Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1550–1559 www.elsevier.com/locate/cnsns

Active control of extended Van der Pol equation B.R. Nana Nbendjo a

a,*

, R. Yamapi

b

Department of Physics, Laboratory of Nonlinear Modelling and Simulation in Engineering and Biological Physics, Faculty of Sciences, University of Yaounde´ I, P.O. Box 812, Yaounde, Cameroon b Department of Physics, Faculty of Sciences, University of Douala, P.O. Box 24157, Douala, Cameroon Received 6 December 2005; received in revised form 28 January 2006; accepted 29 January 2006 Available online 3 April 2006

Abstract We study in this paper the active control of a driven class of Van der Pol oscillator which exhibits three limit cycles. We begin by investigating the dynamics and stability analysis of the system under active control. We also analyze the effects of a time periodic perturbation included in the control process. In all these cases the domain of control gain parameters leading to a good control is obtained and verified numerically. Ó 2006 Elsevier B.V. All rights reserved. PACS: 02.30.Yy; 02.60.x; 05.45.a; 05.45.Gg Keywords: Active control; Self-excited system; Floquet theory

1. Introduction Recently, the control of linear and nonlinear structures have been a subject of particular interests [1–8]. This is due to the fact that nonlinear oscillator under certain conditions can lead to various configurations depending on the types of nonlinearity and associate potential. For example, Refs. [3,4] considered the behavior of the catastrophic single well-Duffing oscillator under active control, while Refs. [5,6] extended the study to a nonlinear oscillator with a single well potential, a catastrophic-double-potential and a tristable non catastrophic potential. In the context of the active control of nonlinear structures, our recent contributions have mainly focussed on the active control with delay of vibrations in a double-well-Duffing oscillator [7]. Various aspects were analyzed: the active control of vibrations, snap-through instability and chaos in a bistable Duffing oscillator. The derivation of the range of control parameters which leads to a good control, the effects of the time delay between the detection of vibration and action of the control. In Ref. [8], we considered the dynamics and active control of a driven multi-limit-cycle Van der Pol oscillator. The effects of the control parameter on the behavior of a driven multi-limit-cycle Van der Pol model were analyzed and it appears that with the appropriate choice of coupling parameter, one can suppress the chaotic vibrations. *

Corresponding author. Tel.: +237 9509029. E-mail addresses: [email protected], [email protected] (B.R. Nana Nbendjo), [email protected] (R. Yamapi).

1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.01.016

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We extend our investigation in this paper by considering the stability of controlled dynamics in a driven Van der Pol model. Our aim is to derive the range of control gain parameters which leads to a good control and the stability of the control design. We first concentrate on the dynamical behaviors of the model under the control. For this aim, we use the harmonic balance method to derive the amplitude of the oscillatory states. The stability analysis of the control process will be carried out by using the Whittaker method. The effects of a time periodic perturbation included in the control process are derived using analytical and numerical investigations. The paper is organized as follows. In the next section, after presenting the model under the active control, we derive the amplitude of the oscillatory states as function of the gain control parameters by means of harmonic balance method, and analyze the stability of the control process using the Whittaker method. In Section 4, we analyze the effects of the amplitude of the time periodic perturbation include in the active control process. We conclude in the last section. 2. Active control and its stability 2.1. The model and mathematic formulation The Van der Pol oscillator with a nonlinear damping function of higher polynomial order that we consider in this paper is described by the following nonlinear equation: €x  lð1  x2 þ ax4  bx6 Þ_x þ x ¼ 0;

ð1Þ

(dots denote times derivative). The quantities a, b and l are positive parameters. This model was proposed by Kaiser [9,10] and represents both a system to simulate certain specific processes in biophysical systems and model which exhibits an extremely rich bifurcation behavior. Nevertheless, especially in biology, such system is of interest to describe the coexistence of two stable oscillatory states. For example this situation can be found in some enzyme reactions presented in Refs. [11,12]. 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 week electromagnetic fields [13,14]. According to the values of the parameters a and b, we have found analytically and numerically that the model described by Eq. (1) can give rise respectively to one stable limit cycle or three limit cycle among which two are stable and one is unstable [8]. Indeed, we have derived from Eq. (1) a limit cycle’s map showing some ranges of the parameters a and b 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 refer to as biorhythmicity. The unstable limit cycle represents the separatrix between the basins of attraction of the two stable limit cycles. For instance, for a = 0.144 and b = 0.005, the two stable limit cycle’s amplitudes are A1 = 2.6930 and A3 = 4.8395 with their related frequencies x(A1) = 1.0011 and x(A3) = 1.0545 while the unstable limit cycle’s amplitude is A2 = 3.9616 with the frequency x(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. With the external drive, we have found the following results [8]: the driven Van der Pol model exhibits various types of bifurcation mechanisms as the amplitude of an external drive evolves, leading to undesired behaviors. We have found that with the external drive, the model exists only for certain values of the amplitude (such as numerical instability), and we derived numerically the boundary of the acceptable value of these amplitude (see Fig. 3, E0 and w are respectively the amplitude and frequency of the external drive). The acceptable values of E0 can be chosen in the region located at the lower part of the curve. We have also found that the external drive lead to the quenching of one limit cycle solution. Due to these undesired behaviors, the system can become unstable if we find the amplitude E0 in the region located at the top of the boundary given by the curve. One can use the active control strategy to solve the problem. To illustrate that, the governing equations of the controlled systems are given by €x  lð1  x2 þ ax4  bx6 Þ_x þ x ¼ cz þ E0 cos wt; z_ þ a1 z ¼ b1 x;

ð2Þ

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Fig. 1. Region of (a, b) leads to one or three limit cycles.

Fig. 2. Phases portrait 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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20 18 16 14

E0

12 10 8 6 4 2

0.2

0.4

0.6

0.8

1

1.2

1.4

Ω Fig. 3. Boundary of the admissible value of the amplitude of the external excitation in the (X, E0) plane with the parameters l = 0.1, a = 0.144 and b = 0.005.

where E0 and w are respectively the amplitude and frequency of the external excitation, z is the control force, a1 the control speed parameters, b1 and c the control gain parameters. Practically, this type of control can be carried out in biology by making use of a microchip, putting together to the original biological system. 2.2. Effects of the control on the behaviors of the model We follow the harmonic balance method [15,16] to find the amplitude of the oscillatory states. Setting x ¼ a1 cos wt þ a2 sin wt;

ð3Þ

and inserting (3) into the second equation of system (2), it comes after some algebraic manipulations that the control force z can be transformed to the alternative form defined as z¼

a21

b1 fða1 a1  wa2 Þ cos wt þ ða1 a2 þ wa1 Þ sin wtg; þ w2

Then, with the Eq. (4), the first equation of the system (2) can be writing as follows:     cb1 cb1 a1 2 4 6 €x  l 1   x þ ax  bx x_ þ 1  2 x ¼ E0 cos wt; lða21 þ w2 Þ a1 þ w2

ð4Þ

ð5Þ

Expressing the solution x as defined by Eq. (3), we find that the amplitude A ðA2 ¼ a21 þ a22 Þ of the oscillatory state of a driven Van der Pol model under control satisfies the following nonlinear equation:  2  2 a1 b c cb1 1 2 1 4 5 6 A aA bA  þ   E20 ¼ 0; ð6Þ A2 1  2 1 2  w2 þ l2 w2 A2 1  a1 þ w 8 64 lða21 þ w2 Þ 4 Using the Newton–Raphson algorithm, we have found and plotted in Figs. 4 and 5 the variation of the amplitude A as function of control gain parameters c and b1. The parameters used throughout the paper are: a = 0.144; b = 0.005; a1 = 1.0; l = 0.1; w = 1 and E0 = 1.0. It is important to note that the choice of E0 is not arbitrary, but in order to be in the unstable region for the uncontrolled Van der Pol model. Fig. 4 shows the variation of A versus c for several different values of the coefficient b1 and it appears that as the coefficient b1 increases, the amplitude A(c) decreases. In Fig. 5, we also provide the amplitude A versus b1 with several different values of c and find the same phenomena. It is important to note that the horizontal line in these figures represents the value of amplitudes A without control. It appears that when the control is applied with a good control parameter in the system, the amplitude of vibration is smaller that the case without control. Fig. 6 shows the comparison between analytical and numerical results. In summary, we observe that when

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B.R. Nana Nbendjo, R. Yamapi / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1550–1559 6

5

A

4

3

β1=0.5 β1=1.0 β1=1.5

2

1

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

γ

Fig. 4. Effects of the coefficient b1 on the amplitude response A(c) with the parameters: a = 0.144; b = 0.005; a1 = 1.0; E0 = X = 1.0 and several values of b1.

6

5

A

4

3

γ=0.5 γ=1.0

2

γ=1.5 1

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

β1 Fig. 5. Effects of the coefficient c on the amplitude response A(b1) with the parameters defined in Fig. 4 and several values of c.

the coefficients c and b1 vary, the configuration of a driven Van der Pol model under active control changes and leads to the stable and unstable limit cycle solutions in some region of the gain coefficients. The comparison between analytical and numerical results is presented in Fig. 7. Due to the presence of unstable solutions in a driven Van der Pol oscillator, it is interesting to analyze the stability of a driven Van der Pol model under the control. 2.3. Stability of the control To carry out the stability analysis of the control process, we consider the alternative Eq. (5) and derive the linear variational equation of the control system around the oscillatory states as follows:

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6

5

A

4

3

2

1

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

γ Fig. 6. Comparison between analytical (with line) and numerical results (with dots). The parameters used are defined in Fig. 4 with b1 = 0.5.

6

5

A

4

3

2

1

0

0

0.5

1

1.5

2

2.5

β

3

3.5

4

4.5

5

1

Fig. 7. Comparison between analytical (with line) and numerical results (with dots). The parameters used are defined in Fig. 4 with c = 0.5.

   cb1 cb1 a1 2 4 6 3 2 €  l 1   x þ ax  bx _ þ 1  2 þ lð2x  4ax þ 6bx Þ_x  ¼ 0; lða21 þ w2 Þ a1 þ w 2 

ð7Þ

where  = x  xs is the perturbation variable and xs the oscillatory states described by Eq. (3). The stability of the control process is equivalent to the boundary of (t) which depends on c and b1. From Eq. (3), the trajectory xs can be written as xs ¼ A cosðwt  /Þ;

ð8Þ

where A depends on the coefficient c and b1 as we have seen in the previous subsection. If we set s = wt  /, the variational Eq. (8) becomes € þ ½2k þ F ðsÞ_ þ GðsÞ ¼ 0;

ð9Þ

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where k¼

lH 2w

l ½I cos 2s þ J cos 4s þ L cos 6s; 2w   1 ca1 b GðsÞ ¼ 2 1  2 1 2 þ M cos sin 2s þ N sin 4s þ P sin 6s ; w a1 þ w F ðsÞ ¼ 

H ¼1

cb1 1 3a 5  A2 þ A4 þ bA6 ; 8 16 lða21 þ w2 Þ 2

1 1 15 1 3 I ¼  A2 þ aA4  A6 ; J ¼ aA4  bA6 ; 2 2 32 8 16   1 5 6 6 2 4 L ¼  bA ; M ¼ lw A  aA þ bA ; 32 6   1 3 3 N ¼ lw aA4  bA6 ; P ¼  lbwA6 . 2 2 16 To investigate the stability boundaries of the control process, let us use the transformation   Z 1 s  ¼ g expðksÞ exp  F ðs0 Þds0 2 0

ð10Þ

and rewrite Eq. (9) in the following appropriate Hill equation [15,16], € g þ ½a0 þ 2a1s sin 2s þ 2a1c cos 2s þ 2a2s sin 4s þ 2a2c cos 4s þ 2a3s sin 6s þ 2a3c cos 6s þ 2a4c cos 8s þ 2a5c cos 10s þ 2a6c cos 12sg ¼ 0.

ð11Þ

where ( ) 1 ca1 b1 l2 a0 ¼ 2 1  2  ð2H 2 þ I 2 þ J 2 þ L2 Þ ; w a1 þ X2 8   1 M lI l2 ð2HI þ IJ þ JLÞ;  ¼  ; a a1s ¼ 1c 2 w2 w 8w   1 N 2lJ l2 ð4HJ þ I 2 þ 2ILÞ; a2s ¼  ; a2c ¼  2 2 w w 16w   1 P 3lL l2 a3s ¼ ð2HL þ IJ Þ;  ¼  ; a 3c 2 w2 w 8w a4c ¼ 

l2 ð2IL þ J 2 Þ. 16w

a5c ¼ 

l2 JL; 8w

a6c ¼ 

l2 2 L. 16w

Following the Floquet theory [17,18], the solution of Eq. (11) may be either stable or unstable. However, this equation exhibits six main parametric resonances, the stability boundaries of the control process are found around the six main parametric resonances defined at a0 = n2 (with n = 1, 2, 3, 4, 5, 6). Then, the solution of Eq. (11) in the nth unstable region may be assumed in the following form: g ¼ expðl0 sÞ sinðns  rÞ; where l0 is the characteristic exponent and r a constant. Then, (t) can be rewritten as follows:   l 1 L ðI sin 2s þ sin 4s þ sin 6sÞ sinðns  rÞ. ðtÞ ¼ exp ðk þ l0 Þs þ 4w 2 3

ð12Þ

ð13Þ

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3 2.5

E0

2 1.5 1 0.5 0

0.5

1

1.5

2

γ

2.5

3

3.5

4

Fig. 8. Analytical stability boundaries in the (c, E0) plane. The parameters used are defined in Fig. 4.

We note that the control process is achieved when (t) goes to zero with increasing time. Following the Floquet theory [15,16], the solution of Eq. (11) may be either stable or unstable. The stability boundaries are found around the above six main parametric resonances. Substituting Eq. (12) into Eq. (11) and equating the coefficients of cos ns and sin ns separately to zero, we obtain the following expression for the characteristic exponent qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l20 ¼ ða0 þ n2 Þ þ 4n2 a0 þ a2n ; ð14Þ with a2n ¼ a2nc þ a2ns . Since the synchronization process is achieved when  goes to zero with increasing time, the real parts of k ± l0 should be negative. Consequently, the synchronization process is stable under the condition H n ¼ ða0  n2 Þ2 þ 2k2 ða0 þ n2 Þ þ k4  a2n > 0; a2n

a2nc

ðn ¼ 1; 2Þ.

ð15Þ

a2ns .

and ¼ þ From the theory of Hill equation, one expects that, if the conditions (15) are satisfied for all the values of n, c and b1, the variable (t) tends to zero with increasing time, resulting on the stability of the control process in a driven Van der Pol system under control, if not, the control process is unstable. It is important to note here that the control process is unstable means that the variable (t) never goes to zero, but has a bounded oscillatory behaviors or grows infinitely. In the first main parametric resonance (n = 1), Fig. 8 shows the stability boundaries of the control process in the region (E0, c) plane. 3. Effects of the time periodic perturbation on the control process Another possibility of optimization of the control would be to introduce a perturbation into the control design so that, this could allow us to direct the dynamics of the system while playing on this perturbation. Consider in this case the perturbation as a sinusoidal excitation, the equation governing the dynamics of the system becomes €x  lð1  x2 þ ax4  bx6 Þ_x þ x ¼ cz þ E0 cos wt; z_ þ a1 z ¼ b1 ðx  f0 cos wtÞ;

ð16Þ

where f0 is the amplitude of the control excitation. Our aim in this section is to use the contribution of parameters f0 to optimize the control process, for this purpose, we first derive the new amplitude equation. We still express the solution of Eq. (16) as in Eq. (2). Substituting (3) into the second equation of (15) we obtain z¼

a21

b1 fða1 a1  wa2  a1 f0 Þ cos wt þ ða1 a2 þ wa1  wf 0 Þ sin wtg. þ w2

ð17Þ

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Using Eqs. (3), (15) and (16) after some algebraic manipulations, we express the amplitude equation in the form  2  2 a1 b1 c cb1 1 2 1 4 5 2 6 2 2 2 2 A 1 2  A þ aA  bA w þl w A 1 a1 þ w 2 8 64 lða21 þ w2 Þ 4  2  2 cb1 a1 f0 cb1 wf 0  E0   ¼ 0. ð18Þ lða21 þ w2 Þ lða21 þ w2 Þ

A

6

5

f0=0.5

4

f =0.4

3

f0=0.3

2

f0=0.2

0

f0=0.1

1

f0=0.0 0 0

0.5

1

1.5

2

2.5

γ

3

3.5

4

4.5

5

Fig. 9. Effects of the amplitude f0 on the amplitude A versus c with the parameters defined in Fig. 4 with b1 = 0.5.

8

7

6

A

γ = 0.5 5

γ = 0.4

4 γ = 0.3 3

γ = 0.2 γ = 0.1

2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

f0 Fig. 10. Effects of c on the amplitude A versus f0 with the parameters used are defined in Fig. 4 with b1 = 0.5.

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To view the effect of f0 in the dynamics of the control systems we present in Fig. 9 the evolution of amplitude A as function of c for several values of f0. It appears that for a certain value of c the amplitude is smaller than the case where f0 = 0 and for a critical value of c this amplitude is greater, along with the fact that the amplitude of vibration increases with f0. Fig. 10 presents the evolution of A as function of the amplitude f0 for several values of c. It is viewed that the presence of the excitation in the control process affects the dynamics of the system and globally decreases the value of the amplitude of vibration for a good choice of control gain parameters. This means that depending on the choice of control gain parameters along with the value of the amplitude of excitation in the controlled process, one can direct the dynamics of the system in good or bad direction. 4. Conclusion We have studied the possibility to use the active control strategy to modify the dynamics of the extended Van der Pol oscillator. The dynamics of the system under control have been study and the stability boundary of these stationary solutions have been obtained in the space of control gain parameters both in autonomous and non autonomous case. It appears that with the good choice of control gain parameters one can modify the dynamics response of our system under the control process. It also comes that the excitation in the control process can be used to stabilize the system, to modify the dynamics responses of the system and it can also be the source of instability in the control design. References [1] Fuller CR, Eliot SJ, Nelson PA. Active control of vibration. London: Academic; 1997. [2] Zhang L, Yang CY, Chajes MJ, Cheng AHD. Stability of active-tendon structural control with time delay. J Engng Mech Div ASCE 1993;119:1017. [3] Hackl K, Yang CY, Cheng AHD. Stability, bifurcation and chaos of non-linear structures with control-I. Autonomous case. Int J Non-Linear Mech 1993;28:441. [4] Cheng AHD, Yang CY, Hackl K, Chajes MJ. Stability, bifurcation and chaos of non-linear structures with control-II. Nonautonomous case. Int J Non-Linear Mech 1993;28:549. [5] Tchoukuegno R, Woafo P. Dynamics and active control of motion of a particle in a /6 potential with a parametric forcing. Physica D 2002;167:86. [6] Tchoukuegno R, Nana Nbendjo BR, Woafo P. Linear feedback and parametric controls of vibration and chaotic escape in a /6 potential. Int J Non-Linear Mech 2003;38:531. [7] Nana Nbendjo BR, Tchoukuegno R, Woafo P. Active control with delay of vibration and chaos in a double-well-Duffing oscillator. Chaos Soliton Fract 2003;18:345. [8] Yamapi R, Nana Nbendjo BR, Enjieu Nkadji HG. Dynamics and active control of a motion of a driven multi-lmit-cycle Van der Pol oscillator. Int J Bifurcation Chaos, in press. [9] Kaiser F. Coherent oscillation in biological systems: interaction with extremely low frequency fields. Radio Sci 1982;17(5S):17S. [10] Kaiser F. Coherent oscillations in biological systems. Specific effectsin externally driven self-sustained oscillating bio-physical systems. Berlin Heidelberg: Springer-verlag; 1983. [11] Kaiser F, Eichwald C. Bifurcation structure of a driven multi-limit-cycle Van der Pol oscillator (I) the superhamonic resonance structure. Int J Bifurcation Chaos 1991;1(2):485. [12] Eichwald C, Kaiser F. Bifurcation structure of a driven multi-limit-cycle Van der Pol oscillator (II) symmetry-breaking crisis and intermittency. Int J of Bifurcation Chaos 1991;1(3):711. [13] Kaiser F. Theory of resonant effects of RF and MW energy. In: Gromdolfo M, Michaelson SM, Rindi A, editors. Biological effects of an dosimetry of nonionizing radiation. New york: Plenum Press; 1983. p. 251. [14] Kaiser F. The role of chaos in biological systems. In: Banet TW, Phl HA, editors. Energy transfer dynamics. Berlin: Springer; 1987. p. 224. [15] Nayfeh AH, Mook DT. Nonlinear oscillations. Newy york: John wiley and Sons; 1979. [16] Hayashi C. Nonlinear oscillations in physical systems. Newy york: McGraw-Hill Inc; 1964. [17] Yamapi R, Woafo P. Dynamics and synchronization of coupled self-sustained electromechanical devices. J Sound Vibr 2005;285:1151. [18] Yamapi R. Synchronization dynamics in a ring of four mutually inertia coupled self-sustained electrical systems. Physica A, in press.