DOI 10.1007/s11071-010-9780-9 O R I G I N A L PA P E R
Analytics of heteroclinic bifurcation in a 3:1 subharmonic resonance
Mohamed Belhaq · Abdelhak Fahsi
Received: 8 April 2010 / Accepted: 6 July 2010
© Springer Science+Business Media B.V. 2010
Abstract Analytical approximation of heteroclinic bifurcation in a 3:1 subharmonic resonance is given in this paper. The system we consider that produces this bifurcation is a harmonically forced and self-excited nonlinear oscillator. This bifurcation mechanism, re- sulting from the disappearance of a stable slow flow limit cycle at the bifurcation point, gives rise to a syn- chronization phenomenon near the 3:1 resonance. The analytical approach used in this study is based on the collision criterion between the slow flow limit cycle and the three saddles involved in the bifurcation. The amplitudes of the 3:1 subharmonic response and of the slow flow limit cycle are approximated and the colli- sion criterion is applied leading to an explicit analyt- ical condition of heteroclinic connection. Numerical simulations are performed and compared to the ana- lytical finding for validation.
Keywords Heteroclinic bifurcation · 3:1 subharmonic resonance · Frequency-locking · Perturbation analysis · Double reduction technique
M. Belhaq ( )
Laboratory of Mechanics, University Hassan II-Casablanca, Casablanca, Morocco e-mail:
mbelhaq@yahoo.frA. Fahsi
FSTM, University Hassan II-Mohammadia, Mohammadia, Morocco
1 Introduction
In this work, we give an analytical approximation of heteroclinic bifurcation to a stable slow flow limit cy- cle near the 3:1 subharmonic resonance in a forced van der Pol–Duffing oscillator. This bifurcation takes place when pairs of saddles form three connections destroy- ing a limit cycle and giving rise to synchronization be- tween the limit cycle and the subharmonic response in which the response of the system follows the 3:1 sub- harmonic frequency.
To approximate heteroclinic connection, numeri- cal simulations are usually used. For instance, a het- eroclinic bifurcation near a 4:1 resonance has been analyzed in a nonlinear parametric oscillator using the fourth-order Runge–Kutta method [1, 2]. Later, a Ph.D. thesis [3] has been devoted to investigate this 4:1 resonance problem using continuation methods.
Recently, an extensive and a detailed numerical path- following and simulation was carried out to analyze bifurcation to heteroclinic cycles in three and four cou- pled phase oscillators [4]. Note that bifurcation to het- eroclinic cycles are of great interest because of their applications in synchronization phenomena in many fields in physics and biology [5–7]. While heteroclinic bifurcations for cycles have been studied extensively from numerical view point, analytical approximations of these bifurcations in periodically forced nonlinear oscillators remains to be explored.
In this paper, we propose an analytical treatment
of heteroclinic bifurcation of cycles near the 3:1 res-
onance. This is done by using the so-called collision criterion based on the coalescence, at the bifurcation, between the slow flow limit cycle and the three sad- dles involved in the bifurcation. This criterion was widely applied in a series of a paper [8–12] to capture homoclinic and heteroclinic bifurcations in two- and three-dimensional autonomous systems. It was shown that the collision criterion is equivalent to the Mel- nikov method [10]; the collision criterion is accessible via approximations of limit cycles, the Melnikov ap- proach, on the other hand, aims directly at the separa- trices. Moreover, it was concluded that the application of the collision criterion gives a satisfactory approxi- mation when using trigonometric functions in approxi- mating the limit cycle, whereas the criterion coincides with the Melnikov method when using the Jacobian elliptic functions [8–10]. It should be emphasized that the collision criterion was also applied to formally de- rive an analytical approximation of homoclinic bifur- cation in three-dimensional systems [11, 12].
The rest of the paper is organized as follows: In Sect. 2 we use, in a first step, the multiple scales method to derive the slow flow and the amplitude- frequency response near the 3:1 subharmonic reso- nance. Then, a multiple scales method is performed, in a second step, on the slow flow to approximate the limit cycle. In Sect. 3, we investigate the heteroclinic bifurcation using the collision criterion and we com- pare the analytical finding with the numerical simula- tions obtained by examining the phase portraits. Sec- tion 4 concludes the work.
2 3:1 Subharmonic response and slow flow limit cycle
The system we consider that exhibits a 3:1 heteroclinic connection is a forced van der Pol–Duffing oscillator given in the dimensionless form as
¨
x + ω
02x −
α − βx
2˙
x − γ x
3= h cos ωt (1) where ω
0is the natural frequency, α, β are the damp- ing coefficients, γ is the nonlinear component and h, ω are the amplitude and the frequency of the external excitation, respectively. The purpose is to approximate the amplitude-frequency response of the 3:1 resonance as well as the slow flow limit cycle of (1). Thus, the
3:1 resonance condition is expressed by introducing a detuning parameter σ according to
ω
20= ω
3
2+ σ (2)
and we use the method of multiple scales [19]. Intro- ducing a book-keeping parameter ε in (1) and (2), we obtain
¨ x +
ω 3
2x = h cos ωt + ε
− σ x + (α − βx
2) x ˙ + γ x
3(3) We seek a two-scale expansion of the solution in the form
x(t ) = x
0(T
0, T
1) + εx
1(T
0, T
1) + O ε
2(4) where T
i= ε
it . Substituting (4) into (3) and equating coefficients of the same power of ε, we obtain the fol- lowing set of linear partial differential equations D
20x
0(T
0, T
1) +
ω 3
2x
0(T
0, T
1) = h cos ωT
0(5)
D
20x
1(T
0, T
1) + ω
3
2x
1(T
0, T
1)
= −2D
0D
1x
0− σ x
0+
α − βx
02D
0x
0+ γ x
03(6) where D
ij=
∂j∂Tij
. The solution to the first order is given by
x
0(T
0, T
1) = r(T
1) cos ω
3 T
0+ θ (T
1)
+ F cos ωT
0(7) where r and θ are, respectively, the amplitude and the phase of the response and F = −
8ω9h2. Substituting (7) into (6), removing secular terms and using the expres- sions
drdt= εD
1r + O(ε
2) and
dθdt= εD
1θ + O(ε
2), we obtain the slow flow modulation equations of am- plitude and phase
dr
dt = Ar − Br
3− (H
1sin 3θ + H
2cos 3θ )r
2, dθ
dt = S − Cr
2− (H
1cos 3θ − H
2sin 3θ )r (8)
Fig. 1 Amplitude–frequency response curve near 3:1 reso- nance. Solid line for stable, dashed line for unstable and circles for numerical simulation,
α=0.01,
β=0.05,
γ=0.1,
h=1
where A =
α2−
βF42, B =
β8, C =
8ω9ξ, S =
3σ2ω−
9ξ F4ω2, H
1=
9ξ F8ω, and H
2=
βF8. Equilibria of the slow flow (8), corresponding to periodic solutions of (1), are de- termined by setting
drdt=
dθdt= 0. This leads to the amplitude–frequency response equation
A
2r
4+ A
1r
2+ A
0= 0 (9) and the phase–frequency response relation
tan 3θ = (A − Br
2)H
1− (S − Cr
2)H
2(A − Br
2)H
2+ (S − Cr
2)H
1(10) where A
2= B
2+ C
2, A
1= − (2AB + 2SC + H
12+ H
22) and A
0= A
2+ S
2.
Figure 1 shows the variation of the amplitude–frequen- cy response curve, as given by (9), for the given para- meters α = 0.01, β = 0.05, γ = 0.1 and h = 1. The solid line denotes a stable branch and the dashed line denotes an unstable one. The stability analysis has been done using the Jacobian of the slow flow system (8). Results from numerical simulation (circles) using Runge–Kutta method are also plotted for validation.
For extensive numerical simulations, see [13, 14].
To approximate the slow flow limit cycle, corre- sponding to quasiperiodic solution of (1), we intro- duce a new small book-keeping parameter μ to per- form a second perturbation analysis on the following slow flow Cartesian system corresponding to the polar form (8)
du
dt = Sv + μ
Au − (Bu + Cv)
u
2+ v
2+ 2H
1uv − H
2u
2− v
2, dv
dt = − Su + μ
Av − (Bv − Cu)
u
2+ v
2+ H
1u
2− v
2+ 2H
2uv
(11) where u = r cos θ, v = − r sin θ. Notice that the double perturbation procedure has been successfully applied in previous works to capture the dynamic in quasiperi- odic Mathieu equations [15–18]. We rewrite (11) in the form
du
dt = Sv + μf (u, v), dv
dt = − Su + μg(u, v) (12)
Using the multiple scales method, we expand the so- lutions of (12) as
u(t ) = u
0(T
0, T
1, T
2) + μu
1(T
0, T
1, T
2) + μ
2u
2(T
0, T
1, T
2) + O
μ
3,
(13) v(t ) = v
0(T
0, T
1, T
2) + μv
1(T
0, T
1, T
2)
+ μ
2v
2(T
0, T
1, T
2) + O μ
3where T
i= μ
it . Substituting (13) into (12) and col- lecting terms, we get
– Order μ
0:
D
02u
0+ S
2u
0= 0, Sv
0= D
0u
0(14) – Order μ
1:
⎧ ⎪
⎨
⎪ ⎩
D
02u
1+ S
2u
1= − 2D
0D
1u
0+ D
0(f (u
0, v
0)) + Sg(u
0, v
0),
Sv
1= D
0u
1+ D
1u
0− f (u
0, v
0)
(15)
– Order μ
2:
⎧ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎨
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎩
D
02u
2+ S
2u
2= −2D
0D
2u
0− D
0D
1u
1− SD
1v
1+ D
0u
1∂f∂u
(u
0, v
0) + v
1∂f∂v(u
0, v
0) + S
u
1∂g∂u(u
0, v
0) + v
1∂g∂v
(u
0, v
0) , Sv
2= D
0u
2+ D
1u
1+ D
2u
0−
u
1∂f∂u
(u
0, v
0) + v
1∂f∂v(u
0, v
0)
(16)
where D
ji=
∂j∂Tij
. A solution to the first order is given by
u
0(T
0, T
1) = R(T
1)cos(ST
0+ ϕ(T
1)),
v
0(T
0, T
1) = − R(T
1) sin(ST
0+ ϕ(T
1)) (17) Substituting (17) into (15), removing secular terms and using the expressions
dRdt= μD
1R + O(μ
2) and
dϕ
dt
= μD
1ϕ + O(μ
2), we obtain the following slow slow flow system on R and ϕ :
dR
dt = AR − BR
3, dϕ
dt = − CR
2(18)
Then, the first-order approximate periodic solution of the slow flow (12) is given by
u(t ) = R cos νt,
v(t ) = − R sin νt (19)
where the amplitude R and the frequency ν are ob- tained by setting
dRdt= 0 and given, respectively, by R =
A
B , (20)
ν = S − CR
2(21)
Using (19), the modulated amplitude of quasiperiodic solution of (1), is approximated by
r(t ) =
u
2(t ) + v
2(t ) = R (22) This indicates that the quasiperiodic modulation area is reduced, at this first-order approximation, to the unique curve given by
r
min= r
max= R (23)
where r
minand r
maxare the maximum and the min- imum of the amplitude modulation, respectively. In Fig. 2 is plotted the amplitude-frequency response curve as well as the quasiperiodic modulation domain obtained numerically (double circles connected with a vertical line). The horizontal line lying in the middle of circles is the first-order approximation of the modula- tion amplitude to the slow flow limit cycle as given by (22). Since the frequency of this limit cycle must be
Fig. 2 Amplitude–frequency response curve and modulation amplitude of slow flow limit cycle. Numeric: double circles, an- alytic to the first order: horizontal line. Values of parameters are fixed as in Fig.
1positive, the condition ν ≥ 0 from (21) imposes this line to end at a certain value of the frequency ω corre- sponding to the limit ν = 0. This is consistent with nu- merical integration because at this limit the amplitude of the limit cycle hits the subharmonic response curve and disappears via a heteroclinic connection. Notice that the first-order approximation fails to capture the quasiperiodic modulation domain (double circles).
At the second-order approximations, a particular solution of system (15) is given by
u
1(T
0, T
1, T
2) = R
2(T
1, T
2)
3S (H
1cos(2ST
0+ 2ϕ(T
1, T
2)) − H
2sin(2ST
0+ 2ϕ(T
1, T
2))), v
1(T
0, T
1, T
2) = R
2(T
1, T
2)
3S (H
1sin(2ST
0+ 2ϕ(T
1, T
2)) + H
2cos(2ST
0+ 2ϕ(T
1, T
2))) (24) Substituting (17), (18) and (24) into (16) and re- moving secular terms gives the following partial dif- ferential equations on R and ϕ:
D
2R = 0, D
2ϕ = − 2R
23S
H
12+ H
22(25)
Then, the second-order approximate periodic solution of the slow flow (12) is now given by
u(t ) = R cos νt + R
23S (H
1cos(2νt ) − H
2sin(2νt )), v(t ) = − R sin νt + R
23S (H
1sin(2νt ) + H
2cos(2νt ) (26) where the amplitude R and the frequency ν are now obtained by setting
dRdt= D
1R + D
2R = 0 and given, respectively, by
R = A
B , (27)
ν = S −
C + 2R
23S
H
12+ H
22R
2(28)
Using (26), the modulated amplitude of quasiperiodic solution is now approximated by
r(t )=
R2+ R4 9S2
H12+H22 +2R3
3S(H1cos(3νt )−H2sin(3νt ))
(29) Thus, the envelope of this modulated amplitude is de- limited by r
minand r
maxgiven by
r
min=
R
2+ R
49S
2H
12+ H
22− 2R
33S
H
12+ H
22, (30) r
max=
R
2+ R
49S
2H
12+ H
22+ 2R
33S
H
12+ H
22(31) In Fig. 3 is plotted the modulated amplitude of slow flow limit cycle (solid lines), as given by (30), (31).
The comparison between this analytical prediction and the numerical simulations (circles) indicates that the second-order analytical approximation can capture the modulation domain. Figure 3 shows that outside the synchronization area, quasiperiodic behavior takes place. When approaching the synchronization area, the upper limit of the modulation amplitude collides with the unstable frequency response curve producing a heteroclinic connection.
Fig. 3 Modulation amplitude of slow flow limit cycle. Numer- ical simulation: double circles, analytical approximation to the second order: double lines. Values of parameters are fixed as in Fig.
13 Collision criterion and heteroclinic bifurcation An analytical approximation of the heteroclinic bifur- cation curve is obtained by applying the collision cri- terion between the slow flow limit cycle and the three saddles. At the collision, this criterion may merely be given by the condition
r
max= r
s(32)
where r
maxis the upper limit of the modulation ampli- tude of the slow flow limit cycle given by (23) (first- order approximation) or by (31) (second-order approx- imation) and r
sdenotes the amplitude of the unstable periodic orbit corresponding to the slow flow saddles obtained from the amplitude-frequency response (9).
This analytical criterion of heteroclinic bifurcation is illustrated in Fig. 4 by the curves labeled H
a1(for the first order) and labeled H
a2(for the second order).
It is worth noticing that the condition ν = 0 in (21), corresponding to infinite period of the slow flow limit cycle, offers another analytical criterion of hete- roclinic connection, as was shown in previous works [8, 9].
To test the validity and the accuracy of the analyt-
ical prediction, we perform numerical simulations by
integrating (11) using the Runge–Kutta method. The
obtained numerical heteroclinic bifurcation curve, la-
beled H
n, is plotted in Fig. 4 for comparison with the
analytical curves H
a1and H
a2. It can be seen from this
Fig. 4 Bifurcation curves for the 3:1 subharmonic resonance.
SN: saddle-node bifurcation,Ha1
: analytical heteroclinic bifur- cation curve to first order,
Ha2: analytical heteroclinic bifurca- tion curve to second order,
Hn: numerical heteroclinic bifurca- tion curve. The values of parameters are fixed as in Fig.
1comparison that for small values of the amplitude ex- citation h, the curves H
a1,2and H
ncome closer.
In Table 1, we compare the values of the frequency ω corresponding to heteroclinic connection obtained by analytical predictions ω
1,2apicked from H
a1and H
a2, respectively, and through numerical calculation ω
npicked from H
n. Note that the curves labeled SN denote the saddle-node bifurcation locations for the 3:1 subharmonic cycles.
Figure 5 illustrates examples of phase portraits of the slow flow (11) for some values of ω picked from Fig. 3. For small values of ω, a slow flow limit cycle born by Hopf bifurcation exists and attracts all initial conditions. The related phase portraits are shown in subfigure (a) for ω = 2.72. The subfigure (b) for ω = 2.80 indicates the coexistence of a stable periodic orbit (stable equilibrium born by saddle-node bifurcation) and quasiperiodic (stable limit cycle) responses. As the forcing frequency varies, the stable slow flow limit cycle approaches the saddles and disappears via a het- eroclinic bifurcation. The corresponding phase portrait is shown in subfigure (c) for the value ω = 2.86256.
This mechanism gives rise to frequency-locking, in which the response of the system follows the 3:1 sub- harmonic frequency (see subfigure (d) for ω = 2.90).
We point out that in this study, we have focused on the heteroclinic bifurcation corresponding to the case where the limit cycle disappears leaving the cycles lo- cated outside the triangle connection, as shown in Fig.
Table 1 Comparison between analytical predictions (ω
1afor the first order,
ω2afor the second order) and numerical calculation (ω
n) of heteroclinic bifurcation, for different values of
h. Valuesof parameters are fixed as in Fig.
1ω1a ω2a ωn
h=
1 2.894 2.886 2.863
h=
0.5 2.901 2.897 2.879
h=
