Periodic or chaotic bursting dynamics via delayed pitchfork bifurcation in a slow-varying controlled system

Accepted Manuscript

Periodic or chaotic bursting dynamics via delayed pitchfork bifurcation in a slow-varying controlled system Yue Yu, Zhengdi Zhang, Xiujing Han PII: DOI: Reference:

S1007-5704(17)30307-6 10.1016/j.cnsns.2017.08.019 CNSNS 4304

To appear in:

Communications in Nonlinear Science and Numerical Simulation

Received date: Revised date: Accepted date:

27 February 2017 5 August 2017 19 August 2017

Please cite this article as: Yue Yu, Zhengdi Zhang, Xiujing Han, Periodic or chaotic bursting dynamics via delayed pitchfork bifurcation in a slow-varying controlled system, Communications in Nonlinear Science and Numerical Simulation (2017), doi: 10.1016/j.cnsns.2017.08.019

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Highlights • The different bursting dynamics via delayed pitchfork bifurcation are discussed. • The slow-varying control item can be taken as a variable bifurcation parameter to investigate

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bursting dynamics.

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• Novel chaotic bursting oscillations may appear by Shilnikov connections or boundary crisis.

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Periodic or chaotic bursting dynamics via delayed pitchfork bifurcation in a slow-varying controlled system Yue Yu1∗ , Zhengdi Zhang2 , Xiujing Han2 1) Xinglin

of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, P.R. China

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2) Faculty

College, Nantong University, Nantong 226007, P.R. China

Abstract:

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In this work, we aim to demonstrate the novel routes to periodic and chaotic bursting, i.e., the different bursting dynamics via delayed pitchfork bifurcations around stable attractors, in the classical controlled L¨ u system. First, by computing the corresponding characteristic polynomial, we determine where some critical values about bifurcation behaviors appear in the L¨ u system. Moreover, the transition mechanism among different stable attractors has been introduced including homoclinic-

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type connections or chaotic attractors. Secondly, taking advantage of the above analytical results, we carry out a study of the mechanism for bursting dynamics in the L¨ u system with slowly peri-

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odic variation of certain control parameter. A distinct delayed supercritical pitchfork bifurcation behavior can be discussed when the control item passes through bifurcation points periodically. This

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delayed dynamical behavior may terminate at different parameter areas, which leads to different

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spiking modes around different stable attractors (equilibriums, limit cycles, or chaotic attractors). In particular, the chaotic attractor may appear by Shilnikov connections or chaos boundary crisis,

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which leads to the occurrence of impressive chaotic bursting oscillations. Our findings enrich the study of bursting dynamics and deepen the understanding of some similar sorts of delayed bursting phenomena. Finally, some numerical simulations are included to illustrate the validity of our study. Keywords: L¨ u system, Delayed pitchfork bifurcation, Bursting dynamics, Chaotic control.

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Corresponding author. Tel.:+86 513 55003306; Fax: +86 513 55003306. E-mail address: [email protected].

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1. Introduction Many dynamical systems in physics, chemistry, biology and geophysics involve multiple time scales [1-4], which often behave in periodic state characterized by a combination of relatively large

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amplitude and nearly harmonic small amplitude oscillations, conventionally described as bursting dynamics [5-7]. Generally, we say the system is in quiescent state (QS) stage when all the variables are at rest or exhibit small amplitude oscillations. The effect of multiple time scales may lead the systems to spiking state (SS), in which the variables may behave in large amplitude oscillations. Bursting phenomena can be observed when the variables alternating between QS and SS. Two important

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bifurcations can be found associated with the bursting dynamical behaviors: bifurcation of a quiescent state that leads to repetitive spiking and bifurcation of a spiking attractor that leads to quiescence [8-10].

Among various bursting patterns, bursting for homoclinic connections or chaotic attractors be-

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longs to a kind of complex oscillation patterns, and is one of the most attractive bursting patterns. Gu [11] provided experimental evidence of the existence of chaotic bursting and its transitions. Viana

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et al. [12] showed that chaotic bursting may be triggered by a loss of transversal stability of the low-period periodic orbits embedded in the invariant manifold. Izhikevich and Han et al. [13, 14]

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reviewed bursting in systems with multiple time scales, and pointed out that chaotic bursting is also linked to saddle-focus homoclinic orbit bifurcation and horseshoe structure [15-17]. Ge and Xu [18]

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pointed out that chaotic bursting observed in coupled neural oscillators is in fact a kind of chaotic itinerancy. Platt et al. [19] showed that, in a synaptically coupled system, whether chaotic bursting

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can be observed is decided by coupling strength. In particular, intermittency, resulted from various mechanisms such as fold bifurcation, subcritical Hopf bifurcation and inverse period-doubling bifurcation, is a common route to chaotic bursting, based on which, intermittent chaotic bursting patterns are intensively studied, e.g., see [20-24]. In this paper, we design a slow-varying L¨ u controller, the variable of which changes on much smaller time scale for investigating the dynamics of the whole system. Different types of bursting oscillations as well as the mechanism will be presented and some new phenomena, such as delayed pitchfork bursting of point/point or point/cycle type, or some chaotic bursting dynamical behaviors will be explained in details. We aim to explore the specific route to delayed bursting, i.e., periodic 3

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and chaotic bursting via delayed supercritical pitchfork bifurcation and its relationship with different attractors. The rest of this paper is organized as follows. In Sec. 2, we construct the non-autonomous dynamical system based on the L¨ u oscillator. In Sec. 3, we consider stabilities and bifurcations by

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considering the slow time-varying item as a bifurcation parameter, where the transitions between different types of attractors and chaotic multi-stability are focused on . Then, in Sec. 4, we turn to the bursting phenomena induced by this time-varying control, in which some interesting dynamical behaviors such as periodic or chaotic bursting dynamics are explored. Finally, in Sec. 5, we conclude

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the paper.

2. Mathematical model of controlled L¨ u oscillator

Upon the Chua’s circuits, showing chaotic oscillations in experiment for suitable parameters,

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many revised systems are established, among which the one presented by L¨ u and Chen [25, 26] can

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be written in the form

x˙ = a(y − x), y˙ = cy − xz,

(1)

z˙ = −bz + xy.

where a, b > 0, c ∈ R. Note that Eq. (1) is symmetric with respect to z-axis, which is invariant

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under the transformation (x, y, z) → (−x, −y, z). Different types of oscillations can be observed in L¨ u system associated with different parameters. It is easily found that Eq. (1) has trivial equilibrium

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point of E0 = (0, 0, 0) for c < 0, which is a stable node. When c > 0, the stable E0 turns into an √ √ unstable saddle-node, and two symmetric nontrivial equilibria E± = (± bc, ± bc, c) appear by a supercritical pitchfork bifurcation [27]. In our study, we include a periodically alternated electrical power to the parameter of c, i.e., c = α + βcos(ωt), where α is a constant control gain, β and ω respectively represent the amplitude and the frequency of the periodic parametric excitation, so as to investigate the occurrence of bursting

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3

subcritical

2.5

GH2(1,2)

supercritical

1.5

Homo

1

LPC

0.5

GH1(5/2,1/2)

subcritical 0 0

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b

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Fig. 1. Partial bifurcation diagrams of the FS on two parameter plane of (a, b) for c = 1 about the Hopf bifurcation.

dynamics. The mathematical model of this time-varying controlled system can be expressed as

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x˙ = a(y − x),

y˙ = [α + βcos(ωt)]y − xz,

(2)

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z˙ = −bz + xy.

Note that the case of time-varying frequency coupled with constant magnitude in system (2) has

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been well studied. However, when the parametric frequency of ω get into a smaller magnitude, i.e., 0 < ω  1, different dynamical behaviors, such as periodic and chaotic bursting patterns can be

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presented in controlled system. With the assumption of small order for the frequency of ω, it can be concluded that the parameter

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of c varies slowly. The controlled oscillator of Eq. (2) can be regarded as the coupling of two differential subsystems. One is the fast subsystem (FS), which has been presented in Eq. (1), while the slow subsystem is represented by c0 (t) = −βωsin(ωt). With the variation of the slow process, bursting phenomena as well as the mechanism of spiking modes will be presented by the bifurcation analysis in the Eq. (1) as a function of c.

3. Bifurcation analysis We start by shortly introducing some of the results about stabilities and bifurcations of the L¨ u system (i,e., the FS). Here we focus on the transition boundary and multi-stability, i.e., the 5

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0.5

(a)

1

x

0

sup−H+

E

(b)

Homo

E0

0

+

Homo (LPC)

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−0.5

x

2

LPC boundary

sup−H−

−1

−

sup−H−

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E−

E

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stable Limit cycle oscillation altitudes

−2 −0.5

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1

c

1.5

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1.01

1.02

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c

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Fig. 2. The stability and bifurcation diagram of the FS related to slow variable c for the fixed parameters of a = 2 and b = 1. (a) One parameter bifurcation diagram with the variation of c. (b) A local enlargement of (a).

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coexistence of equilibrium point, periodic or chaotic attractors, which is important for the following analysis in Sec. 4 about different generation mechanisms of bursting.

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3.1. Structure and the Hopf bifurcation

Now that a pitchfork bifurcation of equilibria in Eq. (1) appears at c = 0, we particularize

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the Hopf bifurcation on such equilibria. By analyzing the characteristic equation of Eq. (1), we infer that the E0 does not exhibit a Hopf bifurcation on the parameter plane, while at c > 0, Hopf

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bifurcations may occur with c = (a + b)/3 for non-trivial equilibria of E± , which leads two symmetric limit cycles around E± , respectively.

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By means of the calculation algorithm of normal form, we obtain the first Lyapunov coefficient

of Hopf bifurcation which determines the stability of the bifurcating limit cycle. According to the results of the sign of first Lyapunov coefficient, it can be concluded that the Hopf bifurcation around the non-trivial equilibria is subcritical at 0 < a/c < 1 or 5/2 < a/c < 3, and supercritical for 1 < a/c < 5/2. Furthermore, the L¨ u system has not three independent parameters, but it only depends on two, for instance a and b. There are infinitely many homothetic L¨ u systems parameterized by c, with exactly the same dynamical behavior. Thus, we draw a partial two-parameter bifurcation set for the L¨ u system on the (a, b) plane for the fixed parameter of c = 1. 6

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Fig. 3. Numerical simulations diagrams of the phase space of the FS for the fixed parameters a = 1, b = 2 and c = 1. Portrait orbits on the center manifold of x2 − 2z 2 = 0.

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As shown in Fig. 1, the Hopf curve of the two nontrivial equilibria has a first-order degenerate point (generalized Hopf bifurcation) GH1 = (5/2, 1/2). This Hopf bifurcation is subcritical on the

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right of GH1 , and supercritical on the left. The curve of limit point bifurcation of cycles (LPC) emerges from GH1 when the Hopf curve is crossed from the left to the right. This LPC curve ends

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at a point Homo ≈ (1.85, 1), where a homoclinic connection of the saddle presents a degeneration. A bifurcation of infinite codimension occurs at another generalized Hopf bifurcation point GH2 = (1, 2).

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The Hopf bifurcation is supercritical between the points GH1 and GH2 , which means a stable limit cycle is born when the Hopf curve is crossed from the right to the left. The Hopf bifurcation is

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subcritical again above the point GH2 . We can easily translate the above results at c = 1 to the other L¨ u systems for c > 0. The

bifurcation set on the (a, b) plane is exactly the same as shown in Fig. 1, i.e., a homothetic copy of Hopf bifurcation set with scale factor of c can be obtained. Furthermore, numerical simulations corresponding to supercritical Hopf bifurcation along with its bifurcating stable limit cycles are shown in Fig. 2, where c is used as the unique control parameter. It is seen in Fig. 2(a) that with the fixed values of a = 2 and b = 1, supercritical Hopf bifurcations (sup − H± ) around the two nontrivial

equilibria occur at c = 1, and a cluster of stable limit cycles can be presented with the increase of bifurcation parameter of c on the right of sup − H± point, the details of which are presented in Fig. 7

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Fig. 4. Partial bifurcation diagrams of the FS on two parameter plane of (a, b) for c = 1 about the homoclinic bifurcation.

2(b), where oscillation amplitudes of limit cycles bifurcated from sup − H− are simulated around E− .

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Both the limit cycles via supercritical Hopf bifurcations sup−H± may approach to the saddle and interact with each other at the bifurcation point homo with the parameter of c ≈ 1.0485 to form a

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stable large amplitude limit cycle via the saddle homoclinic bifurcation connected with the origin E0 , which is also the point that the LPC curve ends and occurs a degeneration at E0 . Furthermore, one

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of the invariant algebraic surfaces of the L¨ u system can be presented by x2 − 2az = 0 (when b = 2a). Analogously, in the case of the Hopf bifurcation of the nontrivial equilibria, the only polynomial

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center manifold is x2 − 2z 2 = 0 (when a = 1, b = 2, c = 1). Fig. 3 shows the invariant algebraic surface for the Hopf bifurcation of the nontrivial equilibria in the FS, where one can easily obtain

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that the two points E± are of hyperbolic foci type in initial, from which some periodic orbits on the

center manifold may appear. 3.2. Homoclinic connections and Shilnikov chaos As is well-known from the pioneering works of Shilnikov, the existence of Shilnikov homoclinic connections implies the complex dynamics including chaotic dynamics. One of the most important features of the Bogdanov-Takes (BT) bifurcation is that it warrants the existence of a homoclinic orbit in its vicinity. In Ref. [30, 31], the BT bifurcation combining bifurcation mechanism and Melnikov method has been discussed. By means of normal form methods, Algaba et al. [32, 33] presented

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Fig. 5. Phase diagrams of the FS for the fixed parameters of a = 3/2, c = 1. (a) Phase diagram of ”growing” limit cycles at b = 1.2, b = 1.3, b = 1.4 . (b) Loop like homoclinic connections at b = 1.5.

the occurrence of a degenerate homoclinic-type BT bifurcation of infinite codimension in the Lorenz system, in which some cascades of BT bifurcation of periodic orbits have also been detected.

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Consider the linearization matrix of the l¨ u system at the origin    −a a 0    0 c 0   0 0 −b

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whose characteristic polynomial is given by

     

F (λ) = λ3 + (a + b − c)λ2 + (ab − ac − cb)λ − abc

(3)

(4)

From the computation of characteristic polynomial (4), we infer that the condition of abc = 0

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can not be fulfilled so that there is no BT bifurcation for the L¨ u system itself [34]. However, a linear change in time and state variables demonstrates the L¨ u system is particular case of the Lorenz system. Thus, the results of detection of homoclinic orbits achieved for the Lorenz system allow to trivially obtain information on the dynamics presented by the L¨ u system. In this way, as a special case of the Lorenz system, we can trivially achieve the corresponding results of homoclinic orbits bifurcation generation from the L¨ u system. Emerged from the curve of BT bifurcation in the Lorenz system, the curve of the codimensiontwo bifurcation can be presented on the (a, b) plane for c = 1 (see in Fig. 4). This completes the results on the Hopf bifurcation discussed in Sec. 3.1, where the homoclinic orbit bifurcation curve 9

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exists and is marked as a green line. There are two degeneracies according to the computation of normal form. One degeneracy of the type does not produce new bifurcation curve, while the other is the intersection point between the homoclinic curve and the curve of limit point bifurcation of cycles (Homo point in Fig. 1). Note that the homoclinic curve emerged from generalized Hopf point

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GH1 = (5/2, 1/2), where the Hopf bifurcation around the nontrivial euilibria undergoes a first-order degeneracy. It also can be observed that the Hopf and homoclinic bifurcation intersect at another generalized Hopf point GH2 = (1, 2), where the Hopf bifurcation undergoes a degeneracy of infinite codimension.

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On the other hand, for all the other positive values of c, a homothetic copy of this bifurcation set as Fig. 4 can be obtained with scale factor of c. Moreover, the corresponding bifurcation set for the other negative values of c can be obtained by presenting a symmetry with respect to the origin from Fig. 4 (change the plane (a, b) to (−a, −b)). The corresponding portraits of two clusters of limit cycles starting from different initial points are presented in Fig. 5(a), where the parameters a and c

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are fixed at a = 3/2, c = 1, and b is assigned as 1.2, 1.3 and 1.4 respectively. With the variation of

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the parameter b, these two symmetric limit cycles grow close to those orbits homoclinic and approach to the saddle (the origin), turning into a homoclinic orbit through homoclinic boundary curve. The

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loop like homoclinci orbit is presented for the value of b = 1.5, as shown in Fig. 5(b). The above results demonstrate the existence of Shilnikov homoclinic connections related to some

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complex dynamics in the L¨ u system. As is well known, in each neighborhood of Shilnikov homobinic bifurcations, infinitely many periodic orbits of saddle type exist. It has been verified that these

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periodic orbits are contained in suspended horseshoes that accumulate onto the homoclinic orbit. And in this way, the occurrence of Shilnikov chaos is guaranteed. 3.3. Transition on multi-stability and boundary crisis The routes to chaos are important characteristic of nonlinear oscillators. With the exception of the presence of Shilnikov chaos we demonstrate above, there are some other important routes to chaos. Among them, crisis route to chaos is considered as a significant phenomenon, which represents a collision between the chaotic attractor and coexisting unstable periodic orbits (or unstable equilibrium). The L¨ u system can also provide this transient chaotic nature. Now we consider the non-stable periodic orbit, emerged from subcritical Hopf bifurcation of the 10

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x

0

E+

c=0.97 sub−H−

E−

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−1

1.5

sub−H+

boundary crisis

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0

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c

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(b)

1

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1

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x

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Fig. 6. The stability and bifurcation diagram of the FS related to slow variable c for the fixed parameters of a = 2.8 and b = 0.2. (a) One parameter bifurcation diagram with the variation of c. (b) Strange attractors at c = 0.97 indicating the occurrence of chaotic crisis.

nontrivial equilibria as has been discussed. The two non-trivial equilibria E± will lose stabilities by

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subcritical Hopf bifurcations at 0 < a/c < 1 or 5/2 < a/c < 3, which leads the occurrence of two unstable limit cycles. Numerical simulations related to sub-Hopf bifurcation sub − H± at c = 1 are

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shown in Fig. 6(a), where a = 2.8, b = 0.2 and c is the control parameter. In particular, there is a parameter interval in the left area close to the sub-Hopf bifurcation point, i.e., c ∈ [0.97, 1], where

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the maximum Lyapunov exponent is greater than zero and the corresponding chaotic behaviors can

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be presented in Fig. 6(b).

As shown in Fig. 6(a), the interval of [0.97, 1] belongs to the area of chaotic multi-stability, i.e., the coexistence of the chaotic attractor and the two nontrivial equilibrium attractors. When the

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control parameter c decreases and approaches the critical value of c = 0.97, the distance between the strange (chaotic) attractor and basin of attraction of the attractor decreases. At critical boundary value of c = 0.97, the chaotic attractor exhibits crisis and disappears.And in this way, the occurrence of chaotic transient of crises route is guaranteed. Then the L¨ u system loses its multi-stability and becomes bi-stable (two stable nontrivial equilibria).The other mode of destruction of the multistability can be observed when c increases the critical value of Hopf bifurcation of c = 1. For this case, the multi-stability is destroyed due to the instabilities of the two nontrivial equilibria. Thus the system has the only chaotic attractor and turns into a mono-stable chaotic attractor.

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take−off

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delay interval

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Fig. 7. Phase diagrams for Eq. (2) at a = 2, b = 1, α = 0.3, β = 0.7, and ω = 0.01. (a) On the space of (x, y). (b) On the space of (α + βcos(ωt), x) overlapped with bifurcation diagrams.

4. Dynamical mechanism of delayed pitchfork bursting phenomena

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We have briefly analyzed stabilities and bifurcations of the L¨ u system. Based on this, in what follows we turn to the non-autonomous case of Eq. (2) with the modulation frequency ω  1, and

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explore attractive bursting patterns therein.

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4.1. Delayed supercritical pitchfork bursting in point/point type Since c = α + βcos(ωt) is set as a slow varying variable, we can illustrate a slow passage

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effect through a pitchfork bifurcation as the slow variable passes through the zero periodically. As is known to all that dynamical mechanism related to this problem is the delayed pitchfork bifurcation

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phenomenon. In the following, the slow manifold and its bifurcation mechanism of bursting in this type will be presented in detail. When we fix the parameters in Eq. (2) at a = 2, b = 1, α = 0.3, β = 0.7, and ω = 0.01, the

corresponding phase portraits are presented in Fig. 7(a). The slow manifold on plane of (α+βcos(ωt), x) is superimposed to the phase diagram to get the clear statement on this dynamic, as is shown in Fig. 7b. The trajectories slowly and periodically pass through the supercritical pitchfork bifurcation, and the L¨ u system undergoes a catastrophic transition from the stable E0 to the other stable equilibriums. In such a case, after increasing through pitchfork bifurcation of P B point, the trajectories will still follow the unstable origin E0 for some time interval (delay interval in Fig.7(b)). After a while,

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E−

+

0.5 E

0

0 −1

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0 x

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1

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z

1

Fig.8. Portrait diagrams of bursting oscillations in point/point type from two different initial values with spiking mode around E− and spiking mode around E+ , respectively.

the transition from E0 to other attractors (the equilibria) can be observed, i.e., a delayed bifurcation

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behavior has be created, which leads to repetitive spiking mode.

Note that the origin E0 is the only attractor for c < 0, so the trajectory will eventually switch to

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E0 when c decreases through the P B point from positive to negative. Yet, it may be quite different for the case when the slow variable c increases through the P B point from negative to positive.

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As has already analyzed, at the right of c > 0, there are bistable equilibrium attractors E± , which provide possible transition as the pitchfork bifurcation delay appears. Thus, such bursting dynamics

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can be regarded as ”delayed pitchfork bursting of point-point type”. Furthermore, the supercritical pitchfork bifurcation results in two coexisting stable equilibria, the

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trajectories of bursting will tend to the nontrivial equilibrium E+ or E− , which depends on the initial

values sensitively. In fact, with the same parameters in Fig. 7, by choosing a set of symmetric initial values of (x0 , y0 , z0 ) and (−x0 , −y0 , z0 ), we can present a pair of stable periodic bursting solutions surrounding with E+ or E− respectively, as shown in Fig. 8. Such particular evolution behaviors may be common in dynamical systems with multiple equilibrium states , i.e., bursting dynamics around multiple stable branches depend on the selection of initial values.

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E+

E−

1 take−off

1 x

z

Limit cycles 0.5

0

E0 PB

delay interval sup−H−

−1

E0

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−1

x

−0.5

E−

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0

0

0.5 α+β cos(ω t)

1

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(b)

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1.5

2

(a)

Fig. 9. Phase diagrams for Eq. (2) at a = 2, b = 1, α = 0.3, β = 0.74, and ω = 0.01. (a) On the space of (x, y). (b) On the space of (α + βcos(ωt), x) overlapped with bifurcation diagrams.

4.2. Delayed supercritical pitchfork bursting in point/cycle type

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On the other hand, if we increase the amplitudes of the controlled item, i.e., the slow variable c is across the critical value of sup-Hopf bifurcation, the two symmetric limit cycles appear which

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leads to the new generation forms of spiking mode, in which the trajectories switch between the stable cycle and stable equilibrium point.

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For example, we modulate the excitation amplitude β from 0.7 to 0.74, in order to investigate the corresponding dynamical behaviors. The phase portraits are presented in Fig. 9(a), and the slow

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manifold is superimposed to the phase diagram to get a clear idea about the dynamical behaviors on plane of (α + βcos(ωt), x).

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In such a case, after increasing through pitchfork bifurcation of P B point, the trajectories will

still follow the unstable origin E0 for some time interval. Then, a delayed transition from E0 to other equilibrium attractor E− can be observed. Upon then, a supercritical Hopf bifurcation will

occur, where the trajectories begin to oscillate in fairly large amplitudes approaching to limit cycle attractor (see in Fig. 9(a)). To put it simply, when the trajectories periodically pass through the supercritical pitchfork bifurcation value, a delayed pitchfork bursting behavior can be presented. Such dynamical behavior does not disappear until the limit cycle oscillation may be created by supercritical Hopf bifurcation, which leads to repetitive complex spiking formations. We can refer to the above behaviors as delayed pitch14

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E+

E− Limit cycles

0.5 E0

0 −1

−0.5

0 x

0.5

1

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z

1

Fig. 10. Portrait diagrams of bursting oscillations in point/cycle type from two different initial values with spiking mode around E− and spiking mode around E+ , respectively.

fork/ sup-Hopf bursting of point-cycle type, for the repetitive spiking process appears by pitchfork

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bifurcation, while it terminates by sup-Hopf bifurcation, and the accompanying stable attractors are a stable equilibrium and a limit cycle. Similar to the above, we can present two symmetric spiking

as is shown in Fig. 10.

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modes in point/cycle type around E− and E+ respectively, starting from two different initial values,

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4.3. Delayed supercritical pitchfork bursting for Shilnikov connections Further increase of the excitation amplitude of β will result in the occurrence of Shilnikov

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homoclinic connections, which implies the presence of complex bursting dynamics in the controlled L¨ u system. As mentioned, under the influence of the homoclinic orbits, the two stable limit cycles

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bifurcated from the two nontrivial equilibriums (see in Fig. 5(a)) may interact with each other, i.e., the homoclinic connection of the two nontrivial equilibria arise (see in Fig. 5(b)). For instance, from the phase portrait as well as the related time histories plotted in Fig. 11 at

β = 1, one may easily find that there are obvious differences from those portraits introduced before. These bursting dynamics are related to E0 and E± , and the amplitude may suddenly increase to connect with the homoclinic loop orbit. We superimpose the slow manifold on plane of (α+βcos(ωt), x) to the phase portraits to detect the corresponding bursting mechanism in Fig. 12. As shown in Fig. 12, the trajectory will still follow the unstable origin E0 for some time delayed interval. Then a catastrophic transition from E0 to other stable equilibrium attractors can be ob15

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E0 −1

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Fig. 11. Phase diagrams for Eq. (2) at a = 2, b = 1, α = 0.3, β = 0.8, and ω = 0.01. (a) Phase portrait. (b)

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Time histories.

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x

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Homo

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Fig. 12. Delayed pitchfork bursting for Shilnikov connections on the space of (α + βcos(ωt), x) overlapped with bifurcation diagrams.

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2

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Fig. 13. Bursting oscillations for chaotic crisis from two different initial values, when a = 2.8, b = 0.2, α = 0.3, β = 0.7, and ω = 0.01 in Eq. (2). (a) Chaotic spiking mode around E− . (b) Chaotic spiking mode around E+ .

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served. After then, supercritical Hopf bifurcations of sup − H± will occur, where the trajectories begin to oscillate in fairly large amplitudes approaching to limit cycles around the nontrivial equi-

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librium E± . Such dynamical behavior does not disappear until Shilnikov connections (homoclinic bifurcation) occurs, which leads to the dramatic change of the amplitude corresponding to generating

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a global cycle like a relaxation oscillation. Obviously, in such case, there are three transition behaviors, i.e., delayed pitchfork bifurcation

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behavior, sup-Hopf bifurcation behavior and Shilnikov homoclinic connections, leading to repetitive bursting dynamics. We can refer to such oscillation modes as delayed pitchfork/sup-Hopf/Shilnikov

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connection bursting of point/cycle/relaxation oscillation type, for the initial spiking process appears by delayed pitchfork bifurcation and Hopf bifurcation, while it terminates by Shilnikov connections. 4.4. Delayed supercritical pitchfork bursting for chaotic crisis Now we consider bursting dynamics for the cases with subcritical Hopf bifurcation of the nontrivial equilibria at 0 < a/c < 1 or 5/2 < a/c < 3 and confronting the fact in the occurrence of chaotic boundary crisis . As has been analyzed, there are multi-stable chaotic area and the monostable chaotic area, which provides two possibility where the pitchfork bifurcation delay terminates. Therefore, different chaotic transition behaviors may be obtained. First, we deal with the case when the delayed pitchfork bifurcation behavior terminates at the 17

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multi-stable chaotic area, which is related to the generation of chaotic bursting. Fig. 13 shows the symmetric examples of chaotic multi-stable bursting oscillations for the fixed parameters of a = 2.8, b = 0.2, α = 0.3, β = 0.7, and ω = 0.01 from two symmetric initial values . It is seen in Fig. 13, this case of the chaotic bursting looks generated by crisis boundary like half a ”butterfly” spreading its

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wings and perching on the stable branch (E− or E+ ). Moreover, we superimpose the slow manifold on plane of (α + βcos(ωt), x) to the phase portraits in Fig. 14, to detect the corresponding bursting mechanism.

As shown in Fig. 14, when the delayed pitchfork behavior terminates at the multi-stable chaotic

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area, the trajectory tends towards the equilibrium of E− , and then undergoes a fast transition to the chaotic attractor. Then a cluster of chaotic signals related to the chaotic attractor appear in the diagram. When the slow parameter passes back to pitchfork bifurcation, and enters the bi-stable area, the chaotic attractor disappears by boundary crisis, which thus lead the trajectory to jump to the focus attractor of E− (see in Fig. 13(a)). In fact, the trajectory can switch to E+ or E− , which

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depends on the initial values sensitively (see in Fig. 13(b)). Then the chaotic signals in the time

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histories disappear. When the slow parameter decreases through pitchfork bifurcation point, there is a soft transition from E− to E0 , and then the system turns to quiescent state. After reaching its

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minimum, the slow parameter begins to increases. When it increases through pitchfork point again, the delayed behaviors take place again and the next evolution starts.

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We have shown that the chaotically active phase of the bursting is induced by a delayed pitchfork bifurcation and disappears by a boundary crisis. According to the classification scheme of bursting

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, this type of chaotic bursting can be classified as “delayed pitchfork bifurcation/boundary crisis” bursting.

Next, further increase of the parameter of β will lead to the fact that the delay terminates at the

mono-stable area. Chaotic bursting can still be obtained if the delayed behavior terminates at the mono-stable area. Note that there is the only strange attractor in the mono-stable area, and thus the trajectory may switch to the chaotic attractor after delay, which will further lead to novel bursting oscillations around the strange attractor. For instance, from the phase portrait as well as the related time histories plotted in Fig. 15 at β = 1, one may find that there exist obvious difference between the two chaotic bursters in Fig. 13

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1 E+

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Fig. 14. Delayed pitchfork bursting for chaotic crisis corresponding to Fig. 13(a), on the space of (α+βcos(ωt), x) overlapped with bifurcation diagrams.

and in Fig. 15, though both of them are related to the chaotic attractors. Moreover, we superimpose

corresponding bursting mechanism.

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the slow manifold on plane of (α + βcos(ωt), x) to the phase portraits in Fig. 16, to detect the

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As shown in Fig. 16, when the delay behavior terminates at the mono-stable chaotic area, the trajectory undergoes a fast transition to the chaotic attractor, as the chaotic attractor is the only

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attractor therein. When the slow parameter passes through back and enters the bi-stable area, the chaotic attractor disappears by boundary crisis, which thus lead the trajectory to jump to the

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attractors of E+ and E− in two ways. Then the chaotic signals in the time series disappear. When the slow parameter decreases through pitchfork bifurcation point, there is a soft transition from E−

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to E0 , and then the system turns to quiescence. After reaching its minimum, the slow parameter begins to increases. When it increases through ρP B , the delayed behaviors take place again and the next evolution starts.

5. Conclusion Delayed pitchfork bursting behavior is a kind of complex bursting pattern, and uncovering the essence of the complexity is an important issue in bursting dynamics. In this paper, we reveal some novel periodic or chaotic bursting oscillations, in which the delayed supercritical pitchfork bifurcation behaviors have been found. And soon the system undergoes a catastrophic transition 19

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1

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Fig. 15. Bursting oscillations for chaotic crisis from two different initial values, when a = 2.8, b = 0.2, α = 0.3,

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β = 1, and ω = 0.01 in Eq. (2). (a) Phase portrait. (b) Time histories.

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x

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Fig. 16. Delayed pitchfork bursting for chaotic crisis corresponding to Fig. 15(a), on the space of (α+βcos(ωt), x) overlapped with bifurcation diagrams.

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around some different attractors including limit cycles or chaotic attractors. Moreover, we have explored dynamical mechanisms underlying the appearance of the chaotically active phase of the bursting. Our study shows that, the termination of bifurcation delay may leads the trajectory to switch to

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the differen attractors, which explains the occurrence of delay-induced bursting oscillations. On the other hand, the strange attractor may suddenly appear by two routes, i.e., Shilnikov connections or boundary crisis, which results in a transition from the chaotically active phase to quiescence state. In a word, there are the rich transitions between equilibrium point and limit cycles or chaotic attractors

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induced by delayed supercritical pitchfork bifurcation in this controlled oscillator, which gives rise to the periodic or chaotic bursting. Our results enrich the possible routes to bursting dynamics as well as the underlying mechanisms. It should be interesting to study such dynamics in other nonlinear

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systems with two or more degrees, and we consider to discuss that in forthcoming paper.

Acknowledgments

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This work is supported by the National Natural Science Foundation of China (Grant Nos.

References

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11472115, 11472116 and 11572141).

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[1] Izhikevich, E.M.: Neural excitability, spiking and bursting. Int. J. Bifurc. Chaos 10, 1171-1266

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(2000)

[2] J. Rinzel. Bursting oscillation in an excitable membrane model, in: B.D. Sleeman, R.J. Jarvis (eds.), Ordinary and Partial Differential Equations, Springer, Berlin, 1985, pp. 304-316.

[3] Li X.H., Hou J.Y., Chen J.F.: An analytical method for Mathieu oscillator based on method of variation of parameter. Commun. Nonlinear Sci. Numer. Simulat. 37 (2016) 326-353. [4] Sriram, K., Gopinathan, M.S.: Effects of delayed linear electrical perturbation of the BelousovZhabotinsky reaction: a case of complex mixed mode oscillations in a batch reactor. React. Kinet. Catal. L. 79, 341-349 (2003)

21

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[5] Lu, Q.S., Gu, H.G., Yang, Z.Q., Shi, X., Duan, L.X., Zheng, Y.H.: Dynamics of fiering patterns, synchronization and resonances in neuronal electrical activities: experiments and analysis. Acta Mech. Sin. 24, 593-628 (2008)

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[6] Roberts, A., Widiasih, E., Jones, C.K.R.T., Wechselberger, M.: Mixed mode oscillations in a conceptual climate model. Physica D 292-293, 70-83 (2015)

[7] Li X.H., Hou J.Y.: Bursting phenomenon in a piecewise mechanical system with parameter perturbation in stiffness. International Journal of Non-linear Mechanics. 81 (2016) 165-176.

Trends Neurosci. 20, 38-43 (1997)

AN US

[8] Lisman, J.E.: Bursts as a unit of neuronal information: making unreliable synapses reliable.

[9] Liepelt, S., Freund, J.A., Schimansky-Geier, L., Neiman, A., Russell, D.F.: Information processing in noisy burster models of sensory neurons. J. Theor. Biol. 237, 30-40 (2005)

M

[10] Miller, B.R., Walker, A.G., Shah, A.S., Barton, S.J., Rebec, G.V.: Dysregulated information processing by medium spiny neurons in striatum of freely behaving mouse models of Huntington’s

ED

disease. J. Neurophysiol. 100, 2205-2216 (2008)

PT

[11] H.G. Gu.: Experimental observation of transition from chaotic bursting to chaotic spiking in a neural pacemaker. Chaos 23, 023126 (2013);

CE

[12] Viana, R.L., Pinto, S.E., Grebogi, C.: Chaotic bursting at the onset of unstable dimension variability. Phys. Rev. E 66, 046213 (2002)

AC

[13] E.M. Izhikevich, Neural excitability, spiking and bursting. Int. J. Bifurcat. Chaos. 10 (2000) 1171-1266.

[14] X.J. Han, F.B. Xia, P. Ji, Q.S. Bi, J. Kurths: Hopf-bifurcation-delay-induced bursting patterns in a modified circuit system. Commun. Nonlinear Sci. Numer. Simulat. 36 (2016) 517-527. [15] Terman, D.: Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math. 51, 1418-1450 (1991) [16] Han, S.K., Postnov, D.E.: Chaotic bursting as chaotic itinerancy in coupled neural oscillators. Chaos 13, 1105-1109 (2003) 22

ACCEPTED MANUSCRIPT

[17] Zhang, F., Zhang, W., Lu, Q.S., Su, J.Z.: Transition mechanisms between periodic and chaotic bursting neurons. In: Wang, R.B., Gu, F.J. (eds.) Advances in Cognitive Neurodynamics (II), pp. 247-251. Springer, Netherlands (2010)

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[18] Ge, J.H., Xu, J.: Stability switches and fold-Hopf bifurcations in an inertial four-neuron network model with coupling delay. Neurocompu- ting, 110 (2013) 70-79.

[19] Platt, N., Spiegel, E.A., Tresser, C.: On-off intermittency: A mechanism for bursting. Phys. Rev. Lett. 70, 279-282 (1993)

AN US

[20] Duan, L.X., Fan, D. G., Lu, Q.S. Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling. Cognitive Neurodyn.7 (2013) 341-349 [21] Gu, H.G., Xiao, W.W.: Difference between intermittent chaotic bursting and spiking of neural firing patterns. Int. J. Bifurcat. Chaos 24, 1450082 (2014)

M

[22] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcation of

ED

Vector Field. Springer, New York (1983)

[23] Grebogi, C., Ott, E., Yorke, J.A.: Crises, sudden changes in chaotic attractors, and transient

PT

chaos. Physica D 7, 181-200 (1983)

[24] Grebogi, C., Ott, E., Yorke, J.A.: Super persistent chaotic transients. Ergod. Theor. Dyn. Syst.

CE

5, 341-372 (1985)

AC

[25] Sparrow, C.: The Lorenz Equations. Springer-Verlag, New York, Heidelberg, Berlin (1982) [26] Wiggins, S.: Global Bifurcation and Chaos: Analytical Methods. Springer, New York (1988) [27] Berglund, N., Gentz, B.: Pathwise description of dynamic pitchfork bifurcations with additive noise. Probab. Theor. Rel. Fields 122, 341-388 (2002) [28] L, Z., Duan, L.: Control of codimension-2 Bautin bifurcation in chaotic L system. Commun. Theor. Phys. 52 (2009) 631-636 [29] Yu,Y., Zhang, S.: Hopf bifurcation analysis of the Lsystem. Chaos Solitons Fract. 21 (2004) 1215-1220 23

ACCEPTED MANUSCRIPT

[30] Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory.Springer, New York (2004) [31] Algabaa, A., Fern´ andez-S´ anchezb, F., Merinoa M, Rodr´ıguez-Luisb A. J.: : Centers on center manifolds in the Lorenz, Chen and L systems. Commun. Nonlinear Sci. Numer. Simul. 19, (2014)

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772-775 [32] Algabaa, A., Dom´ınguez-Moreno, F., Merinoa M., Rodr´ıguez-Luisb A. J.: Takens-Bogdanov bifurcations of equilibria and periodic orbits in the Lorenz system. Commun. Nonlinear Sci. Numer. Simul. 30 (2016) 328-343

AN US

[33] Algabaa, A., Fern´ andez-S´ anchezb, F., Merinoa M, Rodr´ıguez-Luisb A. J.: Chens attractor exists if Lorenz repulsor exists: the Chen system is a special case of the Lorenz system. Chaos 23, (2013) 033108

[34] Algabaa, A., Fern´ andez-S´ anchezb, F., Merinoa M, Rodr´ıguez-Luisb A. J.: The L¨ u system is a

AC

CE

PT

ED

M

particular case of the Lorenz system. Phys. Lett. A. 377 (2013) 2771-2776

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