Chaos, Solitons and Fractals 23 (2005) 809–816 www.elsevier.com/locate/chaos

Active control with delay of catastrophic motion and horseshoes chaos in a single well Duffing oscillator B.R. Nana Nbendjo, Y. Salissou, P. Woafo

*

Laboratoire de M
ecanique, Facult
e des sciences, Universit
e de Yaound
e I, B.P 812 Yaound
e, Cameroon Accepted 5 May 2004

Abstract In this paper, the control of escape and Melnikov chaos of an harmonically excited particle from a catastrophic (unbounded) single well /4 potential is considered. In the linear limit, the range of the control gain parameter leading to good control is obtained and the effect of time delays on the control force is taken into account. The approximate critical external forcing amplitudes for catastrophe and chaos are obtained by using the energy and Melnikov methods. The control efficiency is found by analysing the behaviour of the external critical forcing amplitude of the controlled system as compared to that of the uncontrolled system. Ó 2004 Elsevier Ltd. All rights reserved.

1. Introduction The prediction of escape from a potential well is an interesting problem in physical and engineering sciences. It is studied in particular in naval architecture, civil and aerospace engineering (see [1–3]). For example in naval architecture, there is currently a flood of activities directed towards an improved understanding of the capsize of vessels [2]. In Ref. [1], the authors studied the prediction of escape from a potential well under harmonic excitation through a method based on the harmonic-balance formalism. They plotted the approximate critical forcing amplitude as a function of the forcing frequency and then compared the result to those obtained by numerical integration of the motion equation. In Ref. [3], the authors studied the stability of an active control of a linear system. In that study, the important effects of time delays on the control force have been taken into account and led to the conclusion that time delays are sources of instability. The study of Zhang et al. [3] was restricted to linear dynamics. However, in many situations, the nonlinear dynamics dominates the behaviour of physical systems giving rise to multi-stable potentials or catastrophic monostable potentials. In this paper, we derive the conditions for the control of escape from the potential well and Melnikov chaos in a system whose dynamics is that of a particle moving in a catastrophic single well Duffing potential. Section 2 presents the model and gives the results of the stability analysis. Section 3 considers the control of the catastrophic escape from the potential well and the effects of time delays on the onset of Melnikov chaos. We conclude in Section 4.

*

Corresponding author. E-mail address: [email protected] (P. Woafo).

0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.05.041

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B.R. Nana Nbendjo et al. / Chaos, Solitons and Fractals 23 (2005) 809–816

2. Model and stability 2.1. The model We consider a single-degree-of-freedom model with a nonlinear soft spring, a linear damper and a linear displacement feedback control system. The model is described by the following differential equations system: €x þ k_x þ x20 x þ cx3
bz ¼ F0 cos Xt;

ð1aÞ

z_ þ az ¼ C1 xðt
tx Þ þ C2 x_ ðt
tx_ Þ;

ð1bÞ

where x, x_ and €x are displacement, velocity and acceleration, respectively; x0 and k are, respectively, the natural frequency and the damping coefficient of the structure, c is the nonlinear parameter of the potential, assumed negative in this paper; F0 and X stand, respectively, for the amplitude and the frequency of the external forcing; z is the control force variable; b, C1 , and C2 are the control gain parameters; a the control speed parameter while tx and tx_ are, respectively, the time delays for the displacement and the velocity feedback force in the system. In equation (1b), x and x_ qffiffiffiffiffiffi2ffi are, respectively, written as function of t
tx and t
tx_ .
x The system without control has a potential with the stable equilibrium point x ¼ 0 and two unstable points  c 0 (see Fig. 1). Therefore, depending on the values of the external excitation, the system can escape over the potential barrier and dramatically suffers an unbounded motion. Thus, we need to apply a control strategy to increase the threshold values of F0 leading to catastrophic motion. But before we consider such a problem, let us analyse the stability of the controlled system. Throughout the paper, we use the following values for the system parameters x0 ¼ 1, c ¼
0:5, k ¼ 0:04, X ¼ 1 and b ¼
1. 2.2. Stability of the control system The autonomous system is obtained for F0 ¼ 0. In this case, equation (1) becomes in the linear limit: 8 < x_ ðtÞ ¼ vðtÞ; v_ ðtÞ ¼
kvðtÞ
x20 xðtÞ þ bzðtÞ; : z_ ðtÞ ¼ C2 vðt
tx_ Þ þ C1 xðt
tx Þ
azðtÞ:

ð2Þ

To study the stability of this system, we apply the Lyapunov concept by examining the fundamental solution est . The characteristic equation is given by: s3 þ ða þ kÞs2 þ ðak þ x20
bC2 e
stx_ Þs þ ax20
bC1 e
stx ¼ 0:

ð3Þ

To obtain the stability boundary in the plane of the control gain parameter (C1 , C2 ), we use the D-subdivision method (see Ref. [3]). According to that method, the stability boundary in the plane ðC1 ; C2 Þ are determined by the points that yield either to a root s ¼ 0 or a pair of pure imaginary roots of Eq. (3).

0.5 0.45 0.4 0.35

V(x)

0.3 0.25 0.2 0.15 0.1 0.05 0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

x Fig. 1. The catastrophic single well U4 potential.

2

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811

Substituting s ¼ 0 into equation (3), one finds: C1 ¼

ax20 : b

ð4Þ

Setting s ¼ ib (where b is a real constant) into the characteristic equation (3), one finds the system of equations  ðb cos btx ÞC1 þ ðbb sin btx_ ÞC2 ¼
ða þ kÞb2 þ ax20 ð5Þ ðb sin btx ÞC1
ðbb cos btx_ ÞC2 ¼ b3
ðak þ x20 Þb which leads to C1 ¼

D1 D

ð6Þ

C2 ¼

D2 D

ð7Þ

with    b cos btx bb sin btx_   D ¼  b sin btx
bb cos btx_    
ða þ kÞb2 þ ax20 bb sin btx_    D1 ¼  3 b
ðak þ x20 Þb
bb cos btx_     b cos btx
ða þ kÞb2 þ ax20   D2 ¼  b sin btx b3
ðak þ x20 Þb 

ð8Þ ð9Þ ð10Þ

The stability boundary in the plane ðC1 ; C2 Þ can be found from the bifurcation curve defined by the parametric equations (6) and (7), and the bifurcation line defined by Eq. (4). Setting tx ¼ tx_ ¼ t0 , the stability boundaries are plotted in Fig. 2. They consist of the straight line (Eq. (4)) and the curve associated with each time delay. The stability area is the region of the plane limited by Eqs. (4), (6) and (7) (for instance for t0 ¼ 0:6, the stability area is shaded in the figure). Thus as the time delay increases, the stability area decreases.

3. Control of escape and Melnikov chaos 3.1. Control of the catastrophic escape The total energy of the system is given by: 1 1 1 E ¼ x_ 2 þ x20 x2 þ cx4 : 2 2 4

Fig. 2. Stability boundary in the space ðC1 ; C2 Þ for a ¼ 2 and various values of time delay.

ð11Þ

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The potential barrier (the potential energy at the nearest unstable equilibrium point) is given by V0 ¼


1 x40 : 4 C

ð12Þ

Using the harmonic-balance method, we assume that the displacement of the system has the form: x ¼ A cosðXt þ uÞ

ð13Þ

which has the same frequency as the external excitation. From Eqs. (11) and (13), the maximum energy of the system is given by: 1 1 ðx2
X2 Þ2 : Emax ¼ A2 X2
2 4C 0

ð14Þ

According to Virgin et al. [1], the best criterion to obtain the escape from a potential well is the condition Emax P V0 instead of the condition that stipulates that the maximum displacement should reach the nearest unstable equilibrium state. Therefore, the critical load characteristics is obtained for Emax ¼ V0 . Substituting Eq. (13) into the generating equations (1), the amplitude A satisfies the following nonlinear algebraic equation. 9c2 6 3c 2 A þ ðx0
X2
bDÞA4 þ ½ðx20
X2
bDÞ2 þ ðkX þ bEÞ2 A2 ¼ F02 ; 2 16

ð15Þ

where D ¼ A1 cos Xtx þ A2 cos Xtx_
B1 sin Xtx
B2 sin Xtx_ ; E ¼ A1 sin Xtx þ A2 sin Xtx_ þ B1 cos Xtx þ B2 cos Xtx_ ; aC1 A1 ¼ 2 ; a þ X2

X2 C2 A2 ¼ 2 ; a þ X2

XC1 B1 ¼ 2 ; a þ X2

ð16Þ

aXC2 B2 ¼
2 : a þ X2

Taking into consideration the fact that the threshold limit is obtained where Emax ¼ V0 , one finds that the critical force leading the system to catastrophic motion satisfies the equation " # 2 1 2 2x20
5X2
8bD 2 2 2 þ ðkX þ bEÞ : ð17Þ F0c ¼ ðX
2x0 Þ 2c 8 Fig. 3 shows the variations of F0c versus C2 . The evolution of F0c versus C1 is qualitatively the same. Thus, as the control gain parameters increase, the critical force for escape from the potential well increases. Numerical simulation has been done by integrating directly Eqs. (1) using the fourth order Runge Kutta method. The corresponding results are also represented in Fig. 3 (dotted line) and show a fairly good agreement with the analytical results (thin line). To deal with the effects of time delays on the escape, we have plotted in Fig. 4 the variation of the critical force as function of time delays for C1 ¼ 0:5 and C2 ¼ 0, assuming tx ¼ tx_ ¼ t0 . The horizontal line represents the results in the case where there is no delay. We find that for a certain choice of t0 , F0c is greater than that of the case without delay. Thus there are regions

3 2.5

Foc

2 1.5 1 0.5 0

0

1

2

L

3

4

5

2

Fig. 3. Escape boundary in the ðF0c ; C2 Þ plane for a ¼ 2.

B.R. Nana Nbendjo et al. / Chaos, Solitons and Fractals 23 (2005) 809–816

813

0.7 0.6 0.5

Foc

0.4 0.3 0.2 0.1 0 0

2

4

6

8

10

12

14

16

18

20

to Fig. 4. Evolution of the critical force for escape as a function of time delay with C1 ¼ 0:5, C2 ¼ 0:5 and a ¼ 2.

where time delays affect negatively F0c (smaller than that obtained when t0 ¼ 0) and others where time delays render the control from catastrophic escape more efficient. 3.2. Control of the Melnikov chaos In Ref. [4], Cheng et al. used the Melnikov theory to derive the condition for the occurrence of horseshoe chaos in a monostable Duffing oscillator subjected to an active control without delay. In Ref. [5], the authors derived the condition for the appearance of chaotic escape in a double well Duffing oscillator subjected to an active control when delay is taken into account. In fact, the Melnikov theory helps us to define the conditions for the existence of the so-called transverse intersection points between perturbed and unperturbed separatrices or the appearance of the fractality on the basin of attraction. One would know how the time delays affect the control strategy or in what range of the control gain parameters the heteroclinic chaos in our model could be inhibited. To deal with such a question, let us express the dynamical system in the form U_ ¼ F ðU Þ þ eGðU ; tÞ;

ð18Þ

where in the state vector notation U ¼ ðx; v ¼ x_ Þ, F ¼ ðv;
x20 x
cx3 Þ and G ¼ ð0;
kv þ bz þ F0 cosðXtÞÞ. The control force z can be obtained by integrating Eq. (1b). The unperturbed Hamiltonian system (defined by the flow U_ ¼ F ðU Þ) possesses an heteroclinic orbit connecting the two unstable points of the potential. This orbit is given by rffiffiffiffiffiffiffiffiffi  
x20 x0 x0 ¼  tanh pffiffiffi t ; c 2 ð19Þ   x20 x 0 2 v0 ¼  pffiffiffiffiffiffiffiffiffi sec h pffiffiffi t :
2c 2 The corresponding additional excitation to Hamiltonian system due to the control is bz0 where z0 is obtained from Eq. (1b) as rffiffiffiffiffiffiffiffiffi   Z t Z t
x20 x0 C2 x20 x0 ffi expð
atÞ expð
atÞ expðasÞ tanh pffiffiffi ðs
tx Þ ds þ pffiffiffiffiffiffiffiffi expðasÞ sec h2 pffiffiffi ðs
tx_ Þds: ð20Þ z0 ¼ C 1 c
2c 2 2
1
1 The Melnikov function [6] is defined by Z þ1 MðsÞ ¼ F ðU ðtÞÞ  GðU0 ðtÞ; t þ sÞ dt;

ð21Þ


1

where s is a phase angle. Carrying out the integration, we find that pffiffiffi pffiffiffi 2 2kx30 bx30 C1 bC2 x40 2XpF0   cosðXsÞ
pffiffiffi k1 ða; tx Þ
k2 ða; tx_ Þ þ pffiffiffiffiffiffi MðsÞ ¼ 3c 2c 2c
c sinh pffiffiX 2 x0

ð22Þ

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B.R. Nana Nbendjo et al. / Chaos, Solitons and Fractals 23 (2005) 809–816

with k1 ða; tx Þ ¼

Z

þ1


1 þ1

k2 ða; tx_ Þ ¼

Z


1

    Z t  x0 x0 expð
atÞ sec h2 pffiffiffi t expðasÞ tanh pffiffiffi ðs
tx Þ ds dt; 2 2
1     Z t  x0 x0 2 2 expð
atÞ sec h pffiffiffi t expðasÞ sec h pffiffiffi ðs
tx_ Þ ds dt: 2 2
1

ð23Þ

Let us first restrict the analysis to the case tx ¼ tx_ ¼ t0 ¼ 0 and a ¼ px0ffiffi2 (for mathematical constraints, the integrals in Eqs. (20) and (23) can be integrated analytically only for a ¼ px0ffiffi2). Then Eq. (20) becomes rffiffiffiffiffiffiffiffiffi C1
x20 C2 x20 ffi ðtanhðatÞ þ 2e
at arctanðeat Þ
1Þ ð24Þ ð1
2e
at arctanðeat ÞÞ  pffiffiffiffiffiffiffiffi z0 ¼  a c a
2c and Eq. (22) gives pffiffiffi pffiffiffi     2 2 kx30 bx30 2 p2 F0 x20 pX pX p ffiffiffiffiffiffiffiffi ffi
ðC þ
aC Þ 2
cos ech MðsÞ ¼ : 1 2 3 a2 c c 4 2a a2
2c

ð25Þ

Using the Melnikov criterion for the appearance of the intersection between the perturbed and unperturbed separatrices [6], it is found that chaos is suppressed when the following condition C2 P

C1 2ka
K þ  a 3 2
p42 b

is satisfied. The quantity K is given by   pffiffiffiffiffiffi F0 pX
c pX  : cos ech K¼ 2 2a 2bax0 2
p4

ð26Þ

ð27Þ

We have plotted in Fig. 5 for F0 ¼ 0:15 and X ¼ 1 the domain in the space parameters ðC2 ; C1 Þ where chaos is suppressed. It corresponds to the shaded region. To complement and validate the analytical predictions, we have simulated numerically the set of Eq. (1) to look for the effects of the control parameters and time delays on the onset of the fractality in the basins of attraction. For the system without control and with the set of parameters used, Fig. 6 shows that the fractality first appears at F0 ¼ 0:15. Considering now the case where the control is applied and taking the value of the control gain parameters in the domain where the control is efficient (for instance ðC1; ; C2 Þ ¼ ð
0:3; 0:3Þ and ðC1; ; C2 Þ ¼ ð
0:5; 0:4Þ), we find that the fractality is no more for F0 ¼ 0:15 (see Fig. 7a). When the control gain parameters are in the domain where the control is not efficient (for instance ðC1; ; C2 Þ ¼ ð
0:1;
0:1Þ and ðC1; ; C2 Þ ¼ ð0:1; 0:1Þ), we observe that fractal structures exist (see Fig. 7b). Figs. 6 and 7 are obtained when the time delays are not taken into account. When they are considered, we integrate numerically K1 ða; t0 Þ and K2 ða; t0 Þ. Fig. 8 shows the variation of the critical force F0c as a function of time delay tx ¼ tx_ ¼ t0 with the control gain parameters taken in the domain where the control is efficient in the absence of time delays (for instance C1 ¼ 0:1, C2 ¼ 1). The critical force above which chaos is observed is derived as

Fig. 5. Region in the space ðC1 ; C2 Þ for the control of chaos with a ¼ 2 and F0 ¼ 0:15. Horeshoes chaos does not exist in the shade region.

B.R. Nana Nbendjo et al. / Chaos, Solitons and Fractals 23 (2005) 809–816

815

Fig. 6. A fractal basin boundary diagram for the uncontrolled system with F0 ¼ 0:15.

Fig. 7. (a). Basins of attraction for the case where the control is efficient with a ¼ p1ffiffi2 and F0 ¼ 0:15. (b). Basins of attraction for the case where the control is inefficient with a ¼ p1ffiffi2 and F0 ¼ 0:15.

  pffiffiffiffiffiffi
c sinh pffiffi2Xx 0 pffiffiffi F0c  2Xp

! pffiffiffi 2 2kx30 bx30 C1 bC2 x40
þ pffiffiffi k1 ða; t0 Þ þ k2 ða; t0 Þ : 3c 2c 2c

ð28Þ

It decreases when the time delay increases. This means that in the presence of time delays chaos appears for smaller values of F0 as compared to the system without delay.

816

B.R. Nana Nbendjo et al. / Chaos, Solitons and Fractals 23 (2005) 809–816 6 5 4

Foc

3 2 1 0 0

1

2

3

to

4

5

6

7

8

Fig. 8. Evolution of the critical force for chaos as a function of time delay with C1 ¼ 0:15, C2 ¼ 1 and a ¼ p1ffiffi2.

4. Conclusion In this paper, we have analysed the question of the control of the escape and horseshoes chaos of an harmonically excited particle in a catastrophic (unbounded) single well U4 potential. The stability of the system under control has been studied using the Lyapunov concept and D-subdivision method. The approximate critical force leading to catastrophic motions has been obtained analytically and verified numerically. The domain in the control gain parameters space where the Melnikov chaos is suppressed has also been derived. The effects of time delays on the critical forces for catastrophe and Melnikov chaos appear to be important and should be taken into account when designing control devices.

References [1] Virgin NL, Plaut RH, Cheng CC. Prediction of escape from a potential well under harmonic excitation. Int J Nonlinear Mech 1992;27:357–65. [2] Thompson JMT. Chaotic phenomena triggering the escape from a potential well. Proc Soc Lond A 1989;421:195–225. [3] Zhang L, Yang CY, Chajes MJ, Cheng AHD. Stability of active-tendon structural control with time delay. J Eng Mech Div ASCE 1993;119:1017–24. [4] Cheng AHD, Yang CY, Hackl K, Chajes MJ. Stability bifurcation and chaos of non linear structures with control-II. Int J Nonlinear Mech 1993;28:549–65. [5] Nana Nbendjo BR, Tchoukuegno R, Woafo P. Active control with delay of vibration and chaos in a double-well Duffing oscillator. Chaos, Solitons & Fractals 2003;18:345–53. [6] Melnikov VK. On the stability of the center for some periodic perturbations. Trans Moscow Math Soc 1963;12:1–57.