The Modified Van der Pol Oscillator

Transcription

1 IMA Journal of Applied Mathematics (1987) 38, The Modified Van der Pol Oscillator F. N. H. ROBINSON Clarendon Laboratory, Oxford [Received 6 January, 1987] The extensive mathematical results known for the exact Van der Pol equation are widely used to discuss practical electronic and mechanical oscillators for which this equation may be only a very approximate description. An experimental and mathematical investigation shows that these results, nevertheless, give an excellent qualitative description of a more general type of oscillator, with a different nonlinearity, and require only relatively small, and calculable, modifications to the numerical results for the amplitude and period. 1. Introduction THE equation x + a(x 2 -l)x+x=x cos cot (1) with a small, was introduced by Van der Pol (1927) as an approximate description of the triode anode-bend oscillator and its response to small signals. With X = 0, there is a limit cycle with x(t) ~ 2 cos t + \a cos 3t + Oia 2 ) and, when (o is near an odd integer and a is small enough, the limit cycle oscillations lock to a small signal. Since then, there have been numerous mathematical studies of the Van der Pol equation: typical results will be found in the texts by Blaquiere (1966), Guckenheimer & Holmes (1983), and Jordan & Smith (1977). In particular, Kevorkian & Cole (1981) give a full treatment of the relaxation oscillations that occur when a is large. In this regime, it is convenient to change the units of time and consider the equation ey + (y 2 -l)y+y = 0 (2) with e small. The mathematical results known for equations (1) and (2) are widely used to discuss real electronic and mechanical oscillators, even though these real systems may be only very roughly described by the Van der Pol equations. It is therefore desirable to know whether the results are appreciably modified if the Van der Pol nonlinear term x 2 or y 2 is replaced by a more general nonlinear function (p(x) or (p{y). Any nonlinear function arising in a real system can certainly be assumed to be analytic. To preserve the symmetry of the equations, <f> must be even. Also, although other cases may be interesting, we shall assume that <p is a monotonically increasing function of the magnitude of its argument. For convenience, we normalize it so that 0(1) = Oxford Univeralty Prat 1987

2 136 F. N. H. ROBINSON When a is small, the modifications to give the solutions of x + a[<p(x) - l]x + x = X cos cot are straightforward. The free oscillation is still dominated by a term A cos t and the amplitude can be obtained from l'i In addition, there will be a change in the amplitudes of the odd harmonics; in particular, harmonics higher than the third may have amplitudes linear in a, and this may in turn alter the capture regions. It is less apparent that the relaxation oscillations associated with (3a) (3b) ey + [(p(y)-l]y+y = 0 (4) will be so insensitive to the precise form of (p(y). This is because, as (), there is a singularity at [y = 1 where the coefficient of y vanishes. The solution of (4) has to be constructed by an asymptotic matching of solutions valid in regions where y > 1 and \y\ < 1. This is, at first sight, a rather delicate procedure. In this paper, this question is resolved by studying the behaviour of an electronic oscillator as e is varied from very small values (10~ 5 ) to larger values (10 3 ), with a variety of fairly precisely defined forms for <p(y). These forms vary from the less abrupt nonlinearities lyj 0 " 8 and lyl 1 ' 06, through y 2 and y 272, to the very abrupt nonlinearity exp 17-5( _y 1). The circuit is described in Section 2. The experimental results presented in Section 3 show that there are no dramatic changes in the solution as <p is varied. As the nonlinearity becomes more abrupt, the amplitude decreases and the period lengthens. For all forms of <p, the period r(e) (for small e) satisfies where the slope s varies slowly with <p, decreasing steadily as <p becomes more abrupt. On the basis of these experimental results and the agreement of the exact Van der Pol case with Kevorkian & Cole's treatment, we construct, in Section 4, an extension of their treatment. This extension yields a simple procedure for calculating A and r(0), and explains the universal validity of the -power-law dependence of x(e) - T(0) on e. 2. The oscillator The circuit, omitting the ±15 V supply rails and the amplifier offset adjustments, is shown in Fig. 1. The three operational amplifiers (TL071) have time constants less than 1 [i sec, which is negligible compared with the period of oscillation; about 20 msec. The structure of the nonlinear conductance is shown in Fig. 2. It uses ten pairs of matched diodes (1N916) and, if the resistances R lt..., R w are chosen appropriately, the current i can be arranged to be proportional to v \v\ r where values of r between 0 and 3 are readily achieved.

3 THE MODIFIED VAN DER POL OSCILLATOR kfl 10 Wl C=1.00uF FIG. 1. The oscillator circuit. Some idea of the accuracy is furnished by the experimental plot of i against v, shown in Fig. 3 for a diode chain designed to give i proportional to v 3 and generate (p(y) = y 2. The points are calculated from -l(t 4 i = u 3 chosen to fit the curve at the single point v = 3. If fl, and R 3,..., R 10 are omitted and R 2 is shorted, the exponential forward current-voltage characteristic of the diodes yields i proportional to (sgn y) exp (N/2itr) and yields the exponential nonlinearity. In response to an input v to A A the output of A 3 is Gv - 4>(u) and if G is set equal to 2 (within a few parts in 10 4 ) it is easy to see that the signal voltage v satisfies R 2 CIv v = 0. (5) If we adopt a unit of time RC (1-2y) where y = T/C and let dv' FIG. 2. The diode chain that generates the nonlinear conductance.

4 138 F. N. H. ROBINSON Fio. 3. The experimental current-voltage characteristic for a diode chain designed to give i proportional to v 3 and generate the exact Van der Pol equation. The points are a cubic fitted at the single point v = 3. we obtain eii + l-2y v=0. (6) The value v x at which 0(u t ) = 1 2y can be obtained from the measured curve <P(u). Finally, with y = v/v x, we obtain equation (4). Thus, by selecting the appropriate diode chain, we can choose <p(y) and, by varying F, we can vary e. Hence, with the necessary attention to the scaling factors, we can observe the waveform y(t) and measure the dependence of the period on e. A buffer amplifier, not shown in Fig. 1, transfers the signal to an oscilloscope, a counter to measure the period, and a digitizer which takes some 1000 samples per cycle with eight-bit accuracy and stores them. The stored waveform can then be slowly recalled to provide hard copy, using a chart recorder. Examples are shown in Figs 4 to 9. Because the digitizer has to handle positive and negative signals, its effective accuracy is only seven bits or 1%, and this generates the spurious ripples seen in these figures.

5 THE MODIFIED VAN DER POL OSCILLATOR 139 Apart from the digitizer, the overall accuracy is affected by a number of factors. Apart from the capacitors, the linear elements of the circuit are accurate to better than 0-1%. The capacitors have to be measured, and the various digital and analogue instruments available to us only agree to within +1%. Over the range of primary interest 10~ 4 < < 0-05, where 100 pf < F < 0-05 (if, the accuracy in e is ±1%. For smaller values of F, yielding e down to 10~ 5, the effect of stray capacitance reduces the accuracy: it is about +10% at e = 10~ 5. The accuracy also decreases for larger F and e because of the form of the scaling factor = (r/c)(i - 2/7C)- 2. The period T, in real time, can be measured to five-figure accuracy, and so this presents no problems. Further inaccuracies arise from the exact form of <P(t>). These errors, together with the effects of time constants associated with the amplifiers and the reverse capacitance of the diodes (both about 1 n sec), will be discussed in Section 5. For the moment, we can assume that E (where its numerical value is required) is accurate to ±1%, y to ±2%, and x, obtained from T by a scaling factor, to ±1%. The waveform traces, except Fig. 4, are taken with F = 30 pf, where e = 3 10~ 5 (±5%). The error here is insignificant since all we need to know is that e is small, even in the context of an asymptotic theory which retains terms of orders e$ and $ but ignores terms of order e and In e. 3. Experimental results We begin with the exact Van der Pol equation ey + (y 2 - \)y + v = 0, which is of some interest in its own right since experimental observations are rarely made on oscillators satisfying the equation so closely. Although our main interest is in the relaxation behaviour for «1, the first trace (Fig. 4) shows the waveform for the rather large value = This waveform is intermediate between the -2 FIG. 4. A digitized and stored recording of the waveform for the Van der Pol oscillator with e =

6 140 F. N. H. ROBINSON y=2 ^ \ 0 y=-1 y=-2 FIG. 5. The waveform for a Van der Pol oscillator with e = The points marked on the upper part of the trace are from Kevorkian & Cole's formula. sine-wave characteristic for large e (small a) and a true relaxation oscillation. Figure 5, with e = 3 10~ 5, shows the typical relaxation behaviour, with jumps from y = +1 to y = 2 and from y = -1 to y = +2. The points on the upper part of the trace are calculated from Kevorkian & Cole's formulae, and the entire waveform is in good agreement with their predictions. We note in passing that the amplitudes A = 2 for both c = and y=2.38 \f 1 y ' y=o y=-i n Outer region Inner Transition \ _ region i region ^ ^ y=-2.38 Transition region \ _^ ^ ^ Outer region FIG. 6. The waveform with e = and <p = L)'l 1 ' 06 - The points are calculated from the theory in Section 4.

7 THE MODIFIED VAN DER POL OSCILLATOR 141 e = 3 10~ 5 are the same as the calculated amplitude of the sinusoidal oscillation with a«l. Figure 6 shows the waveform, when e = 3 10~ 5, obtained with the less abrupt nonlinearity in ey + (\y\ l)y + y = 0. The general behaviour is similar to the Van der Pol case; the jumps still begin at y = 1 and y = -1, but their termini, which fix the amplitude, are at y = T The analysis in the next section predicts 2-38, and the points in the upper part of the trace are calculated using this analysis. Again we note that the amplitude is near the value 2-33 calculated for the sine-wave when a «1. Figure 7, with e = 3 10~ 5, shows the similar waveform obtained with the even less abrupt nonlinearity in ey + (\y\ 0 ' 8 - l)y +y = 0. Here the amplitude is 2-6, and our calculated value is The more abrupt nonlinearity in ey + ({y] 2 ' 72 l)y + y = 0 yields Fig. 8 with e = 3 10~ 5. The calculated amplitude is in almost exact agreement. Finally, the very abrupt exponential nonlinearity exp 17-5([y 1) gives Fig. 9, and the amplitude 1-2 is in close agreement with the calculated value We notice that, as the nonlinearity becomes more abrupt, the waveform approaches the limiting form of a square wave of unit amplitude. In Fig. 10, the period r(e) obtained from the experimental period T = (1-2Y)RCT is plotted against e$ over the range 0 < e < 006, where we expect an asymptotic theory, ignoring terms of order e and e In e, to be a good approximation. The results a, b, c, and d are for \y\ os, \y\ 106, y 2, and \y\ 2T1, and e is for the exponential nonlinearity. The experimental accuracy in this range is ±1% in both T and e. The straight lines are least-squares fits to the points with e$<01; for all the lines, the correlation coefficient is greater than y=2.55 y=1 ^ y=-i 0 i y=-2.55 FlO. 7. The waveform with e = 3 10" 5 and <p = \y\

8 142 F. N. H. ROBINSON y= y=-1 y= FIG. 8. The waveform with e = 3 10~ 5 and <f> = \y\ 2Tl. In Table 1, the experimental results are compared with the results calculated using the analysis in the next section. For comparison, we also give the amplitude A a calculated for the sinusoidal oscillations with a «1. The agreement is good apart from a 6% discrepancy in T(0) for the exponential nonlinearity. This is not surprising, since <f>(y) is not very accurately characterized for this case. There is an obvious trend throughout the series in the amplitude, limiting period r(0), and the slope s. We also note that an oscillator with a rather y= \ y=-1 ; _ y= = ^ FIG. 9. The waveform with e = 3 1(T 5 and <p = exp 17-5( v - 1).

9 THE MODIFIED VAN DER POL OSCILLATOR 143 e FIG. 10. Experimental points for T plotted against el for the nonlinearities a: >r 8 ; b: lyl'- 06 ; c: y 2 ; &. \y\ 112 ; e: exp 17-5(^1-1). The lines are least-squares fits to the points with e$<0-l, i.e. e>0-03. Nonlinearity: Experimental amplitude A: Calculated amplitude A: Sinusoidal amplitude A a : Experimental period r(0): Calculated period T(0): Experimental slope s: \y\ * TABLE y \y\ 2zy exp 175 (Lv -l) ill-defined nonlinearity certainly less abrupt than Ly 08 yields T(0) = 0-84 and also obeys the -power law with s = Analysis In Fig. 6, following the general plan of Kevorkian & Cole (1981: pp ), the solution in one half-cycle is divided into four regions, an outer region with

10 144 F. N. H. ROBINSON y > 1, an inner region with y < 1, a transition region near y = 1 which joins these two regions, and a transition region near y = -A which joins the inner region to the beginning of the next outer region. We shall not reproduce all the details of Kevorkian & Cole's calculation, but rather indicate its general structure and the places where modifications are required. The matching of the solutions in the outer region to the solutions in the inner region depends on the time scale in the inner region being much shorter, i.e. of order e, than in the outer region. Thus, in the inner region, we can use a new time variable with a shift of origin. If we take t = 0, where the dominant term in the outer solution gives y = 1, the match near y = 1 can then be attempted as t *0 and t*-*. This proves to be impossible; so, near y = 1, we have to introduce a transition region with a new time variable which is, as we shall see, of the form i=e-i[t-p(e)]. ' This scale is intermediate between t and t*; so the transition region is matched to the outer region as t *0 but f-» o, and to the inner region as t* * and t * constant. The inner-outer match near v = A presents no problems, and a transition expansion is not required in this region. In the outer region, ej} is treated as a small perturbation: we write so that the zeroth-order and first-order equations are y(t, e) = u o (t) + Hx(0 + (7a) [0(K O )-1]«O + «O = O, (7b) ^ l ] U } + U l + ^- = O. (7c) Equation (7c) allows us to express u x in terms of u 0 and a single constant of integration. Equation (7b), with t= 0 at u 0-1, yields ( u 4>(v) 1 t=-\ ^-^ dy. (7d) The points shown in Figs 5 and 6 are obtained from (7d). In the inner region, with t replaced by t*, we have here we treat ey as the perturbation, and write l] d^+^ = O; (8a) y(t,e)= g0 (n + P(e) gl (n+-; (8b) where the order in e of the perturbed term is left undecided since it will later have to be chosen to achieve a match to the transition solution. At this stage, all we

11 THE MODIFIED VAN DER POL OSCILLATOR 145 require is that 1»/3» e. If we express <p(y) as then the zeroth-order equation has a first integral ^ + <P(g 0 ) - go = constant. (8d) If this is to match either the transition solution or the outer solution, which both vary more slowly, we require Thus equation (8d) becomes jj^+*feo)-*o=<p(l)-l. With P» e, the equation of next-highest order is with a first integral This has a solution TT + [$( 0) ~ l]gi = ^i- Ho- where h x is a constant and g lp is a particular integral of (8f) proportional to A:,. The transition solution near v = 1 must satisfy the full equation (4); but now we can expand this solution about v 1 and express it as y(t, e) = 1 + a,(e)/i(r) + a 2 ( )/ 2 (f) +..., where the orders a x and a 2, and the relation between i and /, have to be chosen so that, when <p(y) is expanded as 1 + X(y 1) + n{y I) , the functions/i and f 2 remain nonsingular at v = 1 as e * 0. The results (8e) (8f) (8g) (9a) a 1 = ', a 2 = ', t = e~'[f p(e)] (9b) obtained by Kevorkian & Cole (1981: eqns and ) are independent of the form of (p(y) as long as it is analytic at v = 1. Thus we obtain the equations

12 146 F. N. H. ROBINSON satisfied by / t and f 2 as (9c) (9d) For the exact Van der Pol equation, we have A = 2 and \i = 1. The solution of (9c), which remains finite as t-* -, where it must match to the outer solution, can be expressed in terms of an Airy function, and it tends to at a positive value t 0. The solution of (9d) can be expressed in terms of a constant of integration and the same Airy function. The general form of the solutions in the various regions is independent of the exact form of <p(y), though there will be numerical differences. These general forms, together with their appropriate time scales, are illustrated in Fig. 11. These solutions then have to be matched together by choosing the constants of integration, the time shifts, and the expansion parameter /3(e) in the inner solution. The match between the outer and the transition solutions near y = 1 results in a time shift of order e hi e in (9b), and this can be ignored. The match between the inner and transition solutions in Fig. 11 clearly occurs near t = t 0, and this gives a time shift, of order e$i 0, between t* and t and therefore t. This match also determines the order /3(e) = e$ and fixes the constant k x in (8f) as 1 0. Clearly, at the final match near A, we require dg o /dt* *0 and so, from equation (8e), A is given approximately as the negative root A o of 1. (10) However, y(t) in the inner region contains a term Pgi = e$gi- Also, at A o, equations (8e), (8g), and (10) give g, =g lp, where g lp is the particular integral of (8f), now given by, y=1 Outer /=0 dt* t'=t/r. Transition Inner Outer y=-a FIG. 11. Asymptotic solutions in the various regions with their time scales.

13 THE MODIFIED VAN DER POL OSCILLATOR 147 Since (p(-a 0 ) > 1, this gives g lp - io[<p(-a o ) - I]" 1, and this makes a further contribution of order eh 0 to A. The period r(e) is now given by A y where the last term arises from the time shift between t* and t. Since and the integral is well behaved at A o, we have where A o, T(0), and s depend on the form of <p(y). We have therefore an explanation of the ^-power-law dependence of r(e) - T(0) on e, valid to the extent that terms of orders erne can be ignored. To calculate A o, we have merely to form <P(y) in (8c) from <p(y), and solve equation (10) to obtain the negative root -A o. The integral in (11) then yields T(0), and the results in Table 1 are obtained by this procedure. The calculation of the slope s is more tedious; it involves the calculation of i 0, and then a detailed study of the match between the transition solution and the inner solution. 5. Discussion We have seen that, when the quadratic term in equation (1) is replaced by a more general even and monotonic function <p, there is no qualitative change in the nature of the limit cycle. If or is small, it remains an almost sinusoidal oscillation of period 2JI; if a is large (e small), it remains a typical relaxation oscillation. In both cases, with 0(1) = 1, the amplitude decreases as <p becomes stiffer and the period of the relaxation oscillation increases. Furthermore, for very small e, the dependence of the period on e is always of the form Thus, this equation gives a useful insight into the behaviour of real oscillators, and the necessary numerical corrections to their amplitude and period are readily calculated using equations (3b) or (10) and equation (11). With one exception, the experimental results are in complete agreement with the calculated results. This exception is the slope s in the relation between r(e) and el. We have not attempted to calculate s for a general form of <p(y), but Kevorkian & Cole (1981: eqn ) give s = for the exact Van der Pol equation with <p = v 2. This differs by 13% from our experimental value 6-21, and this is well outside our estimated experimental error. We cannot attribute this discrepancy to small differences between the actual and the assumed form of the nonlinear term, since, as we have seen, gross changes in <f> have only a small effect on s. Nor can it be attributed to the finite (11)

14 148 F. N. H. ROBINSON time constants of the active circuit elements, which might limit the speed of the jump. A change in the timing resistors R by a factor of 10, which produces a similar slowing down of the response, yields almost exactly the same result for i(e). A small correction due to the finite input capacitance F o of the first amplifier A x has some, but not enough, effect. One's first reaction is that there is some unsuspected defect in the structure of the circuit and, to eliminate this possibility, we have built a number of oscillators with different structures which all give excellent agreement for the expected values of A o and T(0) but also all yield values of s significantly less than Finally, to ensure that errors in the capacitance measurements were not the source of the discrepancy, a capacitance bridge was constructed which reduced the uncertainty in these components to 0-2%, and the circuit of Fig. 1 was rebuilt with components (including the diode chain used to generate <p = y 2 ) held to much tighter tolerances. The input capacitance F o of the first amplifier was measured as F o = 8 pf ± 0-5 pf, and this was added to the inserted capacitance F to calculate e. Each section of the circuit was checked separately to ensure its accuracy, and the response of the overall feedback loop was adjusted to give a time constant 6 of 0-9 /*sec with no overshoot. The value of C was pf and, with the timing resistances R = kq, this gave a period near 0-75 sec. The time constant 6 was varied from 0-9 to 2-5 usec with no effect on s. Also a resistance was temporarily inserted in series with C to check that dielectric loss in this component had no significant effect on s. The period T(0) was measured with F = 0, and the timing resistance R between C and F shorted so that the input capacitance F o would effect r(0) by only one e 10 FIG. 12. The period i(e) versus ei for the precision Van der Pol oscillator. The parameter e is calculated making allowance for the input capacitance of the first amplifier. The solid line is the least-squares fit r(e) = ei The broken line is Kevorkian & Cole's relation.

15 THE MODIFIED VAN DER POL OSCILLATOR 149 part in 10 s. The result T(0) = is within 001% of the expected value , and the amplitude A o = in almost exact agreement with the calculated value 2. These two measurements, which depend on the scaling factors, the accuracy of the nonlinear element, and the exact value of the linear part of the feedback-loop gain, served as a further check on the correspondence between the actual circuit and its assumed properties. The results for this oscillator over the range 0 < e < 10~ 3 are shown in Fig. 12. The solid line is the least-squares fit r(e) = e* to the 22 points, and the broken line is Kevorkian & Cole's relation $. The discrepancy between the two slopes is reduced to 5% but is still significant. Urabe (1963) has made a numerical calculation of r(e) for values of e between ^5 and 1. In Table 2, we compare these results with our experimental values, with the values (K & C) obtained from Kevorkian & Cole's relation, and also with the values (K & C (D)) obtained from this relation when they are corrected for the next significant term e In e in Dorodnicyn's (1947) asymptotic expansion. 0 & 4i l Urabe Expt. TABLE K&C K & C (D) It can be seen that the experimental results are very close to Urabe's numerical results over the whole range. Their accuracy therefore seems to be comparable with either of the two mathematical calculations. From a practical point of view, this is probably not important. For most purposes, the methods we have given for calculating A o and T(0), and the general trends in these parameters (as <p is varied) shown in Table 1, will be adequate. Thus, the exact Van der Pol equation furnishes a useful qualitative insight into the behaviour of real oscillators; if the form of <p(y) is known, then relatively simple calculations will furnish the values of A o and T, both in the sinusoidal region (or«1) and the relaxation region (e «1). REFERENCES BLAQUIERE, A Non-linear System Analysis. New York: Academic Press. DORODNICYN, A. A Prik. Mat. i Mekh 11,

16 150 F. N. H. ROBINSON GUCKENHEIMER, J., & HOLMES, P Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. New York: Springer-Verlag, ISBN JORDAN, D. W., & SMITH, P Non-linear Ordinary Differential Equations. Oxford University Press. ISBN KEVORKIAN, J., & COLE, J. D Perturbation Methods in Applied Mathematics. New York: Springer-Verlag. ISBN URABE, M Proceedings of the International Symposium on Non-linear Differentia] Equations and Non-linear mechanics (La Salle, J. P., & Lefschetz, S., Eds). New York: Academic Press, pp VAN DER POL, B Phil. Mag. 3, Reprinted in Classics of Modern Mathematics (Bellman, R., Ed.). New York: Dover (1961).