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International Journal of Bifurcation and Chaos, Vol. 17, No. 4 (2007) 1343–1354 c World Scientiﬁc Publishing Company

DYNAMICS AND ACTIVE CONTROL OF MOTION OF A DRIVEN MULTI-LIMIT-CYCLE VAN DER POL OSCILLATOR R. YAMAPI∗ Department of Physics, Faculty of Sciences, University of Douala, P.O. Box 24157 Douala, Cameroon [email protected] B. R. NANA NBENDJO Laboratoire de M´ecanique, Facult´e des Sciences, Universit´e de Yaound´e I, B.P. 812, Yaound´e, Cameroon [email protected] H. G. ENJIEU KADJI Institut de Math´ematiques et de Sciences Physiques (IMSP), B.P. 613 Porto-Novo, B´enin [email protected] Received August 17, 2005; Revised March 28, 2006 This paper deals with the dynamics and active control of a driven multi-limit-cycle Van der Pol oscillator. The amplitude of the oscillatory states both in the autonomous and nonautonomous case are derived. The interaction between the amplitudes of the external excitation and the limit-cycles are also analyzed. The domain of the admissible values on the amplitude for the external excitation is found. The eﬀects of the control parameter on the behavior of a driven multi-limit-cycle Van der Pol model are analyzed and it appears that with the appropriate selection of the coupling parameter, the quenching of chaotic vibrations takes place. Keywords: Dynamics; active control; bifurcations; self-sustained system.

1. Introduction In the last decade, the dynamics and active control of nonlinear oscillators have been a subject of particular interest [Fuller et al., 1997; Hackl et al., 1993; Cheng et al., 1993; Tchoukuegno et al., 2002, 2003]. This is due to their importance in many scientiﬁc ﬁelds, ranging from biology, chemistry, physics to engineering. Between these nonlinear oscillators, a particular class is that containing self-sustained components such as the classical Van der Pol oscillator, which serves as a paradigm for smoothly oscillating limit cycle ∗

or relaxation oscillations [Van der Pol, 1922]. In the presence of an external sinusoidal excitation, the dynamics leads to various interesting phenomena: harmonic, sub-harmonic and super-harmonic frequency entrainment [Hayashi, 1964], Devil’s staircase in the behavior of the winding number [Parlitz & Lauterborn, 1987], chaotic behavior in the small range of the control parameters [Parlitz & Lauterborn, 1987; Guckenheimer & Holmes, 1994]. The generalization of the classical Van der Pol oscillator including a cubic nonlinear term (so-called Duﬃng– Van der Pol oscillator) has also been investigated

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in [Venkatesan & Lakshamanan, 1997]. They have shown that the model exhibits chaotic motion between two types of regular motion, namely periodic and quasiperiodic oscillations in the principal resonance region. Also, they have obtained a perturbative solution for the periodic oscillations and carried out a stability analysis of such a solution to predict Neimark bifurcation. Our aim in this paper is to consider the dynamics and active control of a driven multi-limit-cycle Van der Pol oscillator with a nonlinear damping function of higher polynomial order proposed recently by Kaiser [1981]. We would like to shed additional light on the dynamics of such self-excited model. In particular, we try to answer the following questions: • How does the amplitude E0 of the external excitation interact with the amplitudes of the two limit cycles? • As the amplitude E0 evolves, which types of bifurcation occur? • Which are the possible values of E0 and how does the boundary of these values of E0 evolve in the parameter spaces? • Which are the values of control gain parameters that lead to a good reduction of the amplitude or a suppression of chaos? The organization of the paper is as follows: In the next section, the driven multi-limit-cycle Van der Pol model is presented. We derive the amplitude of the oscillatory states in the autonomous and nonautonomous model using respectively the Lindstedt’s perturbation and harmonic balance methods [Nayfeh & Mook, 1979]. We use numerical simulations to ﬁnd various bifurcation structures which appear in the model as the parameters of the system evolve. The boundary of the region of admissible values of the amplitude of the external excitation is found. Section 3 deals with the eﬀects of the control on the dynamics of the driven multi-limit-cycle Van der Pol model. Particular emphasis will be paid to the quenching of chaotic vibrations. Concluding remarks come in the last section.

2. Dynamics of the Model 2.1. The driven multi-limit-cycle Van der Pol model The model considered is a classical Van der Pol oscillator with a nonlinear function of higher

polynomial order described by the following nonlinear equation x ¨ − µ(1 − x2 + αx4 − βx6 )x˙ + x = E0 cos Ωt

(1)

where an overdot denotes time derivative. The quantities α, µ and β are positive parameters, while E0 and Ω are respectively the amplitude and the frequency of the external excitation. This model was proposed by Kaiser [1981] and represents both a system to simulate certain speciﬁc processes in biophysical systems and a model which exhibits an extremely rich bifurcation behavior. The system is a nonlinear self-sustained oscillator which possesses more than one stable limit-cycle solution in the unforced case [Kaiser, 1981]. Models like this are considered rather seldom in the literature. However, such systems are of interest, especially in biology, for example to describe the coexistence of two stable oscillatory states. This situation can be found in some enzyme reactions [Li & Goldberter, 1989]. Another example is the explanation of the existence of multiple frequency and intensity windows in the reaction of biological systems when they are irradiated with very weak electromagnetic ﬁelds [Kaiser, 1983a, 1983b, 1987, 1989]. Besides, the model under consideration oﬀers general aspects concerning the behavior of nonlinear dynamical systems, Kaiser and Eichwarld [1991] have analyzed the super-harmonic resonance structure, while Eichwald and Kaiser [1991] have found symmetry-breaking crisis and intermittency.

2.2. Amplitudes and frequencies of limit cycles We consider in this subsection the autonomous extend Van der Pol oscillator (E0 = 0) and ﬁnd analytically the amplitudes and frequencies of limit cycles, using the Lindstedt’s perturbation method [Hagedorn, 1988]. For this purpose, it is interesting to set τ = wt, where w is the unknown frequency, this permits the frequency and the amplitude to interact. We assume that the periodic solution of Eq. (1) without external force can be performed by an approximation having the form x(τ ) = x0 (τ ) + µx1 (τ ) + µ2 x2 (τ ) + · · · .

(2)

where the function xi (τ ) is a function of τ of period 2π. Moreover, the frequency w can be represented by the following expansion w = w0 + µw1 + µ2 w2 + µ3 w3 + · · ·

(3)

Dynamics and Active Control of Motion of a Driven Multi-Limit-Cycle Van der Pol Oscillator

where wi are unknown constant in this point. Substituting the expressions (2) and (3) in Eq. (1) and equating the coeﬃcients of µ0 , µ1 , µ2 to zero, we obtain

where

Order µ0 w02 x0 + x0 = 0.

(4)

Order µ1 w02 x1 + x1 = w0 (1 − x20 )x˙ 0 − 2w1 w0 x ¨0 4 6 + w0 (αx0 x˙ 0 − βx0 x˙ 0 ). Order

(5)

µ2

w02 x2 + x2 = w0 [(1 − x20 )x˙ 1 − 2x0 x˙ 0 x1 ] − 2w1 w0 x ¨1 2 2 − (w1 + 2w0 w2 )¨ x0 − w1 (1 − x0 )x˙ 0 2 4 + w0 [(α − βx0 )x0 x1 + (4α − 6βx20 )x30 x1 x˙ 0 ] (6) + w1 (α − βx20 )x40 x˙ 0 . Making use of xi (τ + 2π) = xi (τ ) and xi (0) = 0 to determine the unknown quantities in the above equations, we get xi (τ + 2π) = xi (τ ),

x˙ i (0) = 0

(7)

Solving Eq. (4) and using conditions (7), we obtain x0 = A cos τ. w0 = 1,

(8)

where A is the amplitude of the limit cycle. In virtue of Eq. (8), Eq. (5) leads to 5β 6 α 4 A2 A − A + − 1 sin τ x ¨1 + x1 = A 64 8 4 + 2w1 A cos τ 1 3 3α 5 9β 7 A − A + A sin 3τ + 4 16 64 5 α 5 7 A − βA sin 5τ − 10 64 β 7 A sin 7τ (9) 64 It appears that the solvability conditions of x1 are +

α A2 5 βA6 − A4 + − 1 = 0, 64 8 4

(10)

w1 = 0. The general solution of Eq. (9) can be now written as follows x1 = A1 cos τ + B1 sin τ + ψ1 sin 3τ + ψ2 sin 5τ + ψ3 sin 7τ,

(11)

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1 1 3 3α 5 9β 2 ψ1 = − A − A + A , 8 4 16 64 1 α 5 5β 7 A − A , ψ2 = 24 16 64

β A7 . 3072 With the initial conditions x˙ i (0) = 0, one now obtains 219 1 3 βA7 − αA5 + A3 . B1 = 3072 12 32 The periodicity condition for solution x2 (τ ) yields the following relations ψ3 = −

A1 = 0, w2 =

1580 738αβ 10 βA12 − A 196608 49512 72α2 + 309β 64α − 219β 8 A − A6 + 384 3072 3 16α + 3 A4 − A2 . + (12) 192 32

So that, the solution of Eq. (1) without external force is approximated by x(t) = A cos wt + µ[B1 sin wt + ψ1 sin 3wt + ψ2 sin 5wt + ψ3 sin 7wt] + O(µ2 ),

(13)

where the frequency w is given by w = 1 + µw2 + O(µ3 ).

(14)

To ﬁnd the values of the limit cycle’s amplitude A from Eq. (10) and the resulting frequencies ω, we use the Newton–Raphson algorithm. According to the values of the parameters α and β, Eq. (10) can give rise to one or three positive roots which corresponds respectively to one stable limit cycle or three limit cycles among which two are stable and one is unstable. Indeed, we have derived from Eq. (1) a limit cycle’s map showing some ranges of the parameters α and β for which the system under consideration can exhibit one or three limit cycles (see Fig. 1). Such a coexistence of two stable limit cycles with diﬀerent amplitudes and frequencies (or periods) separated by an unstable limit cycle for a given set of parameters is referred to as birhythmicity. The unstable limit cycle represents the separatrix between the basins of attraction of the two stable limit cycles. For instance, when α = 0.144 and β = 0.005, the two stable limit cycle’s amplitude are

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Fig. 1.

Region of (α, β) leads to one or three limit cycles.

A1 = 2.6930 and A3 = 4.8395 with their related frequency ω(A1 ) = 1.0011 and ω(A3 ) = 1.0545 while the unstable limit cycle’s amplitude is A2 = 3.9616 with the frequency ω(A2 ) = 1.0114. The above stable limit cycles and their corresponding basins of attraction can be deduced from a direct numerical simulation of Eq. (1) using the fourth-order Runge– Kutta algorithm as it is shown in Fig. 2.

2.3. The amplitude of the harmonic oscillatory states With the external force, the amplitude of the forced harmonic oscillatory states can be found using the harmonic balance method [Nayfeh & Mook, 1979]. Our aim is to study the interaction between the external excitation and the amplitude of the limit cycles. Assuming that the fundamental component of the solutions has the period of the external excitation, we express the solution x as x = a1 cos Ωt + a2 sin Ωt = Anc cos(Ωt − φ).

(15)

Inserting expression (15) in Eq. (1) and equating the coeﬃcient of the cosine and sine terms separately, one obtains 1 1 2 (1 − Ω )a1 − µΩ 1 − A2nc + αΩA4nc 4 8 5 − βA6nc a2 = E0 , 64 (16) 1 5 1 µΩ 1 − A2nc + αΩA4nc − βA6nc a1 4 8 64 + (1 − Ω2 )a2 = 0, where A2nc = a21 + a22 ,

1 5 1 2 4 6 µΩ 1 − Anc + αΩAnc − βAnc 4 8 64 . tan φ = 1 − Ω2 After some algebraic manipulations we ﬁnd that the amplitude Anc satisﬁes the following nonlinear

Dynamics and Active Control of Motion of a Driven Multi-Limit-Cycle Van der Pol Oscillator

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(a)

(b) Fig. 2. Phase portraits of the two stable limit cycles (a) and their corresponding basins of attraction (b). The black zone represents the attraction to A1 and the white one the attraction to A3 .

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algebraic equation 2 2 2α Ω + 5β 25β 2 14 5αβΩ 12 Anc − Anc + A10 nc 4096 256 128 2αΩ + 5β 4α + 1 1 8 Anc + A6nc − A4nc − 32 16 2 (Ω2 − 1)2 E02 2 + 1+ − = 0. (17) A nc µ2 Ω 2 µ2 Ω 2 We also ﬁnd the behavior of the amplitude Anc when the frequency of the external excitation Ω is varied and the results are reported in Figs. 3 and 4. Figure 3 shows the variation of Anc versus Ω

for several diﬀerent values of the amplitude E0 and there appears a resonance peak. The comparison between analytical and numerical response frequency-curves Anc (Ω) of the driven multi-limitcycle model is shown in Fig. 4. In these two ﬁgures, we ﬁnd that when the external excitation is considered, the model exhibits only one limit-cycle.

2.4. Admissible value of E0 and bifurcation mechanisms In the above subsection, we have found the interaction between the limit-cycles and the amplitude

35 4

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E =1

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E =0.5 0

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E

Anc

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0 0

0

0.2

0.4

0.6

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1.2

1.4

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1

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Ω

Fig. 3. Eﬀects of the amplitude E0 on the amplitude of the harmonic oscillatory states Anc (Ω) with the parameters µ = 0.1, α = 0.144 and β = 0.005.

4

µ

2

5

6

Fig. 5. Boundary of the admissible value of the amplitude of the external excitation in the (µ, E0 ) plane with the parameters µ = 0.1, α = 0.144, Ω = 1 and β = 0.005. 20

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E

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Anc

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Fig. 4. Comparison between analytical (with lines) and numerical (with dots) frequency-response curves.

0.4

0.6

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1

1.2

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Ω

Fig. 6. Boundary of the admissible value of the amplitude of the external excitation in the (Ω, E0 ) plane with the parameters µ = 0.1, α = 0.144, µ = 3 and β = 0.005.

Dynamics and Active Control of Motion of a Driven Multi-Limit-Cycle Van der Pol Oscillator

of the external excitation, it appears that with the external excitation, the quenching of limit-cycle occurs since the driven multi-limit-cycle Van der Pol model exhibits only one limit-cycle. This subsection deals with the choice of the possible or admissible values of the amplitude of the external excitation and the analysis of some bifurcation mechanisms which appear in the model. We solve numerically Eq. (1) and plot the resulting bifurcation diagram as E0 and µ vary. The time period of the periodic stroboscopic bifurcation diagram used to map the transition is T = 2π/Ω. Figures 5 and 6 present the boundary of the admissible value of E0 respectively in the (µ, E0 ) and (Ω, E0 ) planes. We ﬁnd that the possible value of E0 for any ﬁxed value of µ or Ω is the region below the curve. For instance, let us consider the particular case µ = Ω = 1, it appears from Fig. 5 that the domain where the choice of E0 is possible belong to the interval of E0 deﬁned as E0 ∈ ]0; 13.0], while for the case µ = 3 and Ω = 1, we have the interval E0 ∈ ]0; 4.6]. After presenting the region of possible values of E0 , it is important

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to ﬁnd various bifurcation mechanisms which the model exhibits in these regions. Figures 7 and 8 show the bifurcation diagrams versus the amplitude E0 and the µ coeﬃcient, respectively. In Fig. 7, we ﬁnd that as the amplitude E0 increases from zero, the amplitude of the chaotic motions exist until E0 = 0.6 where the period-1 orbit appears after the small windows of period-2 orbit appears. When µ = 2 and µ = 3, the same transition scenario occurs but it should be stressed that the region of chaotic motions increases. For µ = 4, we ﬁnd that only the chaotic orbits appear as the amplitude E0 increases in the region of the admissible value of E0 . Figure 8 shows the bifurcation diagrams when µ varies for two diﬀerent values of E0 . For E0 = 0.5, we ﬁnd that the amplitude of the periodic oscillations exists and bifurcates to the chaotic and nonperiodic motion at µ = 2.4, while for E0 = 1 the bifurcation between the periodic and nonperiodic motion appears at µ = 3.52. Figure 9 presents the phase portrait x˙ versus x of the nonperiodic motion.

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0 −2 −4

µ=1 0

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µ=2 −5

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x

x

1 0

0 −1 −2

µ=3 −5

0

1

2

E0

3

−3

0

0.1

0.2

E0

Fig. 7. Bifurcation diagrams showing the variation of the variable x versus the amplitude E0 for several values of the µ coeﬃcient with the parameters µ = 0.1, α = 0.144, Ω = 1 and β = 0.005.

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x

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E =0.5 0

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E =1 0

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µ

2.5

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3.5

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4.5

Fig. 8. Bifurcation diagrams showing the variation of the variable x versus the µ coeﬃcient for several values of the amplitude E0 with the parameters α = 0.144, Ω = 1 and β = 0.005.

10

8

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4

dx/dt

2

0 −2 −4 −6 −8 −10 −5

−4

−3

−2

−1

0

1

2

3

4

5

x Fig. 9. Phase portrait x˙ versus x showing the nonperiodic motion with the parameters µ = 0.1, α = 0.144 and β = 0.005. The initial conditions used are (x(0); ˙ x(0)) = (2; 2) for (i) and (x(0); ˙ x(0)) = (4; 4) for (ii).

Dynamics and Active Control of Motion of a Driven Multi-Limit-Cycle Van der Pol Oscillator

3. Active Control As shown in the above section, the presence of an external force aﬀects signiﬁcantly the behavior of the extended Van der Pol model. Thus, the presence of chaotic motions are observed. Our aim in this section is to limit the undesired eﬀects of the external excitation by using the appropriate control strategy. Among the control strategies, the active control plays a particular role [Fuller et al., 1997; Tchoukuegno et al., 2003; Soong, 1990; Jezequel, 1995]. In what follows, we focus our attention on the linear oscillator coupled with a previous Van der Pol oscillator. The linear oscillator serves as a control element used to reduce the amplitude of vibration and to suppress the nonperiodic motion in the Van der Pol model. Morgan and Wang had shown that piezoelectric materials can be used as passive electromechanical vibration absorbers by shunting them with electrical networks [Morgan & Wang, 2002]. In this case, let us assume that the model of the structure is given by x ¨ − µ(1 − x2 + αx4 − βx6 )x˙ + x + λ1 y¨ = E0 cos Ωt, (18) y¨ + w22 y + γ2 y˙ − λ1 x = 0, where y is the control force, and λ1 the control gain parameters, w2 the free frequency of the second oscillator and γ2 the damping coeﬃcient. Equations (18) can describe various physical systems like those presented in [Yamapi et al., 2003]. In this case, the Van der Pol oscillator represents the motion of the electrical oscillator with a nonlinear resistor while the linear oscillator describes the motion of the linear mechanical oscillator. The coupling between both parts is ensured by the electromagnetic force due to a permanent magnet, which creates a Laplace force in the mechanical part and a Lenz electromotive voltage in the electrical part.

3.1. Eﬀects of the control on the amplitude of harmonic oscillations The harmonic balance method [Nayfeh & Mook, 1979] is also used to determine the amplitude of the vibration of the system under the control. To do that, we consider x as deﬁned in Eq. (15) and inserting it into Eq. (18) to obtain y = (a1 Λ1 − a2 Λ2 ) cos Ωt + (a2 Λ1 + a1 Λ2 ) sin Ωt, (19)

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where Λ1 = Λ2 =

λ1 (w22 − Ω2 ) , (w22 − Ω2 )2 + γ22 Ω2 (w22

λ1 γ2 Ω − Ω2 )2 + γ22 Ω2

Taking into account (19), the ﬁrst equation of the system (18) becomes ΩΛ2 2 4 6 − x + αx − βx x˙ + x x−µ 1+ (1 + Λ1 )¨ µ = E0 cos Ωt

(20)

Inserting the expression (15) into Eq. (20) and equating the coeﬃcient of the cosine and sine terms separately (knowing that the amplitude of vibration of the system is given by A2c = a21 +a22 ), we ﬁnd after some algebraic manipulations that the amplitude of the oscillatory states satisﬁes the following algebraic equation 2 2 2α Ω + 5β 25β 2 14 5αβΩ 12 Ac − Ac + A10 c 4096 256 128 2αΩ + 5β 5λ1 ΩβΛ2 − A8c − 32 38µ 4α + 1 λ1 αΩΛ2 − A6c + 16 4µ 2 2 2 λ1 ΩΛ2 1 λ1 Ω Λ2 − 2λ1 ΩΛ2 4 1− Ac + − 2 µ µ 2 2 2 (1 − Ω − λ1 Ω Λ1 ) A2c +1 + 1+ µ2 Ω 2 E02 =0 (21) µ2 Ω 2 It is necessary to look at the condition fulﬁlled by the control parameter λ1 , so that the control should be eﬀective. In fact, the control is eﬀective when Ac < Anc , Anc being the amplitude of the oscillations of the uncontrolled system (that is λ1 = 0). To illustrate this fact, in Fig. 10 is plotted the variation of the amplitude Ac computed from Eq. (21) versus the control gain parameter λ1 . The solid horizontal line is the value of the amplitude without control. It is found that Ac < Anc for λ1 ∈ [0.06; +∞[. The eﬀects of the control on the amplitude of the forced Van der Pol oscillator are analyzed. In the same ﬁgure, we have plotted with dots-line the amplitude obtained from a direct numerical simulation of the diﬀerential equation (18). It is found that Ac < Anc for a restricted domain of λ1 deﬁned as λ1 ∈ ]0.055; +∞[. −

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3.2. Quenching of chaotic vibrations

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3

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Ac

2

1.5

1

0.5

0

0

0.5

1

1.5

2

2.5

3

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Ω

Fig. 10. Eﬀects of the control on the amplitude of the Van der Pol oscillator with the parameters µ = 0.1, α = 0.144; Ω = w2 = 1.0; γ2 = 0.2 and β = 0.005.

We have analyzed analytically and numerically the conditions to be satisﬁed by the coupling parameter in order to reduce the amplitude of a driven Van der Pol model under the control. Our aim now is to investigate conditions of suppressing chaos or instability issues in the driven Van der Pol oscillator coupled to the linear oscillator, using numerical simulations of Eqs. (18). Figure 11 shows the eﬀects of the coupling parameter λ1 on the bifurcation structures and the following results are observed. As the coupling parameter λ1 varies from zero to 9, diﬀerent bifurcation structures appear as E0 increases. Among these structures, we have periodic, nonperiodic and chaotic motions (λ1 = 0), nonperiodic and multiperiodic motions (λ1 = 2), and nonperiodic and chaotic motions (λ1 = 5; 7; 9). This means that the undesired behaviors remain in the model since the linear oscillator is not eﬃcient in this range

Fig. 11. Eﬀects of the control on the amplitude of the Van der Pol oscillator with the parameters µ = 0.1, α = 0.144; Ω = w2 = 1.0; γ2 = 0.2 and β = 0.005.

Dynamics and Active Control of Motion of a Driven Multi-Limit-Cycle Van der Pol Oscillator

been found. We have also found that with λ1 = 10, the chaotic oscillations of the primary driven multilimit-cycle Van der Pol model has been transformed to periodic motion, interpreted as a quenching of chaotic vibrations.

80

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References

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0

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2.5

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Fig. 12. Eﬀects of the control on the boundary of the admissible value of E0 with the parameters µ = 0.1, α = 0.144; Ω = w2 = 1.0; γ2 = 0.2 and β = 0.005.

of coupling parameter (λ1 < 10). We also ﬁnd the increase of domains of nonperiodic and chaotic orbit. As λ1 increases further, the chaotic oscillations of the driven multi-limit-cycle Van der Pol oscillator disappear at λ1 = 10 and the system passes to have periodic oscillations. The disappearance of the chaotic oscillations can be interpreted like a quenching of chaotic vibrations. We note that when λ1 = 0, the linear oscillator is not eﬃcient to control the motions of the driven Van der Pol oscillators. The eﬀects of the control on the boundary of the admissible value of E0 in the (µ, E0 ) plane are presented in Fig. 12 and it appears that the boundary of this region increases when the coupling parameter varies from λ1 = 0 to λ1 = 2.

4. Conclusion We have considered in this paper the dynamics and active control of motion of a driven multi-limit-cycle Van der Pol model. The amplitude of the oscillatory state in the autonomous and non-autonomous model have been found using analytical investigations. We have found various bifurcation mechanisms which appeared in the model as the amplitude E0 evolves. The boundary of the admissible values of E0 has been derived using numerical simulations of the equations of motion. The active control is used to limit the presence of undesired behaviors. The appropriate coupling parameter for which the linear oscillator reduces eﬃciently the amplitude of a driven multi-limit-cycle Van der Pol model has

Cheng, A. H. D., Yang, C. Y., Hackl, K. & Chajes, M. J. [1993] “Stability, bifurcation and chaos of non-linear structures with control-II, non-autonomous case,” Int. J. Non-Lin. Mech. 28, 549–565. Eichwald, C. & Kaiser, F. [1991] “Bifurcation structure of a driven multi-limit-cycle Van der Pol oscillator. (II) Symmetry-breaking crisis and intermittency,” Int. J. Bifurcation and Chaos 1, 711–715. Fuller, C. R., Eliot, S. J. & Nelson, P. A. [1997] Active Control of Vibration (Academic, London). Guckenheimer, J. & Holmes, P. J. [1994] Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields (Springer-Verlag, Berlin). Hackl, K., Yang, C. Y. & Cheng, A. H. D. [1993] “Stability, bifurcation and chaos of non-linear structures with control-I, autonomous case,” Int. J. Non-Lin. Mech. 28, 441–454. Hagedorn, P. [1988] Nonlinear Oscillations, 2nd edition, (Clarendon Press, Oxford). Hayashi, C. [1964] Non-Linear Oscillations in Physical Systems (McGraw Hill, NY). Jezequel, [1995] Active Control in Mechanical Engineering (Edition Hermes). Kaiser, F. [1981] “Coherent oscillations in biological systems: Interaction with extremely low frequency ﬁelds,” Radio Sci. 17, 17S. Kaiser, F. [1983a] Coherent Excitations in Biological Systems: Speciﬁc Eﬀects in Externally Driven SelfSustained Oscillating Biophysical Systems (SpringerVerlag, Berlin, Heidelberg). Kaiser, F. [1983b] “Theory of resonant eﬀects of RF and MW energy,” in Biological Eﬀects of an Dosimetry of Nonionizing Radiation, eds. Grandolfo, M., Michaelson, S. M. & Rindi, A. (Plenum Press, NY), p. 251. Kaiser, F. [1987] “The role of chaos in biological systems,” in Energy Transfer Dynamics, eds. Barret, T. W. & Pohl, H. A. (Springer, Berlin), p. 224. Kaiser, F. [1989] “Nichtlineac resonanz und chaos. Ihre relevanz f¨ ur biologische funktion,” Kleinheubacher Berichte 32, p. 395. Kaiser, F. & Eichwald, C. [1991] “Bifurcation structure of a driven multi-limit-cycle Van der Pol oscillator. (I) The superharmonic resonance structure,” Int. J. Bifurcation and Chaos 1, 485–491. Li, V.-X. & Goldbeter, A. [1989] “Oscillatory isozymes as the simplest model for coupled biochemical oscillators,” J. Theor. Biol. 138, 149–174.

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