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Vietnam Journal of Mechanics, VAST, Vol. 29, No. 3 (2007), pp. 207 - 219 Special Issue Dedicated to the Memory of Prof. Nguyen Van Dao VAN DER POL'S OSCILLATOR UNDER THE PARAMETRIC AND FORCED EXCITATIONS NGUYEN VAN DAO Vietnam National University, Hanoi NGUYEN VAN DINH, TRAN KIM Cm Vietnam Academy of Sciences, Hanoi (This paper has been published in: YKpa.iHChKHH MaTeMaT11YH11i1 >Kypnan 2007 , ToM 59, N° 2) Abstract. Van der Pol's oscillator under parametric an.d forced excitations is studied. The case where the system contains a small parametrer being quasilinear and the general case (without assumption on the smallness of nonlinear terms and perturbations) are studied. In the first case, equations of the first approximation are obtained by means of the Krylov-Bogoliubov-Mitropolskii technique, their averaging is performed, frequencyamplitude and resonance curves are studied, on the stability of the given system is considered. In the second case, the possibility of chaotic behavior in a deterministic system of oscillator type is shown. 1. INTRODUCTION It is well-known that there always exists an interaction of some kind between nonlinear oscillating systems. N. Minorsky stated that "Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interaction" [3]. Different interesting cases of interaction have been investigated by us and published in the monograph [3], using the effective asymptotic method of nonlinear mechanics created by Krylov N. M., Bogoliubov N. N. and Mitropolskii Yu. A. · The present paper introduces our research on the behaviour of a Van der Pol's oscillator under the parametric and forced excitations. The dynamic system urider consideration is described by an ordinary nonlinear differential equation of type (2.1 ). The section 1 is devoted to the case of small parameters. The amplitudes of nonlinear deterministic oscillations and their stability are studied. Analytical calculations in combination with a computer are used to obtain amplitude curves, which show a very complicated form in Figs. 1 - 4. In the section 2 we study the chaotic phenomenon occurring in the system described by equation (2.1) without assumption on the smallness of the parameters. As known, the fundamental characteristic of a chaotic system is its sensitivity to the initial conditions. The diagnostic tool used in this work is the Liapunov exponents. The positiveness of the largest Liapunov exponent will help us to define the values of parameters with which the chaotic motions are occurred. Chaotic attractors and associated power spectra will be presented. 208 Nguyen Van Dao, Nguyen Van Dinh, T'ran Kim Chi 2. THE CASE OF SMALL PARAMETERS In this section, let us consider the case when the parameters are small. The opposite case will be investigated in the next section. The smallness of parameters is characterized by introducing a small positive parameter E. For this case the asymptotic method of Krylov - Bogoliubov - Mitropolskii (K-B-M) [1, 2] is used to seek the approximate solutions aud to study their stability. 2.1. The differential equation of oscillation and its stationary solution The system under consideration is described by the equation x +w2 x = E {~x -1x 3 + (1- kx 2 ) h :i; + 2pxcos2wt + ecos(wl +CJ)} , (2.1) where h > 0 and k > 0 are coefficients characterizing the self-excitation of a pure Van der Pol's oscillator, 2p > 0 is the intensity of the parametric excitation, e > 0 is the intensity of the forced excitation and CJ, 0 ~ CJ ~ 27f is the phase shift between the parametric and forced excitations. Bellow, two subcases will be investigated separately for a weak parametric excitation when p 2 < h 2 and for a strong parametric excitation when p 2 > h 2 . The solution of (2.1) is found in the form x=acos'ljJ, e are where a and form x=-awsin'ljJ, 1/J=wt+e, (2.2) new variables, which satisfy the following equations in the standard E a= - - { ~x - 1x 3 + h(l - kx 2 )x + 2px cos 2wt + e cos(wt +CJ)} sin 'l/J, w iJ = _ ..!_ { ~x - 1x3 + h( 1 - kx 2 )x + 2px cos 2wt + e cos( wt + CJ)} cos 'ljJ . (2.3) · aw Following the K-B-M method, in the first approximation these equations can be replaced by averaged ones 2 E E a=--Jo=-- 2w { 2w hw (ka - - l ) a+pasinW+esin(e-CJ) } , 4 · E E{ ~a--1a 3 3 +pacosW+ecos(e-CJ) } . ae=--90=-2w The '.1-mplitude a and phase a = e = o: 2w (2.4) 4 e of stationary oscillations are determined from the equations Jo~ hw(k~' - 9o =~a - 41a I )a+ pa sin 3 28 + esin(O - a) c 0, (2.5) +pa cos 2e + e cos(e - CJ) = 0. These equations are equivalent to ~.+ ~1a 2 )asine + hw(k: - 1)acose - esinCJ = 0) 2 9 = Jo sine+ 9o cos e = hw ( k: - 1) a sine+ (P + ~ - ~1a 2 ) a cos e + e cos CJ = 0, J =Jocose - 9~sine = 2 (p- (2.6) or J = A sine+ B cos e - E = 0 , 9 = G sine + H cos e - K = 0 , (2.7) Van der Pol's oscillator under the parametric and forced excila lions where A= (p-.6.+~l'a 2 )a, 2 2 B=hw(k: -l)a, H = (p+.6.-~/'a 2 )a , G = hw(k: -l)a, 2.2. The amplitude-frequency relationship and E = esina , 209 (2.8) K = - ecos a . resonan~e curve The characteristic determinants of equations (2.7) are D-IA - G Bl_ H = a2 { (P + .6. - = D1 I E K = ae { D2 = B 2 hw(k: - l)a (p-.6.+~l'a 2 )a 2 hw(k: - l)a (p+.6. - ~/'a2) (p- .6. + ~/'a2) - h2w2(k: - 1 I hw(ka - 1 )a esina I{ = i =-ae{ I r} 2 -e cos a ( 4 3 ) p + .6. - ,,a 2 a (P + .6. - ~/'a 2 ) sin a+ hw ( k: (p- .6. + ~/'a 2 )a I~ ~~a 2 )a = k 4 2 - (2.9) 1) cos a} , esina 2 hw ( : - 1) a - e cos a (p-.6. + ~l'a 2 )cosa+hw(k: 2 - l)sina}. Below, in the (.6., a)-plane we identify the regular region in which the characteristic determinant D is nonzero and the critical region in which D is identically zero. I3y solving equations (2. 7) relatively to sin 0 and cos 0 and eliminating the phase 0 we obtain the amplitude- frequency relationship W(.6., a) = Dr+ D~ - D2 = 2 = a 2e 2 { (p + .6. - 43 ,,a 2) sin a+ hw (ka2 - 1) cos a } + 4 2 2 +a 2e2 { (p - .6. + ~/'a 2 ) cos a+ hw ( k: - 1) sin a} -a 4 { (p - .6. +4 3 /'a 2) (p + .6. - 3 /'a 2) - h2w2 ( 4 k~ 2 - 4 (2.10) 2 2 ) 1 } = 0. The regular part C1 of the resonance curve satisfies (2.10) and lies in t he regular region, where D "I- 0. The critical part C2 of the resonance curve lies in the critical region, where 2 2 2 2 2 2 2 = 0 or p - ( .6. a ) - h w ( = O, D and satisfies: ~/' k: 1) Nguyen Van Dao, Nguyen Van Dinh, Tran Kim Chi 210 + The compatibility conditions D1 =0 D2 = l1a + l1a or(p + 6.. - 0 or (P - 6.. 2 2 sin CT+ hw ( k: ) 2 ) cos CT+ hw ( 1) cos CT= 0, 2 k: 1) sin CT - = 0. + The trigonometrical conditions 2 A 2 +B 2 :;?: E 2 or a 2 3 2) 2 p - 6.. + :::(ya { ( +h 2 2 w ( ka 2 4 - 1) } :;?: e 2 sin 2 CT, (2.11) 2 G 2 + H 2 :;?: K 2 or a 2 {(p 3 2 )2 + h 2 w 2 + 6.. - :::(ya 4 - (ka2 1) } :;?: e 2 cos 2 CT. It is easy to see that the critical region is the resonance curve of Van der Pol's oscillator under the action of the parametric excitation without the forced excitation ( e = 0). For a weak parametric excitation (p 2 < h 2 ), the resonance curve is an oval encircling the-point Ao ( 6.. = l1a 2 , a 2 = a5 = ~) which is the representative point of the self-oscillation of Van der Pol's oscillator. This oval lies completely above the abscissa axis 6... When the parametric excitation is strong enough (p 2 > h 2 ), the critical oval enlarges and cuts the abscissa axis. From the compatibility conditions it follows that 2 . 3 6.. = pcos2CT + 2 1a , 4 ka hw -1 ) = -psin2CT. 4 ( (2 .12) Hence, the compatibility point has c9ordinates 6.. 3 2 2 2 2 { = 6..* = pcos2CT + 4,a*, a =a*= ao p sin 2CT 1 - hyfl + pcos 2CT } The existence condition of the compatibility point is p sin 2CT 2 a*> 0 or hy'l + p cos 20- < (2.1 3) 1. Obviously, this condition is satisfied if sin 2CT < 0 i.e. 7r 2 2 - 2vr=IJ (the case h 2 > 2 + 2vr=IJ is not considered here) then r < 0 and the trinomial A( cos 2o-) always has the same positive sign as its first coefficient, and the condition (2.15) is satisfied with all values of a- in the interval (2.14). If h 2 ~ 2 - 2vr=IJ, then r ~ 0 and the trinomial A(cos2o-) has either two ' simple roots or a double root. The simple roots cos 20- 1,2 are cos20-1,2 = It is noted that A(l) = h 2(1 s + p) > 2~2 (-ph = - cos 2a- ~ ± vr). (2.16) 0, A(-1) = h 2(1 - p), (p = O(c)) and the numerical 2 . ph p 2 satisfy -1 < 2 2 interval [-1, l]. The condition (2.15) leads to average of two roots: 2 cos 20-2 or s 2 < 1. cos 2o- Hence, two roots (2.16) lie in the ~ (2.17) cos 20-1. Combining (2.17) with (2.14) we obtain 0 ~ a- ~ 0-1 or 0-2 ~ a- ~ 7r 2, or 7r ~ a- ~ 7r + 0-1 or 7r + 0-2 ~ a- ~ 37f 2 · In summary, we have +If h 2 > 2 - 2~, then the compatibility point I* exists for every a- . + If h 2 ~ 2 - 2 J1 - p 2, then the compatibility point I* exists only for (2.18) In Fig. 1 the heavy arcs give the values a- with which the compatibility point I* exist s when h 2 ~ 2 - 2~. Since 2 - 2J1 - p 2 is approximately equal to p 2, then + If p 2 < h 2, i.e. when the parametric excitation is weak in comparison with selfexcitation, the critical oval D = 0 lies completely above the abscissa axis ~. The critical point I* always exists. + If p 2 ~ h 2, i.e. when the parametric excitation is strong enough, t he cri t ical oval D = 0 cuts the abscissa axis~. The critical point I * exists only wit h the values of a- lying in the interval (2.18). 212 Nguyen Van Dao, Nguyen Van Dinh, Tran Kim Chi Verifying the trigonometrical conditions by substituting (2.12) into (2.11) we obtain (2.19) Because the right-hand sides of (2.19) are not equal to zero simultaneously, from (2.19) we find e2 or a2 * >- ~ 4p2 . (2.20) Hence, the compatibility point I* is only a critical point when the amplitude a is large enough, i.e. when the forced excitation is still not too strong in comparison with the parametric one. 2.3. Forms of resonance curves To identify the forms of resonance curves we give in advance the values h k, then for each chosen value p we change e and O' to have the resonance curves. For example, with h = 0.1, k = 4, w = 1, the self-excited oscillation of Van der Pol's original systern has an amplitude a6 = 1 and is represented by the point Ao ( .6. = !1, a6 = 2 a) The case of weak parametric excitation (p < h 2 1). ) As it is known, in this case the critical oval D = 0, i.e. the resonance curve of Van der Pol's oscillator under the parametric excitation, runs around the point Ao and lies entirely above the abscissa axis .6.. We take p = 0.05, and O' = 0. For a weak forced excitation, i.e. when e is small enough, the condition (2.20) is satisfied and the critical point I* with 3 coordinates ( .6.* = p + 1, a; = a6 = 1) exists. For enough strong forced excitation, 4 i.e. when e is large enough, point I* is only a trivial compatibility point which does not belong to the resonance curve. In the Fig. 2 the resonance curves 'O', '1 ', '2', '3', '4 ', '5' correspond to the linear case l' = 0, fore= O; 0.015; 0.017; 0.050; 0.100; 0.120, respectively. The curve 'O' is a critical oval. The curve '1' has two branches: branch C' lies near abscissa axis, branch C" lies higher and consists of two cycles, one of them C~' is outside and the other C~ is inside the critical oval. These cycles are connected to one another at the critical node I* on the critical oval. Increasing the forced excitation (e), the lower branch C' moves up. The inner loop C~ of the upper branch is pressed while the outer loop Ci' is expanded, but both loops are tied at the node I*. Fore~ 0.0177, the lower branch C' is connected with the outer loop C~' at the node J and we have the curve '2', where J is a singular point belonging to the regular region DI- 0. Fore larger than 0.0177, the singular point J disappears. Then the lower branch and the outer loop are unified into one branch which lies outside the critical oval. We have the resonance curve '3'. Increasing e further, the inner loop C~ continues to be pressed into I* and disappears when e = 0.1 (see the resonance curve '4'). At this moment I* is a returning point. The curve '5' corresponds to a very large value of forced excitation; the point I* is a trivial compatibility point which lies outside the resonance curve and does not belong to it . Fig. 3a, Fig. 3b show the resonances curves in the nonlinear case 1x3 , I I- 0 with / = -0.l (a) and/= 0.1 (b). The curves 'l', '3', '5' in these figures have the same values of parameters (except 1) of the curves '1', '3', '5' in Fig. 2. 213 Van der Pol's oscillator under the parametric and forced excitations Fig. 2. Resonance curves for / = 0, e 0 (curve 0), e = 0.115 (curve 1) e = 0.0177 (curve 2), e = 0.050 (curve 3), e = 0.100 (curve 4), e = 0.120 (curve 5) (a) 2.0 ,' (b) 2.0 --- I I I 3--' I I 1.0 1.0 I I I ,'o I ,' \ \ ' 0.0 _j::...=...=..:...:.-=----~;=:=::::::::::::===-........:--:.:-:.:-:...=...: -0.1 -0.3 I ,' \ 0.1 -- 0.0 -0.2 0.0 0.2 Fig. 3. Resonance curves for/ = 0.1 (a), / = 0.1 (b), u = 0, e = 0 (curve 0), e = 0.015 (curve 1), e = 0.050 (curve 3), e = 0.12 (curve 5) With the negative value of/ (see Fig. 3a) resonance curves lean toward the left in comparison with the case / = 0 (Fig. 2). Otherwise, resonance curves lean toward the right for the positive value of/ (see Fig. 3b). This situation is common for nonlinear Duffing's systems. b) The case of strong parametric excitation (p 2 > h 2 ). As before we take h = 0.1, k = 4 but p = 0.12. In this case the critical oval 'O' is enlarged and cuts th~ abscissa axis D.. In Fig. 4a (r = 0) and Fig. 4b (r = 0.1), the resonance curves 'l' correspond to ~ = 0, e = 0.06. The curve 'l' has a cycle lying inside the critical oval and is connected with the outside branch by the critical point I*. If only e increases, the inside cycle is tied and then disappears. The critical point I* first becomes a returning point and then an isolated trivial compatibility point. The resonance curve is the only outside branch which is moving up. 1 214 Nguyen Van Dao, Nguyen Van Dinh, Tran Kim Chi -0.2 0.0 -0.2 0.2 0.0 0.2 Fig. 4. Resonance curves for/ = 0 (a), / = O.l(b), and u = O(curves 1), u = 7r 7r 7r (curves 2), u = "6 (curves 3), u = (curves 4) 12 4 Changing a, the critical point moves along the critical oval '0'. In Fig. 4a, Fig. 4b the 7r 7r ; G; resonance curves '2', '3', '4' correspond to the values/= O(a), / = O.l(b) and a= 12 7f respectively. We see that , when a increases the critical point moves down, the critical 4 point I* becomes a returning one and then disappears. Then the resonance curves separate into a cycle lying inside the oval 'O' and a branch lying outside this oval (curves '3', '4 '). 2.5. Stability conditions To have the stability condition we use the variational equations by putting in (2.4) a = a0 + <5a, fJ = ()0 + <5() and neglecting the terms of higher degrees with respect to c5a and <5() !!:_ (c5a) = _ _!__ (afo) <5a _ _!__ (afo) <5fJ, dt 2w aa 0 2w afJ 0 ao!!:_ (<5fJ) dt (2.21) (ago) c5a _ _!__ (ago) <5fJ, aa 0 2w afJ 0 where ao, fJo are stationary values of the amplitude a and phase fJ - the roots of equations = _ _!__ 2w (2.5). The characteristic equation of the system (2.21) is aop2+_!_ {ao (afo) +(ago) } p+~2 { (afo) ' (ago) _ (afo) (ago) } = 2w aa 0 afJ 0 4w aa 0 afJ 0 afJ 0 8a 0 Hence, the stability conditions for stationary solutions ao , fJo is 81 = ao ( C:::) +( 0 - {2pao sin Wo i;) = 0 ao { hw ( k:~ - ,1) + ~kw a6 + p sin 2()0 } + e sin( fJo - - 0 · (2.22) a)} > 0 . From the first equation of (2.5) we find sin Wo, then by substituting into (2.22) we get 81 = hwao(ka 02 - 2) 2 > 0 or a02 > k' (2.23) This condition means that only the oscillations with large amplitudes may be stable. Van der Pol's oscillator under the parametric and forced excitations 21 5 The second stability condition still has the abbreviated form [1, 4] 8W(6., a6) > 0 S 2 -- ao D 8a5 . (2.24) The curves D(6., a6) = 0 and W(6., a6) = 0 divide the plane P(6., a6) into regions. In each region the functions D(6., a6) and W(6., a6) have a determined sign. Moving upwards along a straight line, parallel to the ordinate axis and cutting the resonance curve at a point M, if we go from the region DW < 0 (> 0) to the region DW > 0 ( < 0) then point M corresponds to the stable (unstable) oscillation. Therefore, basing on the sign distribution of the functions D and W in the ?-plane, we can identify the stable and unstable branches of the resonance curve. a6 3. THE CASE OF ARBITRARY PARAMETERS. REGULAR AND CHAOTIC SOLUTIONS Les us go back to the equation (2.1), ignoring the assumption on the smallness of the parameters, i.e. we will consider the following differential equation x +w 2 x = 6.x -1x 3 + h(l - kx 2 )i: + 2pxcos2wt + ecos(wt +er). (3.1) We fix the parameters: w = 0.83, 6. = 0.01, / = 1, h = 1, k = 0.6, p = 0.001, er = 0 and use e as a control parameter. With different values of e, solutions of the equation (3.1) can be regular or chaotic. To identify the regular or chaotic character of a solution, we can use various methods, such as consideration of the sign of the largest Lyapunov exponent, or building the Poincare sections [4, 10]. To construct a Poincare section of an orbit , we 2 use the period T = 7f of the external excitation force. Then, the Poincare section acts w like a stroboscope, freezing the components of the motion commensurate with the period T. If we have a collection of n discrete points on the Poincare section, the corresponding motion is periodic with the period nT. For example, for e = 5.09, the Poincare section consists of three points (Fig. 5a), the motion is periodic with the period 3T; fore= 5.116, the Poincare section consists of six points (Fig. 5 b), the motion is periodic with the period 6T. When the Poincare section does not consist of finite number of discrete points, the motion is aperiodic, it may be chaotic (Fig. 6). x • •• (a) (b) 0 0 "1.5 .J.5 •• • • -3 x 1.5 Fig . 5. Poincare section: e •• -3 1.5 = 5.09 (a), e = 5.116 (b) Nguyen Van Dao, Nguyen Van Dinh, Tran I 300. Therefore, there is a positive Liapunov exponent associated with the chaotic orbit at e= 5.15. The evaluated largest Lyapunov exponent is,\::::::: 0.062 > 0 (Its rnlculation will be mentioned below). 0 Q) 0.5 Fig. 'l. Chaotic attractor and associated power spectra at e = .5.15