Grasman, J. (Johan)

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Grasman, Johan

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Relaxation oscillations of a van der pol equation with large critical forcing term : (preprint)

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stichting mathematisch centrum AFDEL I NG TOEGEPASTE WI SKUNDE (DEPARTMENT OF APPLIED MATHEMATICS) J. GRASMAN TW 173/78 ~ MC JANUARl RELAXATION OSCILLATIONS OF A VAN DER POL EQUATION WITH LARGE CRITICAL FORCING TERM Preprint 2e boerhaavestraat 49 amsterdam PJr..,i,.n;te.d a:t .the. Ma:thema:ti.c.a1. Ce.nbr.e., 49, 2e. Boell.haave6.t'1.a.a:t, Ami.dell.dam. The. Ma:themcuUc.a1. Ce.nbr.e., 6ounde.d :the. 11-:th o0 Fe.bJtu.aJLy 1946, ,l6 a. n.on.-pJtooLt in6:tl...t.c.Ltlon. cum,i.n.g a:t :the. pJtomotion. 06 pwie. ma:thema:ti.C6 a.n.d Lt6 a.ppUc.a:ti.oru.). I:t ,l6 .6pon6oJte.d by :the. Ne.:theJli.a.n.d.6 Govell.n.me.n;t :thJtou.gh .the. Ne.:theJli.a.n.d.6 0Jtga.n.,i.za:ti.on. ooJt :the. Adva.n.c.eme.n..t 06 Pwie. Re6 e.aJtc.h ( Z. W. 0) • AMS(MOS) subject classification scheme (1970): 35B25, 35J25 Relaxation oscillations of a van der Pol equation with large critical forcing term by *) J. Grasman ABSTRACT A van der Pol equation with sinusoidal forcing term is analyzed with singular perturbation methods for large values of the parameter. Asymp-totic approximations of (sub)harmonic solutions with period T =2TT(2n-I), n = 1,2, ••• can be constructed when the amplitude of the forcing term is within an interval that depends on n. These intervals overlap so that two periodic solutions with period T = 2TT(2n±I) may coexist. KEYWORDS & PHRASES: Van der PoZ equation, reZa.xation osciZZation, sub-harmonic entrainment, singu.Zar perturbation. *) This report will be submitted for publication elsewhere. I • INTRODUCTION In this paper we consider a Van der Pol equation for large parameter values with a periodic forcing term of a same order of ·magnitude (1. 1) ix 2 d -- + v (x -1) 2 + x = b(v)cost, dt2 dt V >> 1 with b(v) = O(v). This equation was first investigated with analytical-to-pological methods by LITTLEWOOD [6], who proved the existence of (sub)-harmonic solutions of period (1. 2) T = 2ir(2n-1), n= 1,2, ••.• Littlewood stated that for b = av, a> 2/3 only globally asymptotically stable solutions of period 2,r are found, see also [7]. The proof of this statement has been given by LLOYD [8]. For decreasing a there also occur solutions of period 6,r. As a further decreases the 2ir-periodic solu-tion dissappear; alternately a passes intervals where one subharmonic so-lution of period T = 2ir(2n-1) exists and intervals where two subharmonic solutions of period T = 2ir(2n±l) coexist, n = 1,2, .•.. We will analyse the problem (1. I) for a specific choice of b (,;) and write (I. 3) b = a v + S. Using singular perturbation techniques we will construct asymptotic ap-proximations of (sub)harmonic solutions of (1.1), (1.3) with a= 2/3. The periods of these solutions satisfy (1.2) with n independent of v. In the process of construction of the approximation we will have to impose conditions upon S of- the type (1.4) s . n It turns out that 2 (I. 5) S < S l = S < S • --n n+ --n-1 n This overlapping of invervals differs slightly from Littlewoods results and from numerical results by FLAHERTY and HOPPENSTEADT [2], as we only find intervals with two subharmonic solutions of period T = 2TI(2n±l). This dif-ference is explicable for one part from the fact that we study the case a= 2/3 instead of a< 2/3 in the limit v + 00 , while also the dependence upon the initial values may play a role. We found s2 = 11/6v'3 which refines Littlewood's statement about the existence of solely 2TI-periodic solutions for b sufficiently large. In [3,4] the case a= 0 was also analyzed with asymptotic techniques. There the subharmonics had a period T = 2Tim with m = O(v). The choice a= 0 or a= 2/3 leads to solutions with a completely different asymptotic behaviour, which makes it necessary to consider them as separate problems. In [3] we met an unusual structure of two-variable expansions matched with boundary layer solutions. We will see here that the case a= 2/3 also exhibits an exceptional structure. The global behav-iour of the solution depends strongly on local conditions: each time the solutions passes a neighbourhood of the lines x = ± 1 some quantity is in-creased with a given value. When it reaches a threshold value the solution enters a phase of rapid change characteristic for a relaxation oscillation. This part of the solution is.approximated by a boundary layer type of so-lution. For the regions sketched in figure I separate local approximations have been constructed from the differential equation. Integration constants in there local asymptotic solutions are determined by matching pairs of local solutions of adjacent regions. Thus, in this paper we investigate the equation ( 1.6) 2 d x 2 dx 2 --2 + v(x -1) dt + x = (3 v + S)cost. dt It is expected that the study of this problem with a critical forcing term may bring u:s in the position to deal succesfully with the more complicated problem of O <a< 2/3, It is anticipated that periodic solutions of this problem have a behaviour in which elements of both the case a= 0 and a= 2/3 are present. A formal asymptotic analysis may help us to get a bette~ understanding of the essentials of this problem, which has the reputation of being hard to investigate by rigourous analytical methods, see [2,5 and 7]. 2 0 -1 -2 t X t-------, I I -,, I \ / \ I I / I I 1...-j-L--, B I I n' ,' I •- - 7 I \ I 11 / , __ ., I'- ,,1 I I I I I I I t n r-1--, I f I 11 I 11 C Ii I I I L_ -~.! I I Al: L- - .J t ➔ Fig. I Characteristic regions for a periodi~ solution of (1.6) 2. ASYMPTOTIC SOLUTIONS FOR THE REGIONS A m It is supposed that 1.n the regions A where I < x < 2 the solution m can be expanded as (2. I) :x(t;E:) -1 = xm0 (t) + v xml (t) + •••• 3 Substituting (2.1) into equation (1.6) and equating the terms of order O(v) and O ( I) we obtain (2. 2) 2 = 3 cost, 4 (2. 3) di ·mO - --2- - xmO + S cost. dt Integration of equation (2.2) gives (2 .4) 1 3 = ~3 s1.'nt + co(m). 3 xmO - xmO Since in the regions A the value of the left-hand side of this equation m varies from -2/3 to 2/3, we have to take c0 (m) = 0. For this value of c0 (m) the solution of (2.4) reads (2.5) xm0 (t) = 2 cos{½ arccos(sint)}. Integrating (2.3), while making use of (2.2), we obtain (2 .6) (2. 7) 2 cost 2 3(xm0-t) t m-1 () d + 0 s1.'nt + C (m) xmO T T ~ 1 , When t approaches tm from below, xmO and xml behave as (2 .Bab) where (2.9ab) Km= - ½ + ½ 13 ( -c 1 (m)+ I), I= Jtm xmO(t) dt = 6 13. tm-1 Thus, fort t t the asymptotic solution (2.1) looses its validity. m 3. ASYMPTOTIC SOLUTION FOR THE REGIONS B m We analyse the local behaviour of the solution near (x,t) = (1,t ), m m = 1,2, ••• by introducing a stretching transformation in both the depen-dent and independent variable (3. lab) x = 1 + V (~)v-y, m t = t m -a. + ~ V Substitution into the differential equation yields iv I-2y+a. dV (3.2) -y+2a. m (2V +v -Yv 2) --1!! + I + V --:7 + V d~ m m d~ 3 -3a. 2 -a. ~ V ... ) . (3 v+S) (~v . - 3! + 5 V V -y = m We see that for a.= y = 1/2 the second derivative becomes of the same order of magnitude in v as the leading terms constituting equation (2.2). Multi-1 ' h ' ' h -l / 2 d 1 ' ' f. ' b . h p ying t e equation wit v an etting v to in inity, we o taint e limit equation (3. 3) The function vm0 (~) expresses the local limit behaviour of the solution for v-+ 00 • In order to match the solution of region A it must satisfy m (3.4) for~-+ 00 , see (2.8). Such a function indeed exists and has the form (3.5) = a D~m (-a~) VmO (~) D (-a~) ' Km a = ~4/3, where Dµ(z) is the so-called parabolic cylinder function of orderµ, see WHITTAKER and WATSON [9,p.347]. For z-+ 00 we have that -lz2 µ { 1 µ(µ-I) ... } , D (z) = e 4 z -2z2 + µ while for z-+ - 00 -lz 2 µ (µ-1) D (z) = e 4 zµ {1-2z2 + ... } + µ rz; lz 2 -µ-1 {1+ (µ+1) <r2) r(-µ) e4 z + ... } . 2z Assuming that Km~ 0 the function Vmn(I;) will be regular for finite I;, while for I; + 00 (3.6) On the other hand at region A 1 the solution is approximated by m+ (t-t) -½+½v'3(c1(m+l)_S) m -3 x(t) =I+ --- + (t-t) + O((t-tm) ) 13 m as t ~ t. Consequently, (3.6) matches the local solution for region A m ~I if K = -m Y3 (S-C (m+ I)) I or using (2. 9a) (3. 7) C (m+l) 1 = C (m) - I I with I= 6/J, see (2.). Obviously, we will arrive in the situation that for some m, say m = n, (3.8) K I ~ 0 < K ~ ½ v'3 I n- n (if n = I, inequality (3.8) reads K1 > 0). The parabolic cylinder function Dµ(z) withµ> 0 vanishes for certain value(s) of the argument z. Let z0 be the largest zero. For I; t 1;0 with z0 = a 1;0 we have (3.9) V (r:) (t"-t"o)-1 + .!. a2 cla2r:2_K -.!.) er: r:) mO "' R$ "' "' 3 4 "'O n 2 .,,-.,,0 ' 6 so VnO + - 00 and the local solution at region Bn becomes singular at I;= 1;0• 4. ASYMPTOTIC SOLUTION FOR REGION C At this point the solution enters the boundary layer region C with local coordinate (4. 1) We assume that the solution can be expanded as (4. 2) Substituting (4.1) and (4.2) into (1.6) and equating the terms of order 2 O(v) and of O(v) we obtain, respectively, (4. 3) iw cw/-1) dW0 __ o_ + -- = o, dn2 dTl (4 .4) iw <w/-1) dW1 dW0 __ 1 + --+ 2 WO w -= o. dn 2 dn 1 dn The solution of the first equation matches the local solution for region B if n (4.5) w0 (n) ~ 1 + 1/n as n + - 00 , see (3.9). This condition is satisfied by the class of solu-tions (4.6) I I W0+2 1-W + 3 log 1-W = - n + HO' 0 0 7 where the integration constant H0 is found from matching with higher order terms of the asymptotic solution for region B. It turns out that n (4. 7) I I H0 = 6 log v + 3 log 3. From (3.9) we also deduce that w1 should behave as (4 .8) for n + - 00 , so that the integrated equation (4.4) will have the form (4.9) On the other hand for n >> 1/6 log v we have (4. IOa) w0 (n) = -2 + O(v 112e-3n) (4. IOb) I 2 1 2 2 1 -3n w 1 ( n) = 3 a c-4 a ~0 -Kn - 2) + O ( e ) • The boundary layer solution matches the solution for region A1 if (4.11) -I -1 v ·+o(v ), where x1i(t) are the coefficients of an expansion for region A1 of the form (2. I). 5. PERIODICITY CONDITIONS 8. Let us assume that the periodic solutions we are looking for are I symmetric in the sense that x(t) =-x(t-2T). Then we have completed the local (4.11) to the complementary phase t = t 1 - TT+ approximations. Transposing + r v-1/2 . . A h ~o in region 1 we ave (5. I) 2 - .!_ a2 (.!_ a2 e- 2 -K - .!_) v-1 ( - I ) 3 4 "'O n 2 + O V or (5 .2) K = - .!_ + .!_ v'3 (S+C 1 (1) __ 2I I). n 2 2 Using (2.9a), (3.7) and (5.2) we find (5. 3) S=-1 (2K+l) v'3 n I . 1 2 (n-2) I. From (3. 8) we know that K ranges from O to 9, so S has to satisfy n (5. 4) 3 I 3 ( ! ! -n) < S ~ 3 I 3 Ci~ -n) n = 2,3, •.•• Solutions of period 2TI are found for S > - 7/6 13. 6. SOME REMARKS 9 We havi~ found an interval for S, see (5.4), where an asymptotic approximation of a synnnetric periodic solution of period 2TI(2n-1) can be constructed. The lower bounds S agree quite well with the numerical results -n of FLAHERTY and HOPPENSTEADT [2,fig.3], see table I. They took (a,S) (.475,0) and varied the value of v. In this way intervals for 1/v were constructed, where numerical solutions of period T = 2TI(2n-1) exist. Our S relates to their v as f3 = -.192v. In the limit v -+ 00 this problem differs from the one we study with (a,S) = (2/3,S), but for finite (large) values of v one may expect some agreement. However, our upperbounds S are sys-n tematically above their values. It is possible that there are more condi-tions to be imposed upon (3. They may be of a type we do not meet in our construction procedure. Apart from the different choice of a, we also expect complications because of the strong dependence upon the initial values. It is possible that for S just below S 1 the set of initial n+ values leading to solutions of period T = 2TI(2n+l) becomes so small, that only solutions of period T = 2TI(2n-1) are found in numerical experiments. Our results as given in formula (1.5) support the view that a= 2/3 is a critical value (no subharmonics above this value). We see that for a= 2/3 the system is at the point to have overlap by three S-intervals. 10 Asympt. bounds Numeric. bounds of intervals of intervals n T 13 sn 1/v ·~ 13 11v ~ sn --n --n --n _n 2 61T - 7.2 3.2 .028 - 6.8 • 148 - 1.3 3 101T -12.4 - 2.0 .016 -12.0 .041 - 4.7. 4 14,r -17.6 - 7.2 .012 -16.0 .018 -10.7 5 181T -22.8 -12.4 .009 -21. 3 .012 -16.0 Table I ACKNOWLEDGEMENT The author is gratefull to E.J.M. Veling for reading the manuscript and for some valuable remarks. REFERENCES [1] BAVINCK, H & J. GRASMAN, The method of matched asymptotic expansions for the periodic solution of the Van der Pol equation, Int. J. Nonlin. Mech. 2_ (1974), p.421-434. [2] FLAHERTY, J.E. & F.C. HOPPENSTEADT, Frequency entrainment of a foraed Van der Pol oscillator, to appear in Studies in Appl. Math. [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, to appear in Proceedings of the Third Scheveningen Conference on Differential Equations, Sutmner 1977, W. Eckhaus and E.M. de Jager (eds.), North Holland Math. Studies. [5] LEVINSON, N., A second order differential equation -with singular solutions, Anns. Math. 50 (1949), p.127-152. I I [6] LITTLEWOOD, J.E., On nonlinear differential equations of the second order, III. Acta Math. 2}__ (1957), p. 267-308. [ 7] LITTLEWOOD, J.E., On Van der Pol's equation -with large k, Nonlinear Problems, R.E. Langer (ed.) Univ. of Wisconsin Press, Madison (1963), p.161-165. [8] LLOYD, N.G., On the non-autonomous Van der Pol equation -with large parameter, Proc. Carob. Phil. Soc. 7l:... (1972), p.213-227. [9] WHITTAKER, E.T., & G.N. WATSON, A course of modern analysis., Cambridge at the Univ. Press (1935). 

Relaxation oscillations of a van der pol equation with large critical forcing term : (preprint)

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