Asymptotics of homoclinic bifurcation in a three-dimensional system

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Asymptotics of Homoclinic Bifurcation in a Three-Dimensional System

Article in Nonlinear Dynamics · February 2000

DOI: 10.1023/A:1008353609572

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Asymptotics of Homoclinic Bifurcation in a Three-Dimensional System

M. BELHAQ and M. HOUSSNI

Laboratory of Mechanics, Faculty of Sciences Aïn Chock, Group of Nonlinear Oscillations and Chaos, BP 5366, Maârif, Casablanca, Morocco

E. FREIRE and A. J. RODRÍGUEZ–LUIS

Department of Applied Mathematics II, Escuela Superior Ingenieros, University of Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

(Received: 22 October 1998; accepted: 19 May 1999)

Abstract. An analytical approach to predicting a critical parameter value of homoclinic bifurcation in a three- dimensional system is reported. The multiple scales method is first performed to construct a higher-order approximation of the periodic solution. A criterion based on a collision between the periodic orbit and the fixed point involved in the bifurcation is applied. This criterion developed initially to predict homoclinic bifurcations in planar autonomous systems, is adapted here to derive a critical value of the homoclinic bifurcation in a specific three- dimensional system. To support our analytical predictions and to describe the dynamical behaviour of the system, a complete numerical study is provided.

Keywords: Three-dimensional systems, periodic orbit, multiple scales analysis, homoclinic bifurcation.

1. Introduction

The dynamics of three-dimensional systems near one of their periodic orbits is very rich in terms of bifurcation and stability. Important behaviours include symmetry-breaking, period-doubling, Neimark–Sacker (bifurcation to invariant torus) and specially homoclinic or Shil’nikov bifurcation. See, for example, one of the pioneering works of Shil’nikov [29]

and the works where a three-dimensional electronic circuit is studied by Freire et al. [15] and Algaba et al. [1–3].

Indeed, homoclinic and heteroclinic orbits are of great importance from an applied point of view. For instance, they form the profiles of travelling wave solutions in reaction-diffusion problems (see, for instance, [13, 19]). Their existence can be a source of chaotic dynamics in three-dimensional systems (see, for example, [25, 29]). In static-dynamics analogies, a homoclinic orbit corresponds to a spatially localized post-buckling state [30]. The well- known Chua circuit governed by a three-dimensional differential equation system is now one of the most studied systems because of its richness and simplicity [10]. Khibnik et al. [18]

analyzed Chua’s circuit equations with a smooth nonlinearity and reported the central role of homoclinicity in the model. Nekorkin and Kazantsev [24] investigated the travelling waves in a one-dimensional circular array of Chua’s circuits. It was shown that the problem can be reduced to an analysis of the periodic orbit of a three-dimensional system of ordinary differential equations describing the individual dynamics of Chua’s circuit. Within this context,

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the development of new analytical methodologies to predict bifurcations in three-dimensional systems is an exciting issue and strongly required.

The investigation of homoclinic bifurcation has been receiving much attention from both the analytical and numerical points of view (see, for example, [11, 22, 28]). The classical mathematically rigorous approach to predicting bifurcations of homoclinic orbits is Mel- nikov’s method. This approach mainly requires the distance between the manifolds of the perturbed system to vanish. Recently, another analytical method to predict the homoclinic bifurcations in autonomous self-excited two-dimensional systems was reported in [5]. This approach formally approximating the infinite period of the bifurcating periodic orbits. More precisely, the condition considered at such bifurcations is the limit when the period goes to infinity or the vanishing of the frequency of the periodic solution. In [31], a semi-analytical and numerical process was developed to determine the separatrices and limit cycles of strongly two-dimensional nonlinear oscillators. Conditions under which a limit cycle is created or destroyed were derived. In [4] and [6], a formal analytical criterion to predict homoclinic bifurcations in autonomous two-dimensional systems was reported. This criterion is mainly based on the collision, at the homoclinic bifurcation, of the periodic orbit with the equilibrium involved in the bifurcation. Mathematically speaking, this criterion is equivalent to the Mel- nikov method (for details, see [7]). Note, however, that the collision criterion is accessible via approximations of periodic orbits. The Melnikov approach, on the other hand, circumvents periodic orbits by aiming directly at the separatrices.

In this paper, we apply the collision criterion to formally derive an approximate homoclinic bifurcation exhibited by the three-dimensional system

˙

x = µx−y−xz,

˙

y = µy+x,

˙

z = −z+x²z+y². (1)

System (1) may be thought of as a response control system consisting of a damped linear oscillator inx,y variables and a control variablez. One should note that this oscillator has negative damping for positiveµ. The ‘dot’ denotes the time derivative,x,y andzare scalar variables andµis a scalar parameter. For system (1), symmetry-breaking and period-doubling bifurcations has been analytically investigated by Rand [26], Nayfeh and Balachandran [21], Belhaq and co-workers [8, 9].

The approximate homoclinic bifurcation is provided by a suitable adaptation to system (1) the criterion proposed in [4, 6] for autonomous planar systems.

The origin (x0=0,y0=0,z0=0) of Equations (1) is an equilibrium, stable forµ <0 and unstable forµ >0 so that a Hopf bifurcation occurs atµ=0. As the parameterµincreases from zero, the periodic orbit undergoes a symmetry-breaking bifurcation atµ =µSB1. Asµ increases again, this orbit becomes unstable and a new stable periodic orbit of twice the period appears by period-doubling at µ = µPD2,1. This orbit undergoes a second period-doubling bifurcation atµ = µPD4,1. A detailed numerical study is given in Section 4. A heteroclinic connection between the nontrivial equilibria organizes the branch of the principal periodic orbit. Homoclinic connections of double, quadruple, octuple, etc., pulse also act as organizing centers of the dynamics. The presence of cascades of period-doubling bifurcations, and then of chaotic attractors, is also pointed out.

Using the center manifold theory and a near-identity transformation, Rand and co-workers [26, 27] constructed a first-order approximation of the limit cycle near the Hopf bifurcation.

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The critical valueµPD2,1(≈0.45), corresponding to the first period-doubling bifurcation was approached by studying the stability of the orbit. Nayfeh and Balachandran [21] used the method of multiple scales [20] to obtain, as in [26], the same first-order approximation of the periodic solution. The critical valuesµSB1(≈0.30), corresponding to the symmetry-breaking bifurcation, andµPD2,1(≈0.4405)were numerically approximated using Floquet theory [23].

In [9] a higher-order approximation of the periodic orbit, using a higher-order multiple- scales expansion, was constructed. This expansion was successfully used to predict the bifurcation values µSB1 = 0.31 and µPD2,1 = 0.446. Recently, Belhaq et al. [8] derived an analytical approximation of the critical valueµPD4,1(≈0.486)corresponding to the second period-doubling.

In this work, we specifically present an analytical scheme based on formal asymptotic expansions to approximate the parameter value at which a homoclinic bifurcation takes place in such a system. We first derive a higher-order asymptotic expansion of the periodic orbit by using the multiple scales method. The collision criterion of homoclinicity [6] is then applied and adapted to the system under study in order to predict a homoclinic bifurcation.

Comparisons to numerical results are provided for validating our analytical prediction.

This paper is organized as follows. In Section 2, a higher-order approximation of the periodic solution following Hopf bifurcation is obtained using the multiple-scales method to a higher-order. Section 3 is devoted to the prediction of the homoclinic connection. A detailed numerical study of system (1), which provides comparisons to the analytical approach as well as the full dynamical behaviour of the system, is given in Section 4. Finally, we present some conclusions in Section 5.

2. Asymptotic Expansion

To determine an approximation of the periodic orbit of system (1), we apply the method of multiple scales [20], by seeking a uniformal valid expansion of the form

x = X6

n=1

εⁿxn(T0, T1, T2, T3, T4, T5)+ · · ·,

y = X6

n=1

εⁿyn(T0, T1, T2, T3, T4, T5)+ · · ·,

z = X6

n=1

εⁿzn(T0, T1, T2, T3, T4, T5)+ · · ·, (2) whereTn =εⁿt are the time scales andε is a small positive dimensionless parameter which is the order of the amplitude of the motion. This parameter is artificially introduced to serve as a bookkeeping device, in obtaining the approximate solution, and will disappear naturally, since in the expansion (9), for example, the term εa always appears as a block. Hence, following Equations (6), we see that this term can be written in the original parameter form:

εa =(2/3)p

10ε²µ2 =(2/3)√

10µ. For this reason, one usually setsεequal to unity in the final analysis [22, 23]. The control parameter is expanded asµ=ε²µ2+O(ε³). Substituting this last relation and Equations (2) into Equations (1), taking into account thatx0=0,y0=0, z0 = 0 and equating coefficients of like powers of ε, we obtain, at different orders ofε, the

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following systems of successive approximationsxn,yn,zn: o(ε¹):

D0x1+y1 = 0, D0y1−x1 = 0,

D0z1+z1 = 0. (3)

o(εⁱ, i ≥2):

D0xi +yi = µ2xi−2− Xi

j=0

xjzi−j− Xi j=1

Djxi−j,

D0yi −xi = µ2yi−2− Xi

j=1

Djyi−j,

D0zi +zi = Xi j=0

yjyi−j − Xi j=1

Djzi−j + Xi j=0

xi−j

Xj k=0

zkxj−k

!

, (4)

whereDn=∂/∂Tn. The explicit expression to higher orders of system (4) are detailed in [9].

Using Equations (3) and (4) fori=2,3, the solution up to second order is given by [21]

x(t) = −εasinθ+O(ε³), y(t) = εacosθ+O(ε³), z(t) = 1

2ε²a²

1+ 1

5cos(2θ )+2

5sin(2θ )

+O(ε³), (5)

where a= 2

3

p10µ2, θ =

1− ε²a² 20

t+O(ε³). (6)

These values ofaandθ are obtained from the two conditions D1A=0, 2D2A−2µ2A+9+2i

5 A²A=0, (7)

that make the secular terms in Equations (4) vanish fori =2 andi=3. Now let us investigate a higher-order approximation of the limit cycle. The general solution of system (4) fori =3 is given by

x3 = 3

40(2i−1)A³e^3iT⁰+cc, y3 = 2+i

40 A³e^3iT⁰ −i(9+2i)

10 A²Ae^iT⁰+cc,

z3 = 0, (8)

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where ‘cc’ stands for the complex conjugate of the preceding expressions. Hence, from Equa- tions (5–8), the uniformally valid expansion to third order of the periodic solution is given by

x(t) = −εasinθ−ε³ 3a³

160cos(3θ )+3a³

80 sin(3θ )

+O(ε⁴),

y(t) = εacosθ+ε³ a³

20cosθ+ 9a³

40 sinθ + a³

80cos(3θ )− a³

160sin(3θ )

+O(ε⁴),

z(t) = ε²a² 2

1+ 1

5cos(2θ )+2

5sin(2θ )

+O(ε⁴), (9)

whereaandθ are given by Equations (6).

On the other hand, systems (4) fori =4,5,6, allows one to determine the approximation of the periodic orbit up to fifth order. Indeed, the elimination of the secular terms in system (4) fori=4 leads to the condition

D3A=0 (10)

and then the solution up to fourth order can be written as x4 = 0,

y4 = 0, z4 = 9i−2

20(1+4i)A⁴e^4iT⁰− 2

(1+2i)²AD2Ae^2iT⁰

− 22+51i

20(1+2i)A³Ae^2iT⁰ +11−7i

5 (AA)²−D2(AA)+cc. (11)

The equation that eliminates the secular terms in Equations (4) fori =5 is given by the condition

D4A=µ2KA²A+H A³(A)², (12)

where K= 38

25 + 61

100i, H = −3052+861i

2000 . (13)

Similarly, from the higher-order system we obtain the last condition that cause the secular terms in Equation (4) to vanish fori=6

D5A=0. (14)

Therefore, the set of five conditions to be resolved is given by

D1A=0, (15)

2D2A−2µ2A+9+2i

5 A²A=0, (16)

D3A=0, (17)

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D4A=µ2

38 25 + 61

100i

A²A−3052+861i

2000 A³(A)², (18)

D5A=0. (19)

SubstitutingA = (1/2)ae^iβ (where a and β are real quantities) into Equations (15–19), separating the real and imaginary parts, we obtain

D2a = µ2a− 9

40a³, (20)

D4a = 19

50µ2a³− 763

8000a⁵, (21)

D4β = − 1

20a², (22)

D4β = 61

400µ2a²− 861

32000a⁴. (23)

The solution of Equation (20) is a=C(T4)

r

µ2− 9

40a²exp(µ2T2), (24)

whereaandµ2satisfy the condition a²< 40

9 µ2. (25)

Solving Equation (24) forayields a= C(T4)√µ2exp(µ2T2)

q

1+ ₄₀⁹(C(T4))²exp(2µ2T2)

, (26)

where C(T4) is an arbitrary function of T4. Now the substitution of Equation (26) into Equation (21) provides an ordinary differential equation onC(T4)in the form

1+ ₄₀⁹(C(T4))²exp(2µ2T2)

19

50(C(T4))³−₈₀₀₀⁷⁹ (C(T4))⁵exp(2µ2T2)dC =µ²₂exp(2µ2T2)dT4, (27) in whichT2is to be treated as a constant. This may be integrated to give

(C(T4))²

exp

− σ (C(T4))²α

+Eexp

F T4+ c α

= 19 50exp

F T4+ c α

, (28)

where E = 79

8000exp(2µ2T2), F = 2µ²₂

α exp(2µ2T2), σ = 50

19, α = 763

3040σexp(2µ2T2) (29)

and c denotes a constant of integration. Since this last equation is a transcendental one for C(T4), one cannot obtain a closed form expression for C(T4) to be substituted into the

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previous expression (26) for a. Nevertheless, in order to overcome this difficulty, it turns out that an asymptotic approximation forC(T4)can be obtained by expanding the function exp(−σ/(C(T4))²α)and retaining only the two first terms of the expansion. Expanding, substituting these terms in Equation (28) and resolving inC(T4), we obtain an approximation of C(T4)as follows

C(T4)=

"

σ

α +¹⁹₅₀exp F T4+_α^c 1+Eexp F T4+_α^c

#1/2

. (30)

Hence the previous expression ofa, given by Equation (26), becomes a=

µ2 3040

763 exp(−2µ2T2)+¹⁹₅₀exp 2µ2T2+¹¹⁵⁵²₃₈₁₅µ²₂T4+ _α^c1/2

₁₄₄₇

763 +₈₀₀₀⁷⁶³ exp 2µ2T2+¹¹⁵⁵²₃₈₁₅µ²₂T4+ _α^c1/2 . (31) It follows that the new approximation of the amplitude of the limit cycle is a → 20√

(38/3815)µ2 ast → ∞. This expression verifies, as expected, the condition given by Equation (25). Note that the first approximation of the amplitudea is given by Equation (6).

Consequently, from Equations (9) and (11), and the solution of Equations (4) fori = 5, we obtain the following fifth-order approximation of the periodic solution

x(t) = −εasinθ−ε³ 3a³

160cos(3θ )+ 3a³

80 sin(3θ )

+ε⁵

− 32a⁵

15360cos(5θ )+ a⁵

15360sin(5θ )+11587a⁵

486400 cos(3θ )−3247a⁵

972800sin(3θ )

+O(ε⁶), (32)

y(t) = εacosθ+ε³ a³

20cosθ+ 9a³

40 sinθ − a³

160sin(3θ )+ a³

80cos(3θ )

+ε⁵

−197771a⁵

4620800 cosθ+1289a⁵

30400 sinθ+ 186181a⁵

110899200cos(3θ )

− 180177a⁵

166348800sin(3θ )− a⁵

76800cos(5θ )− 32a⁵

76800sin(5θ )

+O(ε⁶), (33)

z(t) = ε²a² 2

1+1

5cos(2θ )+2

5sin(2θ )

+ ε⁴a⁴ 80

757 38 +161

190sin(2θ )−2187

190 cos(2θ )+cos(4θ )−1

2sin(4θ )

+O(ε⁶), (34)

whereaandθ are now given by the new approximations a=20

r 38

3815µ2, θ =

1− 1

20ε²a²+ 2765 243200ε⁴a⁴

t+O(ε⁶). (35)

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(a)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 X

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Y

N

F F

I T

T I

(b)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 X

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Y

I

T T

F F

N

Figure 1. Comparison of different approximations of periodic orbits for: (a)µ = 0.3; (b)µ = 0.4. LabelN corresponds to the exact orbit obtained by numerical integration.Iindicates first-order approximation,Tdenotes the third-order approximation orbit andFcorresponds to the orbit obtained with the fifth-order approximation.

In Figure 1 we compare the approximations at different orders of the periodic orbit obtained by the multiple-scales technique, Equations (32–35), with the periodic orbit obtained numerically by integrating system (1) for two parameter values ofµ.

3. Homoclinic Double-Pulse Connection

This section is concerned with an analytical prediction of a homoclinic connection occurring in system (1). The strategy followed consists of two steps. First, we formally construct a higher-order approximation of the bifurcating periodic orbit forµ > 0. Then the criterion given in [4, 6], for the planar autonomous systems, is adapted to system (1) and applied to predict an approximate critical value of this homoclinic bifurcation. This criterion is based on

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PD_2,1

PD_2,2 PD_4,1

PD_4,2PD_4,4 PD_4,5 PD_4,6

S_1,1 S_1,2

S_1,3 S_1,4

S_2,1 S_2,2

PD_4,3

▼ ▼

■

●

● ●

●

■

Period

µ

SB1 SB2

SB3 SB4

Hopf

● ■

Hom_2,1 Hom_2,2 Het

PD_8,2 PD_8,1

Figure 2. Partial bifurcation diagram of the periodic orbit emerged from the Hopf bifurcation. In this qualitative figure, the solid line means stable periodic orbit and the dashed line means saddle periodic orbit. We have used the following convention: empty square for the Hopf bifurcation, inverted triangle for the symmetry-breaking bifurcation, filled circle for the period-doubling bifurcation and filled square for the saddle-node bifurcation. The notationPD_i,j means that from that period-doubling bifurcation point, a branch of periodic orbits emerges with a period approximatelyitimes the period of the principal branch (we will call it aniT-orbit). The labeljindicates that it is thejth point of the kind we have mentioned in our description of the complex bifurcation diagram exhibited by system (1). Analogously, we use the notationS_i,j to indicate thejth saddle-node bifurcation found in a branch ofiT-orbits. We denote byHom_i,j thejthi-pulse homoclinic connection of the nontrivial equilibria.

Note that we represent the period of theT, 2T, 4T, 8T-orbits divided, respectively, by 1, 2, 4 and 8.

the idea that, at the homoclinic bifurcation, there exists a timetfor which the periodic orbit and the involved equilibrium collide.

The stationary solutions of system (1), corresponding tox˙ = ˙y= ˙z=0, arex=y=z = 0 which is an unstable focus, and the two equilibria









x₊= −µy₊ y₊=q

zs

1+µ²zs

zs = ¹⁺_µ^µ² ,









x₋= −µy₋ y₋= −q

zs

1+µ²zs

zs = ¹⁺_µ^µ²

. (36)

If 0 < µ < µc ≈ 0.24, then the fixed points (x_±, y_±, zs) are saddles, whereas they are saddle-focussed ifµ > µc.

An analysis of the eigenvalues of the fixed points (x_±, y_±, zs) shows that the variation of the periodic orbit in the direction of the control variable z is the most important when compared to the variation of the amplitude along the other directions. In consequence, we can apply, as a first approach, the criterion of homoclinic bifurcation only in thez-direction. Then, the criterion reduces to the condition

z(t)=zs, (37)

where z(t) is the approximation of the periodic orbit (see the Appendix) and zs is the coordinate of the fixed point given in Equations (36).

We point out that the approximate periodic solution up to fifth order (Equations (32–35)) has the same shape as the one obtained numerically which is lying in the vicinity of theHom_2,2

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branch (see Figure 2). This observation suggests us using this analytical approximation to investigate the corresponding homoclinic bifurcation. The approximate solution of the periodic orbit in thez-direction at different orders inεis given in the Appendix.

Using the approximation of the periodic orbit inz (see z⁽³⁾ in the Appendix) up to third order, the criterion (37) leads to the condition

−1+5

3µ²=0. (38)

Resolving this last equation, we obtain a first approximation of the homoclinic bifurcation, namelyµH B1=0.774.

Clearly, this approximation is not in agreement with the numerical calculation (µHB = 0.540, see the next section). To improve this analytical prediction, we shall consider the fifth- order approximation of the periodic solution inz(seez⁽⁵⁾ in the Appendix). Similarly, using this last approximation, criterion (37) now leads to the equation

−1+1061

763 µ²+1087104

582169 µ³=0. (39)

Resolving Equation (39) gives a second approximation of the homoclinic bifurcationµHB2= 0.625.

This result is still not good enough, but comes closer to the numerical result. This stim- ulating observation tells us to go further in our calculation and to construct a higher-order approximation of the periodic solution.

Following the same procedure as above and using now the seventh-order approximation in z(seez⁽⁷⁾in the Appendix), we obtain the following new condition

−1+1061

763 µ²+1087104

582169 µ³+5149190570912

2993556660675µ⁴=0. (40)

Hence, resolving Equation (40) leads to the third approximation of the homoclinic bifurcation µHB3 =0.574. At this stage, this result can be considered as a good prediction compared to the result obtained by the numerical simulation given below.

It is clear that to investigate the higher-order approximations µHB4, µHB5, . . ., in order to improve the approximation of the critical value of homoclinicity, the hand computations become very cumbersome to perform. However, we can use the same idea which allowed us to predict a very good approximation of the critical value corresponding to symmetry-breaking bifurcation [9]. It turns out that combining the above three values forµHB1,µHB2and µHB3, we can conjecture the following result

µHB1−µHB2∼=0.15, µHB2−µHB3∼=0.05= 0.15

3 . (41)

Assuming now that this process continues for the other successive approximations ofµHB, we can construct the following trigonometric series

µHBn=µHB1−α

"

1− ¹₃n−1

1−¹₃

#

, (42)

whereα =0.15. The limit of this series asngoes to infinity provides the following critical value of the homoclinic bifurcation

µHB= lim

n−→∞µHBn =0.545. (43)

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This conjecture has improved the approximation of the homoclinicity, but it is clear that it can be justified only by performing further calculations, for instance approximateµHB4 and verify whether or notµHB4−µHB3follows the process given by Equation (41). The numerical study performed in the next section gives the valueµ(Hom2,2)=0.540.

4. Numerical Study

In this section, we describe the dynamical behaviour found in system (1). To do this, we have used the software continuation code AUTO94 [12] as well as DSTOOL [17]. Particularly, the critical valuesµSB1,µPD2,1,µPD4,1and µHBmentioned in the previous sections, correspond, respectively, to the bifurcation pointsSB1,PD_2,1,PD_4,1andHom_2,2described below.

Note that this system has the same symmetry the Lorenz equations have: it is invariant to the change (x, y, z) −→ (−x,−y, z). The origin is always an equilibrium point, and two other equilibria are given by Equations (36). The origin is stable for µ < 0 and it exhibits a Hopf bifurcation forµ = 0 and then becomes a saddle-focus forµ > 0. The nontrivial equilibria are always saddle (two negative eigenvalues) in the region of interest (µ >0).

The stability analysis of the Hopf bifurcation (see, for instance, [14]) reveals that it is supercritical: a stable symmetric periodic orbit emerges forµ >0. The evolution of this periodic orbit is schematized in the bifurcation diagram of Figure 2. In this qualitative figure, we have indicated the Hopf bifurcation by an empty square, the symmetry-breaking bifurcation by an inverted triangle, the period-doubling bifurcation by a filled circle, and the saddle- node bifurcation by a filled square. The notationPD_i,j means that, from that period-doubling bifurcation point, a branch of periodic orbits with period approximatelyi-times the period of the principal branch (we will call it aniT-orbit) emerges. The labelj indicates that it is the jth point of the kind we have mentioned in our description of the complex bifurcation diagram exhibited by system (1). Analogously, we use the notationS_i,j to indicate thejth saddle-node bifurcation found in a branch ofiT-orbits. We denote by Hom_i,j thejthi-pulse homoclinic connection of the nontrivial equilibria. Note that we represent the period of theT, 2T, 4T, 8T-orbits divided, respectively, by 1, 2, 4 and 8.

First, the periodic orbit exhibits a symmetry-breaking bifurcation,SB1(µ = 0.3150232), to become a saddle orbit and a pair of asymmetric stable periodic orbits emerges. The principal orbit recovers its stability in a second pitchfork bifurcation,SB2(µ=0.6939566). This stable symmetric orbit collapses in a saddle-node bifurcation (fold), S_1,1 (µ = 0.721608), with a saddle symmetric orbit, exhibiting a new saddle-node bifurcationS_1,2(µ=0.6334804) where a new stable symmetric orbit appears. This orbit undergoes a symmetry-breaking bifurcation, SB3(µ=0.6391011) to become unstable and a new pair of asymmetric stable periodic orbits emerges. The saddle symmetric orbit is stable again when it exhibits the pitchfork bifurcation SB4 (µ = 0.6625362). The interval of the parameterµ where this orbit is stable is narrow as it suffers a saddle-node bifurcation, S_1,3 (µ = 0.6646371). The saddle symmetric orbit undergoes a new saddle-node bifurcation,S_1,4(µ =0.6641073). Finally, the resulting stable symmetric orbit approaches a heteroclinic connection between the non-trivial equilibria,Het (µ=0.6658963).

The behaviour of the orbit that emerged from the Hopf bifurcation corresponds to the typ- ical wiggle of a periodic orbit around a homoclinic/heteroclinic connection (see, for instance, [16]). Because of the symmetry the system has, symmetry-breaking and saddle-node bifurcations are combined in the principal branch in the way we have just described. In Figures 3a and

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(a) (b)

X Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

X Z

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

(c) (d)

X Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

X Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Figure 3. Phase portraits of the heteroclinic orbit (µ =0.6658963) ((a) and (b)) and the two saddle symmetric periodic orbits that coexist with this global connection ((c) and (d)).

3b, we show the projection of the heteroclinic orbitHeton the(x, y)-plane and on the(x, z)- plane, respectively. The two saddle symmetric periodic orbits that coexist with this global connection are drawn in Figures 3c and 3d.

Now we focus on the pair of asymmetric stable periodic orbits that emerged atSB1. These orbits become saddle when they exhibit a period-doubling bifurcation,PD_2,1(µ=0.4403559).

Note that this flip bifurcation was analytically predicted to occur forµPD1=0.446 by Belhaq and Houssni [9]. From such a bifurcation point, a stable periodic orbit of approximately twice the period of the original orbit emerges. The stability of the pair is recovered in a new flip bifurcation PD_2,2 (µ = 0.6815652) and this branch disappears at SB2. In short, there is a branch of asymmetric periodic orbits connecting SB1 and SB2 where two period-doubling bifurcations occur.

The asymmetric 2T-orbit (in fact a pair, due to the symmetry the system has) born atPD_2,1 becomes non-stable in a flip bifurcation PD_4,1 (µ = 0.4765392) where a 4T-orbit emerges (this period-doubling bifurcation was analytically predicted to occur for µPD2 = 0.486, in [8]). It becomes stable again in a new period-doubling bifurcation PD_4,2 (µ = 0.4942767) and finally approaches a double-pulse homoclinic connectionHom_2,1(µ =0.503992). In this situation there is not a Feigenbaum period-doubling cascade, since the 4T-orbit that emerges atPD_4,1always becomes stable and its branch disappears atPD_4,2.