Relaxation Oscillations and the Mechanism in a Filippov-Type … Vol 29 No 1 Pape… · Relaxation...

ISSN 1749-3889 (print), 1749-3897 (online)International Journal of Nonlinear Science

Vol.29(2020) No.1,pp.3-11

Relaxation Oscillations and the Mechanism in a Filippov-Type System withPeriodic External Excitation

Ying Zhang, Zhengdi Zhang∗

Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, P.R. China(Received 23 January 2020, accepted 12 March 2020)

Abstract: The main idea of this paper is to explore the dynamical behavior of a Filippov system with twoscales. A new Filippov-type model based on the modified Liu model with periodic excitation is employedhere as an example. For the case that the exciting frequency is relatively small compared to the naturalfrequency, the exciting term can be considered as a slow-varying parameter, and thus the equilibrium branchesas well as those bifurcations of two subsystems divided by the non-smooth boundary can be derived. For atypical set of parameters, since several smooth and non-smooth bifurcations occur in the vector field, differenttypes of bursting oscillations, such as fold/fold-sliding and Hopf/fold/fold-sliding bursting, can be observedwith the variation of excitation amplitude, and the mechanism of which can be revealed via overlappingthe transformed phase portrait and the equilibrium branches. Furthermore, three different types of slidingmovement along the non-smooth boundary can be observed, leading to different forms of the spiking states.

Keywords: Filippov systems; bursting oscillations; grazing-sliding bifurcation; switch-sliding bifurcation;bursting mechanism

1 IntroductionSince various patterns of spikes in a neuron were discovered in the slow-fast H-H model which is established by theNobel Prize winners Hodgkin and Huxley[1], the dynamical phenomenon in slow-fast system has attracted much interest[2]. Particularly, two time scales in the system may lead to bursting oscillations, which can be respectively characterizedby the large-amplitude type, denoted by the spiking states (SP ), and the small-amplitude type or at rest, representedby the quiescent states (QS). Instead of revealing the bursting mechanism, most of earlier works had concentrated onnumerical simulations as well as the method for approximate solutions[3, 4] until Rinzel presented the slow-fast analysismethod[5, 6], which divides the original system into two subsystems, namely the slow and fast subsystems, written in theform

x = f(x, y, µ), (Fast Subsystem)y = ϵg(x, y, µ). (Slow Subsystem)

(1)

where 0 < ϵ ≪ 1 represents the ratio of two subsystems and therefore the slow state variable y can be regarded as slow-varying parameter. Then the mechanism of bursting oscillations can be revealed by overlapping the transformed phaseportrait and equilibrium branches on the (x, y) plane[7], in which the quiescent and spiking states as well as bifurcationsare determined by the fast subsystem while the slow subsystem may modulate oscillations. Owing to this slow-fastanalysis method, several patterns of bursting attractors in autonomous systems such as the fold/fold and fold/Hopf type[8]have been presented in recent years.

While for the non-autonomous systems with two scales, for instance the periodic excited oscillators whose excitingfrequency is far less than the natural one, the original method cannot be employed anymore since no obvious slow and fastsubsystems can be isolated while the coupling effect of two time scales remains. To explore the mechanism, new methodneeds to be developed and therefore we modify the original (1) to the following form

∗Corresponding author. E-mail address: [email protected]

Copyright c⃝World Academic Press, World Academic UnionIJNS.2020.03.25

4 International Journal of NonlinearScience,Vol.29(2020),No.1,pp. 3-11

x = f(x, µ, w), (Fast Subsystem)w = A sin(Ωt). (Slow Subsystem)

(2)

Similar to Rinzel’s method, here we consider the exciting term w as a generalized slow variable, thus the equilibriumbranches and the corresponding bifurcation points can be derived with the variation of the slow-varying parameter w,meanwhile the transformed phase portraits can be obtained to explore the bursting mechanism of the system with para-metric or external excitation[9]. Based on this method, several types of bursting oscillations with two scales in frequencydomain has been reported[10–12]. Most of the works with regard to the coupling effect of two scales mainly focus onthe smooth systems, while for the non-smooth systems governed by different subsystems[13], dynamical behaviors arerelatively complicated[14] since the discontinuity of the vector field. Traditional methods cannot be directly applied toexplain the behaviors of the trajectory crossing the non-smooth boundary because several non-smooth bifurcations[15, 16]may occur in the vector field, leading to different forms of the spiking states as well as the quiescent states. Since then,most of the earlier works have focused on the dynamics of Filippov systems with only one scale[17].To explore the effectof two scales in Filippov systems, a modified Liu chaotic system with external excitation is introduced, and different typesof bursting oscillations can be observed with the variation of the exciting amplitude.Transformed phase portrait. In order to explore the mechanism of the bursting oscillations in non-autonomous system,we introduced the concept of the transformed phase portrait, where the slow-varying parameter w is regarded as a gener-alized state variable. Since the equilibrium branches as well as bifurcations of the fast subsystem reveal the relationshipbetween the state variable and generalized state variable w, the transformed phase portrait can help to demonstrate theinfluence of the fast subsystem on the dynamical behaviors with the trajectory of w via overlapping with equilibriumbranches. For the trajectory of the movement, the traditional phase portrait can be defined by Π ≡ [x(t)] | t ∈ R, whilethe transformed phase is then described by ΠG ≡ [x(t), w] | t ∈ R = [x(t), A cos(Ωt)] | t ∈ R.

The structure of this paper is arranged as follows. By means of introducing an external excitation to the Liu chaoticmodel, a typical Filippov-type system with two scales, which may exhibit complex dynamical behaviors, is established insection two, and the equilibrium branches as well as the associated bifurcations are analyzed in section three. Furthermore,the dynamical evolution of the bursting oscillations are presented in section four, whose mechanism are alse revealed viaapplying the overlap of the transformed phase portrait as well as the equilibrium branches.

2 Mathematical ModelIn order to explore the dynamical behaviors of a Filippov system with two time scales, the Liu chaotic system[18] isemployed here as an example. By introducing a non-smooth factor as well as a external excitation, a new mathematicalmodel can be written in the form x = a(y − x),

y = bx− kxz + δsgn(y) + w,z = −cz + hx2.

(3)

where w = Acos(Ωt). A and Ω respectively represent the amplitude and the frequency of the excitation term w in thissystem and the parameters a, b, c, k and h are all real. when the exciting frequency Ω is far less than the natural frequencyωN , i.e., Ω ≪ ωN , leading to two scales in the frequency domain. Moreover, since the non-smooth boundary, denotedby Σ = (x, y, z) | y = 0, divides the phase space into three regions, respectively represented by D0 = y = 0,D+ = y > 0 and D− = y < 0, dynamics governed by different regions as well as the boundary may change thebehaviors of dynamics. To explore the slow-fast dynamics in a Filippov system, we first turn to the bifurcation analysis.

3 Bifurcation analysis

3.1 Bifurcation analysis of two generalized autonomous subsystemsOwing to an order gap between the natural frequency and the exciting frequency, the state variables may oscillate accordingto the natural frequency ωN , while for the excitation term w = Acos(Ωt) during any period t, t ∈ [t0, t0 + 2π/ωN ], itundergoes a variation from w = Acos(Ωt0) to w = Acos(Ωt0 + 2πΩ/ωN ), which is relatively slight compared tothe variation of state variables that even can be ignored. Therefore, the external excitation w can be considered as a

IJNS email for contribution: [email protected]

Y. Zhang, Z. Zhang, Relaxation Oscillations and the Mechanism in a Filippov-type System 5

generalized state variable and hence the two subsystems can be treated as the generalized autonomous system, which canbe expressed as

S+ =

x = a(y − x),y = bx− kxz + δ + w,z = −cz + hx2.

(4)

for y > 0. and for y < 0,

S− =

x = a(y − x),y = bx− kxz − δ + w,z = −cz + hx2.

(5)

the equilibrium points of two subsystems can be uniformly represented by E0(x, y, z) = (X0, X0,hX2

0

c), where the X0

satisfies −kX03h

c +bX0 +δsgn(X0)+w = 0, and the stability of which can be determined by the characteristic equation,written in the form

λ3 + (a+ c)λ2 +

(hX0

2ak

c− ab+ ac

)λ+ 3hX0

2ak − abc = 0. (6)

According to the Routh-Hurwitz Criterion, equilibrium points are asymptotically stable under the certain circum-stances, which can be expressed as

a+ c > 0,3hX0

2ak − abc > 0,

(a+ c)(

hX02akc − ab+ ac

)− 3hX0

2ak + abc > 0.(7)

For stable equilibrium branches qualified by above condition, the stability of which may change with the parametervarying via two types of codimension-1 bifurcations, fold bifurcation (FB) and Hopf bifurcation (HB), etc. the necessarycondition of the former type is

FB : 3hX02ak − abc = 0. (8)

in which the fold bifurcation may take place, leading to the jumping phenomenon between equilibrium points. Note thatwith the bifurcation parameter varying, the Hopf bifurcations may appear when

HB : − (a+ c)(

hX02akc − ab+ ac

)+ 3hX0

2ak − abc = 0. (9)

which may lead to periodic oscillation approximately at the frequency ΩH =√

hX02akc − ab+ ac. Due to the discon-

tinuity of the vector field, when the trajectory travels across the non-smooth boundary, the non-smooth bifurcations mayoccur. Here we turn to the bifurcation analysis on the non-smooth boundary.

3.2 Bifurcation analysis on the non-smooth boundaryBased on the Filippov theory, the non-smooth system can be expressed as

X ∈ F (X) = qF+(X) + (1− q)F−(X). (10)

in which q ∈ [0, 1]. Obviously, the dynamics of two subsystems is governed by F+(X) = S+ and F−(X) = S− for q = 1and q = 0, respectively. While for q ∈ (0, 1), since the equilibrium point exists on the boundary when X0 = 0, denotedby EΣ(0, 0, 0), dynamics may be affected by non-smooth bifurcations. The associated related characteristic equations canbe obtained by the differential inclusion theory[19], written as

λ3 + (a+ c)λ2 + (−ab+ ac)λ− abc = 0. (11)

indicating that the sliding bifurcation may take place for w ∈ [δ,−δ] when δ < 0.

IJNS homepage: http://www.nonlinearscience.org.uk/


-15 -10 -5 0 5 10 15-8

-6

-4

-2

0

2

4

6

8

EB-4

EB+4

LC-1

EB+1

EB+2

EB+3

EB-3

EB-2

EB-1

LC+1

FB-1

FB+1

HB+1

HB-1

y

w=Acos( t)

Figure 1: Equilibrium branches as well as their bifurcations on the (w, y) plane for a = 1.2, b = 2, c = 5, k = 1, h =1.5, δ = −2.5.

3.3 Equilibrium branches as well as their bifurcationsTo investigate the oscillations of the system, we take the parameters a = 1.2, b = 2, c = 5, k = 1, h = 1.5, δ = −2.5as an example. The associated equilibrium branches as well as the bifurcations with the variation of the slow-varyingparameter w can be obtained, shown in figure 1. The solid and the dotted black curves represent the stable and unstableequilibrium points, respectively, while the dotted dash red curves denote those equilibrium branches which the trajectorycan not arrive since there exists the non-smooth boundary.

From the figure 1 one may find that when the dynamics is governed by subsystem S−, the unstable equilibriumbranch EB−

3 turns to stable EB−2 once the fold bifurcation FB−

1 at w = −0.5129 occurs. When w = −6.5937, Hopfbifurcation HB−

1 takes place, resulting in the unstable EB−1 and the stable limit cycle LC−

1 . While for the subsystem S+,fold bifurcation FB+

1 at w = 0.5129 takes place, which connects the unstable EB+3 and the stable EB+

2 , respectively.The stability of equilibrium points will change again once the w meets the critical values at w = +6.5937 where the Hopfbifurcation HB+

1 occurs, causing the unstable equilibrium branch, denoted by EB−1 , and the stable limit cycle LC+

1 .Since the equilibrium branches and the corresponding bifurcations mainly reveal the relationship between the gener-

alized autonomous system as well as the generalized state variable, The dynamical behaviors with two scales, however,still need to be investigated. Now we turn to the evolution of bursting oscillations and the mechanism.

4 Evolution of bursting oscillations and the mechanismHere we fix the frequency of external excitation at Ω = 0.005, implying two scales in frequency domain. Since thebehaviors of dynamics can be affect by the coupling effect of two scales, bursting oscillations with quiescent states andspiking states alternated can be observed with the increase of exciting amplitude. Furthermore, it can be obtained that thesystem keeps unchanged via the transformation (x, y, z, t) ⇒ (−x,−y, z, t+ 2π/Ω), yielding the symmetric property inthe vector filed. Details of numerical simulation results are given in following parts.

4.1 Symmetric fold/fold-sliding/fold/fold-sliding burstingFor A > 0.5129, the fold bifurcations FB−

1 and FB+1 take place, seeing in figure 1, which can cause the trajectory to

jump from one stable equilibrium branch to another, resulting in bursting oscillations, shown in figure 2 for A = 5.0.From the phase portrait in figure 2a two symmetric bursting attractors connected by the non-smooth boundary Σ can beobserved in the vector field. The movement can be divided into four segments with two quiescent states QS1,2 as wellas the two spiking states SP1,2, seeing the time history plotted in figure 2b, from which one can find that the slidingmovement at the non-smooth boundary Σ connects different states. In order to reveal the mechanism, here we turn to theoverlap of transformed phase portrait and the equilibrium branches in figure 2c.

Assuming that the trajectory starts from point P1, where the slow-varying parameter takes its minimum value atw = −5.0, the trajectory is governed by the subsystem S− in the phase space since it is located in region D−. Themovement keeps strictly along the stable equilibrium branch EB−

2 , behaving in the quiescent state QS1. As the time tincreases, the fold bifurcation FB−

1 at w = −0.5129 occurs, causing the trajectory to jump to the equilibrium branchEB−

4 located in region D+. However, when the trajectory moves across the non-smooth boundary Σ, it will be governedby subsystem S+ via the sole stable equilibrium branch EB−

4 located in region D+, which can lead the trajectory to turn



-4 -2 0 2 4

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x10000 10500 11000 11500 12000

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(b)

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QS1

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y

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2.5

5.0

(c)P2

P1

EB+2

EB+3

EB-3

EB-2

FB-1

FB+1

y

w=Acos( t)

Figure 2: Bursting oscillations for A = 5.0. (a) Phase portrait on the (x, y) plane; (b) Time history of y; (c)overlap of thetransformed phase portrait and the equilibrium branches on the (w, y) plane.

-5 -2.5 0 2.5 5-6

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0

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6

(a)

y

x10000 10500 11000 11500 12000-7

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0

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QS4SP4

SP1

QS3

SP4 QS4

QS3

QS2

QS1

(b)

SP1

y

t

SP3

SP2

SP3

10780 10785 10790-8

-6

-4

-2

0

T1

(c)

y

t3 3.5 4

0

2

4

6

(d)

y

x

Figure 3: Bursting oscillations for A = 8.50. (a) Phase portrait on the (x, y) plane; (b) Time history of y; (c) the locallyenlarged part; (d) The poincare map.

back to region D−, resulting in sliding movement strictly along the non-smooth boundary Σ until the w arrives at theintersection of EB+

3 and the EB+4 at w = +2.50 on the non-smooth boundary.

For further increase of w, once the trajectory is governed by subsystem S+, it will jump from the non-smooth bound-ary due to the attraction of the stable equilibrium branch EB+

2 , appearing in repetitive spiking state SP1 around theequilibrium branch EB+

2 . The trajectory, when time t evolves, gradually settles down to the stable EB+2 , denoted by

quiescent state QS2 in figure 2b.When the trajectory arrives at the point P2, at which w reaches its maximum at w = +5.0, half period of the

bursting oscillations is done, while the other half is omitted due to the symmetric. According to those bifurcations thatcause the transitions between quiescent states and spiking states, the movement can be called the symmetric fold/fold-sliding/fold/fold-sliding bursting.

4.2 Symmetric Hopf/fold/fold-sliding/Hopf bursting

Note that further increase of the external excitation amplitude A will lead to the appearance of two Hopf bifurcationsat points HB−

1 and HB+1 , resulting in two stable limit cycles LC−

1 and LC+1 in the generalized autonomous system,

shown in figure 1, the structure of bursting oscillations will evolve, seeing the attractor in figure 3 for A = 8.5. From thephase portrait demonstrated in figure 3a one may find that the trajectory keeps symmetry while the number of oscillationschanges, which can be verified by the time history of x. The movement, seeing the time history in figure 3b, can be dividedinto eight segments, corresponding to four stages of quiescent states and four stages of spiking states. The frequency ofspiking states SP1,3 can be computed at ω = 2π

T1= 2.7414, which matches well with the frequency of related bifurcation

points HB±1 at ωH = 2.7386, seeing figure 3c. Furthermore, the Poincare map related to SP3 on the cross-section

z = 3.5 are plotted in figure 3d, which nearly forms a cycle on (x, y) plane, suggesting that the symmetric spiking statesSP1,3 are quasi-periodic type.

To illustrate the mechanism of bursting oscillations, figure 4 gives the overlap of the transformed phase portrait and theequilibrium branches on the (w, y) plane. The trajectory, starting from w = −8.5, tries to approach the limit cycle LC−

1 ,behaving in quasi-periodic oscillations, denoted by spiking state SP1 in figure 4b. As time evolves, the Hopf bifurcationat HB−

1 occurs, leading to stable equilibrium branch EB−2 , which causes the trajectory to settle down to the stable EB−

2 .Due to the slow passage effect, the oscillations will last for a while with the amplitude gradually decreasing, shown in



-9 -6 -3 0 3 6 9-6

-4

-2

0

2

4

6

(a)LC+

1

LC-1

HB+1

HB-1

EB+2

EB-2

FB-1

FB+1

y

w=Acos( t)-8.4 -8.2

-4.4

-4.0

-3.6

-3.2

-2.8

-2.4

EB-1

(b)

y

w=Acos( t)-5.0 -4.5 -4.0 -3.5 -3.0 -2.5

-4.0

-3.2

-2.4 QS1

SP1

SP4

EB-2

(c)

y

w=Acos( t)6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4

0

2

4

6

EB+1

EB+2

HB+1

LC+1

SP3

(d)

y

w=Acos( t)

Figure 4: The overlap of the transformed phase portrait and equilibrium branches for A = 8.50 on the (w, y) plane in (a);(b) the locally enlarged part; (c) the locally enlarged part; (d) the locally enlarged part.

figure 4a and 4c. Then the trajectory moves strictly along the stable EB−2 and jumps to the non-smooth boundary Σ

because of the fold bifurcation at FB−1 .

Similar to the case for A = 5.0, the sliding movement takes place along Σ for the same reason. When arriving at theregion D+, the trajectory will be attracted by the stable EB+

2 immediately, behaving in spiking oscillations. With theincrease of w, the amplitude of oscillations gradually decreases and the trajectory settles down to the EB+

2 and movesalong the unstable equilibrium branch EB+

2 . The movement of trajectory along the EB+2 lasts for a while and then

behaves in oscillations since the delay effect of the Hopf bifurcation when passing across the Hopf bifurcation pointHB+

1 , which corresponds to the quiescent state QS2 as well as the spiking state SP3 in figure 3b. When w reaches itsmaximum value w = +8.5, half of movement is finished. Since the symmetric property of the system, other half of periodis omitted. According to those bifurcations at the transitions, this type of bursting attractor can be called the symmetricfold/fold-sliding/Hopf bursting.

4.3 Two types of symmetric sliding/Hopf/fold/fold-sliding/Hopf/sliding bursting

Since the bursting attractor may expand in the phase space with the increase of exciting amplitude, oscillations of thetrajectory may cross the non-smooth boundary, leading to different non-smooth dynamical behaviors for which casetwo types of sliding bifurcations may take place. figure 5 presents the details of bursting attractors for A = 9.00 andA = 12.00, from which one may find that difference only exists at the non-smooth boundary. For A = 9.00, themovement of oscillations is strictly according to Σ, seeing time history as well as the enlarged part in figure 5c and 5e,respectively. While for A = 12.00, the trajectory travels across the boundary, which further changes the structure ofspiking states, shown in figure 5d and 5f. To investigate the mechanism of special non-smooth dynamical behaviors, wepresent the overlap of transformed phase portrait and the equilibrium branches.

For A = 9.00, assuming that the trajectory starts from w = −9.00, oscillations of the trajectory appears to moveaccording the non-smooth boundary Σ, shown in figure 6c and 6e, implying the grazing-sliding bifurcation takes place inthe vector field. The movements continue to be in form of oscillations and gradually settle down to the EB−

2 once theHopf bifurcation occurs at point HB−

1 . Since the trajectory is governed by the subsystem S− and no extra bifurcationsoccur in the subsystem, the rest of the movement, which is similar to the former cases, is omitted here for simplicity.

While for A = 12.00, the trajectory starting at w = −12.00 behaves in oscillations along the unstable equilibriumbranch EB−

1 , the amplitude of which is relatively larger compared to the case for A = 9.00 that the oscillations movecross the boundary for a while, seeing figure 6d. Details of the beginning movements are shown in the enlarged parts, fromwhich one may find that the trajectory firstly crosses the non-smooth boundary Σ for a while and then settles down to slideat the boundary Σ in a very short time, shown in figure 6f, after which the trajectory continues to oscillate according to thestable limit cycle LC−

1 since it is governed by the subsystem S−. As the slow-varying parameter w increases, the upperpart of the oscillations move strictly according the non-smooth boundary, indicating that the switch-sliding bifurcationstake place. The rest of the movement is also omitted for the same reason.

For the two cases above, according to the smooth and non-smooth bifurcations involved in the change of the oscillationstructure, the two bursting attractors can be called the grazing-sliding/Hopf/fold/fold-sliding/Hopf/grazing-sliding andswitch-sliding/Hopf/fold/fold-sliding/Hopf/switch-sliding, respectively.



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10000 10500 11000 11500 12000-8

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SP4QS4

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(c)

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0

2

4

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8

(e)

y

t11360 11365 11370-2

0

2

4

6

8 (f)

y

t

Figure 5: Bursting oscillations for A = 9.00 and A = 12.00 (a) and (b) Phase portrait on the (x, y) plane; (c) Time historyof y for A = 9.00; (d) Time history of y for A = 12.00; (e) the locally enlarged part of time history for A = 9.00; (f) thelocally enlarged part of time history for A = 12.00.

-10 -5 0 5 10-7.0

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3.5

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(a) LC+1

LC-1

HB+1

HB-1

EB+2

EB-2

FB-1

FB+1

y

w=Acos( t)-12 -8 -4 0 4 8 12

-9

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-3

0

3

6

9

(b)

LC+1

LC-1 HB+

1

HB-1

EB+2

EB-2

FB-1

FB+1

y

w=Acos( t)-9 -8 -7 -6 -5 -4

-7.5

-5.0

-2.5

0.0

EB-2

EB-1

(c)

LC-1

HB-1

y

w=Acos( t)

-12 -10 -8 -6

-8

-6

-4

-2

0

2(d)

LC-1

HB-1

y

w=Acos( t)-8.74 -8.72 -8.70 -8.68

-7.5

-5.0

-2.5

0.0LC-

1

(e)

EB-1

y

w=Acos( t)-11.85 -11.84 -11.83 -11.82

-0.4

-0.2

0.0

0.2LC-

1

(f)

y

w=Acos( t)

Figure 6: The overlap of the transformed phase portrait and equilibrium branches on the (w, y) plane. (a) for A = 9.00;(b) for A = 12.00; (c)the locally enlarged part for A = 9.00; (d)the locally enlarged part for A = 12.00; (e)the locallyenlarged part for A = 9.00; (f)the locally enlarged part for A = 12.00.



5 Conclusions

For a typical Filippov system with external periodic excitation, since the exciting frequency is far less than the naturalfrequency, different types of bursting oscillations behaving in the combination of large-amplitude and small-amplitudeoscillations or at rest can be observed with the variation of the exciting amplitude, whose bursting patterns are quitedifferent from those in smooth dynamical systems. Several conventional bifurcations, such as fold bifurcations and Hopfbifurcations, can be obtained in the vector field by taking the excitation term as a slow-varying parameter, which mayfurther change the structure of the bursting oscillations.Moreover, the non-smooth boundary, which divides the phase spaceinto three regions, may cause the trajectory to be governed by different subsystems when passing across the boundary,indicating that the non-smooth bifurcations may take place.

For a typical set of parameters, several types of bursting attractors can be observed with the excitation amplitude in-creasing, such as the fold/fold-sliding/fold/fold-sliding bursting and Hopf/fold/fold-sliding/Hopf bursting, and the mech-anism of which can be investigated via employing the overlap of the transformed phase portrait and the equilibriumbranches. From the analysis one may find that fold bifurcations may cause the trajectory to jump to the non-smoothboundary, connecting two quiescent states. Meanwhile the Hopf bifurcations may lead the trajectory to oscillate accord-ing to the limit cycles, behaving in the quasi-periodic oscillations. Furthermore, three types of the sliding movements canbe obtained, implying three types of sliding bifurcations occur in the vector field.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (Grant No. 11872189)

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Relaxation Oscillations and the Mechanism in a Filippov-Type … Vol 29 No 1 Pape… · Relaxation...

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