Grasman, J. (Johan)

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Grasman, Johan

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On the van der pol relaxation oscillator with a sinusoidal forcing term

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stichting mathematisch centrum AFDELING TOEGEPASTE WISKUNDE (DEPARTMENT OF APPLIED MATHEMATICS) J. c;RASMAN ON THE VAN DER POL RELAXATION OSCILLATOR WITH A SINUSOIDAL FORCING TERM TW 207/80 ~ MC SEPTEMBER kruislaan 413 1098 SJ amsterdam BIBLIOTHf:EK MATHEM,-\TISCH GE'NTRUM -AMSTER~AM-PJu.nted at :the Mathemati.c.ai. Cen.t!te, 413 K.IU.Ll6laan, Am-6:teJr.dam. The Ma:thema.tic.ai. CentJr.e, ·6ou.nded :the 11-:th 06 FebJz.u.aJc.y 1946, .u, a. non-p1t.06U w.tUu;Uon al.mbig a:t :the pit.Omo.ti.on 06 pWLe. mathema-ti.C-6 a.nd -it6 a.ppU.c.a,t.i.on6. 1:t .u, -6pon601r.ed by :the Ne:thelli.a.nd6 GoveJr.nment :thlt.Ough :the Ne:the!r.la.nd6 01t.ga.nl.za-ti.on 601t. :the Adva.nc.ement 06 PUJr.e Ruea.1r.c.h (Z.W.O.). 1980 Math~mati,.c$ s'Ul;>j~ct cla$sification: 35B25, 35J25. On the Van der Pol relaxation oscillator with a sinusoidal forcing term by J. Grasman ABSTRACT Asymptotic approximations of subharmonic solutions of the periodically forced Van der Pol relaxation oscillator are constructed with singular per-turbation techniques. These approximations are locally valid and may take the form of a two variable expansion in one region and a boundary layer type of solution in a next region. Integration constants are determined by averag-ing and matching conditions. The construction of the approximations brings about certain restricting conditions on the amplitude of the forcing term. KEY WORDS & PHRASES: Van der Pol equation, relaxation oscillation, subhar-monic entrainment, singular perturbation 1 1. INTRODUCTION In this paper we study the Van der Pol equation with a sinusoidal forc-ing term {1.1) d2 2 dx ~ + v{x -1)- + x = {av+S)cost, dt2 dt for large values of the parameter v and with O <a< 2/3. Using singular perturbation techniques we construct a formal asymptotic approximation of the 2~{2n-1)-periodic solution with n = O{v). In the process of constructing such approximation we arrive upon a set of conditions for a, f3 and v. These condi-tions are such that for a given a, the parameter f3 lies on an interval (1. 2) f3 {a) < f3 < f3 {a) • -n n These intervals overlap, so that for f3 on the interval {8 {a),8 1 {a)) two -n n+ solutions with period T = 2~{2n±1) may coexist. In earlier studies [3,4 and 5] asymptotic solutions for the cases a= 0 and a= 2/3 have been constructed. In our analysis of the present problem we see that in the asymptotic solution elements of both cases can be.distinguished. A periodic solution of {1.1) has a behaviour that is characteristic for singularly perturbed type of problems. Locally the solution has a boundary layer type behaviour like one meets in problems of fluid mechanics. On the other hand the solution also passes a large time interval, where a two time-scales expansion can be made. Finally, we distinguish a sequence of points, determined by the intersections with the lines x = ±1, where the local behaviour of the solution is analyzed by a stretching procedure in both the dependent and independent variable. For a complete picture of the different regions which are successively passed through by the solution we refer to Fig. 1.1. Integration constants in local-ly valid asymptotic solutions are determined by averaging conditions and by matching pairs of local solutions of adjacent regions. 2 2 I A" X A , -, ( )~ ~ ~- -~ ,~ , I I I I I :c I I I I I 0~ t, t, t t-Fig. 1.1. Regions with a local asymp~otic solution 2. ASYMPTOTIC SOLUTION FOR REGION A In this region the solution exhibits an oscillatory behaviour of period 2n in the regular time scale and, at the same time, a slow decrease of its average value. In order to analyze the solution asymptotically we introduce an additional time variable T = (t-t0-n)/v with t 0 = n/2(mod) 2n and consi-der the following two-variable expansion for the solution (see COLE [1]). (2.1) -1 -2 x = xo(t,T) + v xl (t,T) + v x2(t,T) + .••. Substituting (2.1) into (1.1) and equating the terms with equal powers of v we obtain a recurrent system of differential equations for x. (t,T). The first 1 equation reads (2.2) or 3 (2. 3) having a solution of the form (2. 4) 1 3 3 x 0 = 2cos[ 3 arccos{2 a sint + 2 c0 (T)}]. The second equation of the iterative scheme becomes (2.5) or, after integration with respect tot and with the use of (2.3), t (2.6) I - -Go(t,T)dt + 8 sint + cl (T) with (2. 7) The integral of G0 in the right-hand side of equation (2.6) is secular in the sense that for (t-t0-i) + 00 this term would increase in order of mag-nitude so th.at (2 .1) would not hold for a large time interval. This secular-ity is banished by choosing the constant of integration int such that on the average over a 2'IT-interval G0 disappears: (2. 8) or (2. 9a) TV+2'IT f G0 (t,T)dt = 0 TV TV+2'IT I x0 (t,T)dt. TV Since at time t 0 + 'IT the solution starts at the value x = 2, c0 satisfies the initial value 4 (2.9b) cOco) = 2/3 - a. The solution will leave the region A at a time tm = t 0 + 2mn as it ap-proaches the line x = 1, which occurs when c0 reaches the value -2/3+a. From (2.9) it follows that in the slow time scale this will be for -2/3+a 27f (2.10) T(a) = 27f I { I 2/3-a 0 Finally, the third equation reads (2.11) 2 2 cl x 0 cl x 1 2 -'-- + -- + ata, at2 so according to (2.6) we have -3x0 ax 1. t t oG0 = _ 2 f { f 2 .(2.11) (xo-1) x2 = --- - -- + - dt-G }dt - X1XO oT clt OT 1 to+1r to+1r with (2 .12) 2 The term 3G0/3T satisfies the averaging condition and behaves as a+b/(x0-1), therefore its integral is not secular. The averaging condition (2.13) TV+27f I G1 (t,T)dt = 0, TV yields a linear differential equation for c1 : (2.14) TV+27f ac1 + S:_ f OT 21r TV 1 --at 2 X -1 0 TV+27f -8 =-27f I TV s;nt dt. X -1 0 Let c 1 (0) = c 10 , then we have T TV+2'1r = exp{;; f 0 f TV 1 -- 2- dtdT} X -1 0 T T TV+2'1r - JL J exp{-1 J J 2'11" 2'11" 0 0 TV 1 -- 2- dtdT} X -1 0 and so (2.15a) c 1 (T (a)) = q (a){c10 - Sp (a)}, TV+2'1r J TV sint dtdT] 2 X -1 0 T Ca) T exp{211r J 0 TV+2'1r TV+21r (2.15b) p (a) 1 =-2'11" J 0 T(a) TV+21r (2. 15c) q (a) = exp{;; f f 0 TV I + dtdT} f - xo-1 TV TV 1 - 2- dtdT}. x -1 0 s;nt dtdT, X -1 0 Ast approaches the value t 0 ·+ T(a)v each time fort= t = t 0 + 2nm the 5 -1/2 m solution gets closer to the Line x = 1. In av -neighbourhood of (x,t) = (1,tm) x0 would behave as (2. 16a) and x 1 as (2. 16b) -a(t-t) m X = 1 4v2 T = T(a). -1 Clearly, the asymptotic expansion is not valid anymore, as v x 1 increases in order of magnitude. Just before the solution enters such regions we have that asymptotically 1 ~ -1/2 -1 1/2 r:. v Ri -- v2a(t-t )v - C' (T) (t -t -1r-Tv) (t-t ) v /v2a 2 m O m O m ' and so 6 (2.17) 3. ASYMPTOTIC SOLUTION FOR REGION A m Let us assume for a moment that the solution has passed a v- 112 neigh-bourhood of a point (x,t) = (1,t 1) and returns to the interval (1,2). Since m-the two variable expansion (2.1) is not valid anymore and the solution again approaches the line x = 1 at time t = t, we introduce for the time interval m t 1 < t < t the expansion m- m (3 .1) -1 x(t;v) = xm0 (t) + v xm(t) + •••• Similar to (2.1) we obtain a recurrent system of differential equations for x. (t), i = 1,2, •••• The first two equations read mi (3.2a) 2 dxmO (xm0-1) ~=a cost, (3. 2b) 2 d xmO = - - xmO + 8 cost. dt2 Integration yields ( 3. 3a) (3. 3b) = a cint + C {m) 0 dxmO =----dt t I tm-1 Since fort= tm x approaches the value 1, we have C~m) = (3. 4) 1 3 3 = 2 cos{3 arccos(2 a sint + 2 a - 1)}. Ast approaches t from below {3.1) behaves asymptotically as m {3. Sa) X ~ 1 - -21 v'2ti {t-t) + V-lK /{t-t ), m m m so that (3 .Sb) (3. Sc) t m-1 (-C(m) +$+I )/ili, 1 n xm0 (t)dt. Consequently, also the expansion ( 3. 1) looses its validity as t + t • m 4. ASYMPTOTIC SOLUTION FOR REGION B m 7 -1/2 We investigate the local behaviour of the solution in av -neighbour-hood of (x,t) = (1,t) by introducing the transformations m (4.1ab) X = 1 + V (~)v-112 , m Substituting (4.1) into (1.1) and multiplying this equation with v-112 we obtain, after taking the limit v + m, (4. 2) The function vmO(~) expresses the local limit behaviour of the solution for v + m. In order to match the-solution of region A it must satisfy m (4.3) see (3.5). The function (4. 4) vm0 (~) = -aD~ (-a~)/DK (-a~), m m satisfies (4.2) as well as (4.3). In (4.4) Dµ(z) denotes the so-called para-bolic cylinder function of orderµ (see WHITTAKER and WATSON [9, p. 347]) with (4.5) 1 2 µ µ(µ-1) D (z) = exp(--4 z )z {1 - -'--'"'-"-- + ••• } µ 2z2 for z + m. Assuming that Km~ 0 the function vm0 (~) will be regular for 8 finite~, while for~ ➔ 00 (4.6) as ( 4. 7) for z ➔ -oo by (4.8) D (z) µ exp (-¾ z 2 ) z µ { 1 - µ ( µ-21) + ... } 2z h1r 1 2 -µ-1 (µ+1) (µ+2) r(-µ) exp(4 z +µ1ri)z {1 + 2z2 + ... } On the other hand, at region A 1 , the solution is approximated m+ as t t t • Thus, the local solution (4. 4) of B matches the local solution of m m A l if m+ ( 4. 9) Using (3.Sb) we find that ( 4. 10) c(m+1) = 1 C (m) - I • 1 n Let for some m, say m = n, (4.11) K l :5: 0 < K :5: I /fia. n- n n Then the parabolic cylinder function ~(-a~) vanishes for certain value(s) of the argument. Let ~ = ~O be the lowest zero. For ~ t ~O we have that ( 4 .12) so vmO becomes singular and the solution rapidly decreases as~ approaches ~o· 9 5. ASYMPTOTIC SOLUTION FOR REGION C At this point the solution enters the boundary layer region C with local coordinate (5 .1) Assuming that. the solution can be expanded as (5. 2) X we arrive at a recurrent system of equation for the coefficients W.: l (5. 3a) (5. 3b) ') d'-w0 -2+ dn 2 d w1 -2+ dn According to (4.12), (5.2) matches the solution for region B if for n + - 00 n (5. 4ab) w0 :::::: 1 + 1/n, Condition (5.4a) is satisfied by the class of solutions (5 .Sa) while because of (5.4b) the integrated equation (5.3b) must have the form (5. Sb) aw 1 --+ dn As n ➔ 00 the solution leaves the boundary layer region at exponential rate and fort - tn - s0v-l/2 small, but independent of v, (5.2) behaves asympto-tically as (5. 6) X = 10 6. CONDITIONS Jli'OR 21T ( 2n-1) -PERIODIC SOLUTIONS As the solutions of period T = 21r(2n-1), we are looking for, are sym-metric, that is (6. 1) x(t) 1 = -x (t - 2 T), we have completed the local approximations. Transposing (5.6) to the comple--1/2 mentary phase and substituting t = t 0 + 1r + ~0v in (2.1), we obtain the periodicity condition (6. 2) = 2 1 2 1 2 2 1 -1 - 3 a (4 a ~O - Kn - 2) v or (6. 3) From (4.9) and (4.10) it follows 0 that (6. 4) = S - c(m) + (n-m+l)I . 1 n Comparing (2.17) with (3.5) we see that for n-m-+ 00 , but at the rate such that n-m = o(v), there exists a matching relation between c 1 (T) and Cim): (6. 5) Substituting (6.6) into (6.4), while using (2.15), we obtain (6. 7) -1 (21r) I Tv. n From (6.3) and (6.7) we derive (6. 8a) (6.8b) (6. 8c) S = {&(Kn+½) (l+q (a)) - Gn (a) }/S (a), G (a) n s (a) = 1 + q(a) - p(a)q(a). 11 Since K may range from Oto I/&', a periodic solution of period 2TI(2n-1) n n is possible for (6. 9) < { (21 -12ci'+ I ) (l+q(a)) - G (a) }/S(a). n n 7. CONCLUDING REMARKS The formal asymptotic analysis of the Van der Pol relaxation oscillator with a periodic forcing term as we presented in this report forms the last ~art of a series of studies on this problem, see [3, 4 and 5]. In [3] and [5] solutions of period 2TI(2n~1) were found under given conditions for S for the special cases a= 0 and a= 2/3. If for a= 2/3 we consider the range of S given by (6.9) we observe that this special case, studied in [5], is com-pletely covered by the results of this report. As I (2/3) = 613, T(2/3) = n p(2/3) = 0 and q(2/3) = 0, the conditions on s213 read (7.1) For a= 0 we have I (0) = 2TI, T(0) = 3/2 - log 2, p(0) = 0 and q(0) = 1/2, n so (7.2) (3 - 2 log 2)v - 2TI(2n-1) < 3130 < (3 - 2 log 2)v - 2TI(2n-2), which for s0 + 00 matches the conditions on this parameter given by inequali-ties (21) and (22) of [3]. 12 It is our intention to write a final report in which we compare the asymptotic conditions on a, /3 and v with numerical results on this problem, see FLAHERTY and HOPPENSTEADT [2]. For that purpose some integrals such as I (a) and T(a) need to be evaluated by numerical integration or (if possible) n by an analytical expression. Moreover, we will attempt to relate our formal asymptotic results with the outcome of topological-analytical work by GUCKENHEIMER [6] and LEVI [7]. REFERENCES [1] COLE, J.D., Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, Mass., 1968. [2] FLAHERTY, J.E. & F.C. HOPPENSTEADT, Frequency entrainment of a forced Van der Pol oscillator, Studies in Applied Math. 18 (1978), p. 5-15. [3] GRASMAN, J., E.J.M. VELING & G.M. WILLEMS, Relaxation oscillations governed by a Van der Pol equation with periodic forcing term, SIAM J. Appl. Math. l!_ (1976), p. 667-676. [4] GRASMAN, J., M.J.W. JANSEN & E.J.M. VELING, Asymptotic methods for relaxation oscillations, in: Proceedings of the Third Scheveningen Conference on Differential Equations, W. Eckhaus and E.M. de Jager (eds.), North Holland Math. Studies l!_ (1978), 93-111. [5] GRASMAN, J., Relaxation oscillations of a Van der Pol equation with large critical forcing term, Quart. of Appl. Math. 38 (1980) p. 9-16. [6] GUCKENHEIMER, J., Symbolic dynamics and relaxation oscillations, to appear in Physica D: Nonlinear phenomena. [7] LEVI, M., Periodically forced relaxation oscillations, preprint. [8] LITTLEWOOD, J.E., On nonlinear differential equation of the second order, III, Acta Math. 2].__ (1957), p. 267-308. [9] WHITTAKER, E.T. & G.N. WATSON, A course of modern analysis, Cambridge, Cambridge University Press , ( 19 35) • 

On the van der pol relaxation oscillator with a sinusoidal forcing term

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