, Han XJ, Bi QS. Bifurcations and fast-slow behaviors in a hyperchaotic dynamical system. Commun Nonlinear Sci Numer Simulat
2011;16:1998–2005.
[3] Savino GV, Formigli CM. Nolinear electronic circuit with neuron like bursting and spiking dynamics. Biosystems 2009;97:9–14.
[4] Lu QS, Yang ZQ, Duan LX, Gu HG, Ren W. Dynamics and transitions of ﬁring patterns, synchronization and resonances in neuronal electrical activities:
experiments and analysis. Acta Mech Sin 2008;24:593–628.
[5] Yu Y, Zhang C, Han XJ, Bi QS. Oscillations and their mechanism of compound system with periodic switches between two subsystems. Acta Phys Sin
2012;20:200501–7.
[6] Batista CAS, Lopes SR, Viana SL, Batista AM. Delayed feedback control of bursting synchronization in a scale-free neuronal network. Neural Networks
2010;23:114–24.
[7] Izhikevich EM. Bursts as a unit of neural information: selective communication via resonance. Trends Neurosci 2003;26:161–7.
[8] Lewis TJ, Rinzel J. Dynamics of spiking neurons connected by both inhibitory and electrical coupling. J Comput Neurosci 2003;14:283–309.
1184
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
Y. Yu et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 1175–1184
Izhikevich EM. Neural excitablity spiking and bursting. Int J Bifurcation Chaos 2000;10:1171–266.
Wang HX, Wang QY, Lu QS. Bursting oscillations, bifurcation and synchronization in neuronal systems. Chaos Solitons Fractals 2011;44:667–75.
Curtu R. Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network. Physica D 2010;239:504–14.
Duan LX, Lu QS, Cheng DZ. Bursting of Morris–Lecar neuronal model with current-feedback control. Sci China Ser E: Technol Sci 2009;52:771–81.
Simo H, Woafo P. Bursting oscillations in electromechanical systems. Mech Res Commun 2011;38:537–41.
Shorten PR, Wall DJN. A Hodgkin–Huxley model exhibiting bursting oscillations. Bull Math Biol 2000;6(2):695–715.
Bertram R, Butte MJ, Kiemel T, Sherman A. Topological and phenomenological classiﬁcation of bursting oscillations. Bull Math Biol 1995;57:413–39.
Azad AKA, Ashwin P. Within-burst synchrony changes for coupled elliptic bursters. SIAM J Appl Dyn Syst 2010;9:261–81.
Belykh I, de Lange E, Hasler M. Synchronization of bursting neurons: what matters in the network topology. Phys Rev Lett 2005;94:188101.
Buric´ Nikola, Rankovic´ Dragana. Bursting neurons with coupling delays. Phys Lett A 2007;363:282–9.
Leung AYT, Guo ZJ, Myers A. Steady state bifurcation of a periodically excited system under delayed feedback controls. Commun Nonlinear Sci Numer
Simulat 2012;17:5256–72.
Aguirregabiria JM, Etxebarria JR. Fractal basin boundaries of a delay-differential equation. Phys. Lett A 1987;122:241–4.
Kovacic Ivana. On the motion of a generalized van der Pol oscillator. Commun Nonlinear Sci Numer Simulat 2011;16:1640–9.
Khan QJA. Hopf bifurcation in multiparty political systems with time delay in switching. Appl Math Lett 2000;13:43–52.
Atay FM. Delayed-feedback control of oscillations in non-linear planar systems. Int J Control 2002;75(5):297–304.
Illing Lucas, Gauthier Daniel J. Hopf bifurcations in time-delay systems with band-limited feedback. Physica D 2005;210:180–202.
de Oliveira JCF. Oscillations in a van der Pol equation with delayed argument. J Math Anal Appl 2002;275:789–803.
Jeevarathinam C, Rajasekar SMA, Sanjuán F. Theory and numerics of vibrational resonance in Dufﬁng oscillators with time-delayed feedback. Phys Rev
E 2011;83:066205.
Xu J, Chung KW. Effects of time delayed position feedback on a van der Pol-Dufﬁng oscillator. Physica D 2003;180:17–39.
Han XJ, Bi QS. Bursting oscillations in Dufﬁng’s equation with slowly changing external forcing. Commun Nonlinear Sci Numer Simulat
2011;16:4146–52.
Izhikevich EM. Hoppensteadt, classiﬁcation of bursting mappings. Int J Bifurcation Chaos 2004;14:3847–54.