Jump phenomenon in a Van der Pol oscillator

Automatica, Vol. 7, pp. 481-487. Pergamon Press, 1971. Printed in Great Britain.

Jump Phenomenon in a Van der Pol Oscillator* Phenom~ne de saut dans un oscillateur de Van der Pol Sprungph/inomen in einem Van der Pol-OszciUator ~IBaenne craqra B aBxoroae6axeae BaH ~ep IIoa~ A. LE P O U R H I E T t and J.-G. PAQUETI"

Complex hysteresis phenomenon which are assoc&ted with jump phenomenon in a Van der Pol oscillator, are studied qualitatively and quantitatively. S,mmary--This work deals with a new presentation of the jump phenomenon in a Van der Pol oscillator. It is shown how, in the output signal, a mathematical component at the natural frequency of the oscillator can cause complex discontinuities in the frequency response. Therefore some secondary phenomena can be explained, for example hysteresis which characterizes jump phenomenon when we proceed at increasing and decreasing frequency. More quantitative and qualitative details are then given on previous results obtained by M. L. Cartwright and A. W. Gillies.

If the signal x oscillates at the frequency 0), it can be written as follows: x = Xsin0)t+ Ycos0)t. That notation implies that x is approximated by its first harmonic and that, in a sinusoidal steady-state, parameters X and Y are constants; it specifies also the uniqueness for determination of these parameters. In this method, suggested by Van der Pol, X and Y are supposed to be slowly varying functions of time, which means that a and B are sufficiently small [1]. Besides that, neglecting superharmonics in the equation for x ratifies the assumption for B; this assumption reinforces also the role of low-pass filter for the linear part of (1). Then the identification of terms sin0)t and cos0)t of (1) determines X and Y [I-6]. If we let:

1. INTRODUCTION

LET US consider a system which is governed by the Van der Pol non-linear differential equation:

5i-a~ +0)2x + Bx2~=eo0)sin0)t

(1)

in which a and B are positive coefficients. This equation describes, for example, the behaviour of the circuit that is shown in Fig. 1 and studied in Appendix 1.

Pl =4~'(aX2 + y 2 ) , I sin COt

Im

F 2_e2B --~as' Negative

0)2 __ 0)2

/ resistance _-c

(2)

a0)

the amplitude Pl is determined by: Pl[ a2 + 0 1 - 1)2] = FZ

Fro. 1. Van der Pol oscillator.

(3)

which is the equation of the frequency response curves F 1 of the system, in the Pl - a plane. Moreover, it can be shown that the stability conditions of the oscillation are:

* Received 24 August 1970; revised 5 January 1971 The original version of this paper was not presented at any IFAC meeting. It was recommended for publication in revised form by associate editor B. D. O. Anderson. This work has been partially supported by Defence Research Board of Canada (Grant DRB 4003-04). t Department of Electrical Engineering, Laval University, Quebec, Canada.

p,>½ (3pt -- 1)(Pt - 1 ) + 172> 0 . 481

(4)

482

A. LE POURHIET and J.-G. PAQUET

The latter inequality describes the fact that parts of the curve F 1 inside the ellipse E 1 given by the equation

these calculations, by the Andronov's and Witt's method, are given in the work of STOKER [8].* Remark

(3Pl - 1)(pl - 1) + a 2 = 0

(5) Lord Rayleigh's equation

represent unstable oscillations. It can also be easily shown that E1 is the locus of all points with a vertical tangent to the curves F 1 and that the jump phenomenon corresponds to crossing the ellipse. The instability, corresponding to (4), is a focal instability and gives some information, in X - Y plane, on the existence of a limit cycle surrounding the unstable singular point; this is equivalent to considering, in the input signal x, the existence of a supplementary term with a frequency co~.

~-aYc+092x+Byca=eosin09t

(a, B > 0 )

has the same frequency response curves. If we let F 2 = 3Be2 4a ~ '

Pl = ~a(-D2(X2 -~-y2), then we have preceding results [9]. A physical system described by this equation is analysed in Appendix 2.

Let us write signal x as follows: x = X s i n 0 9 t + Ycos09t+ Xosin091t+ Yocos091t. (6)

If we substitute (6) in (I), the identification of terms in sin09t, cos09t, sin091t and cos091t gives four other equations which can be used to determined X, Y, 091 and ( X 2 + y 2 ) . Letting Po = B ( X2 + y2),

these equations can be written as follows:

(-Ol--090, 2Pl +Po = 1,

(7)

2. JUMP PHENOMENON WITH FREE OSCILLATION We assumed that theoretical instability, as indicated when we have a first harmonic approximation is, precisely, due to an inadequate approximation; practically, this is justified since the system is largely stable. The existence of a complex oscillation suggests then to make a two harmonics approximation; this implies that x be given in a form described by (6) with 091 =09 0. The non-linear part of (1) can then be written: BxZYc=ql sin09t +q2 cos09t +q3 sin090t+q4cos090t. (ll)

If we let: Pl [tr2 + (3pl - 1) 2] = F 2.

(8)

X2-~ yz=R2

'

Instead of considering the behaviour of Pl, it is easier to consider

X 2 + Yo2 = R o2,

P=Po+Pl

and if we substitute (6) in the left side of (1 I), the identification of the coefficients of sin09t, coscot, sin09ot and cos09ot gives:

which can be measured directly. (7) and (8) can then be written as: p=l-pl

(9)

4ql _ _ __ _ coY(R 2 + 2R 2) + ,~[3X 2 + y2 + 2Ro2] B

+ 2X Y 17+ 4XXo~'0 + 4X Yo 17o, and ( 1 -- p)[a 2 + (3p -- 2) 2] = F 2 .

(10)

4q2 = ogX(R 2 + 2R2o)+ I?[3 y2 + X 2 + 2R2o] B

From (7), self-oscillation at frequency 090 does not exist if p l > 1/2; in this case, the amplitude is given by (3) and the jump zone by the ellipse E 1. If pl < 1/2, we know that the oscillation is focally unstable; then (10) gives the new frequency response curves F 2 of the system. More details of

+ 2 X Y f ( +4YYo1?o +4YXo.~o ;

* In his book, Stoker analysed the case of a circuit that is forced by input eosintot, in which the amplitude is independent of the frequency. The results obtained, valid only when to is near too, cannot be used when the frequency is larger.

Jump phenomenon in a Van der Pol oscillator q3 and q4 are obtained from qt and q2 with some permutation with X and Xo, Y and Yo, RZ and Ro2. If p is the derivative operator with respect to time, the identification of terms in sintot, costot, sintoot and costoot in both sides of (1) gives [10]: MZ+Q=E,

483

and a, = 4to2Ro2Eto2(9R~- a)(3R~ - a) + (tOo2 - to2)2]. Since a o =0, this determines the presence of a free oscillation term [I0]. Coefficients ~i for i > 1 being complicated, we shall consider as a stability condition the following inequality:

with:

cq>0.

to2 _ to2 _ ap + p2

ato- 2top

0

0

--ato+ 2top

t o ~ - t o 2 - a p + p2

0

0

0

0

--ap+p 2

atoo-2pto o

0

0

-atoo+2pto o

--ap+p 2

M=

X

eoto

Y Z=

Xo

0

q21 ,

Q=

E=

qa q+

Yo

0

The equilibrium state Zo is the solution of the system obtained in letting p = 0 in matrix M and equating to zero the derivatives included in Q; details on stability can be obtained in considering the linearized system near Zo. Let us consider dZ as a small perturbation around Zo; the vector matrix equation describing the behaviour of dZ is."

(M + Jo +J~p)dZ=O,

0

It is important to note that this condition is a necessary condition only, a sufficient condition being related to the complete Routh criterion, including all ~t. With (2) and (9), this inequality can be written as: (9p - 8)(3p - 2) + a 2 > 0 ;

(12)

in which Jo and Jl are Jacobian matrices:

j o _ D ( q l , q2, qa, qa) D(X, Y, Xo, Yo) '

it describes the instability of oscillations corresponding to points on F2 inside the ellipse Ez given by equation: (9p - 8)(3p - 2) + a 2= 0 ,

J1--O(qD q2, qa, q4)

O(~, t', ~o, ~o) at Z0. The stability conditions can be obtained in using the Routh-Hurwitz criterion when applied to the characteristic polynomial of (12), i.e. to determinent: A = [ g + Jo + J~Pl which can be written in the form: s

A = ~ aip~, 0

with

0~0=0

which also describes the locus of all points with a vertical tangent to curves F2, corresponding to jump points on these curves. The following paragraphs show how these jumps can occur from F1 to F2 when frequency is increasing, or from F2 to F1 when frequency is decreasing. Given the symmetry in the (p, Pl) versus a plane this analysis will be done only for a > 0, results for a < 0 being obtained from the preceding ones by permutation. It should be noted that it is the same to use to or tr for discussion, derivative da/dto being always positive as in Fig. 2.

484

A. LE POURHIET and J.-G. PAQUET

-

where PI < 1/2• This gives rise, in the signal, to a free oscillation term and then to the j u m p on a F2-curve. Consequently, it is our intent to look for the locus of starting points of F2-curves, after a j u m p from ellipse Ej. Let p~ be the ordinate of point A where we have a vertical tangent on the Fl-curve in Fig. 3, and G' the corresponding abscissa; (5) and (3) give: a '2 = (3p~ - 1)(1 - p~) and F 2 = 2p~2(1 - p ~ ) . J

C0 1

tg ~ =-6"

Replacing these two quantities in (10), we obtain the equation giving ordinate p~ of the starting point B of F2-curve corresponding to the same value of F 2 : (1 - p~)['(3p~ - 1)(1 - p~) + (3p~ - 2) 2] =2p~2(1-p~). On Fig. 3, C1- locus is given, corresponding to these points. This locus crosses ellipse E2 with an infinite slope and points of C1 which are inside E 2 are instable points. C1 does not exist except for 0 < g < x/5/4 and the slopes at these points are respectively 0 and - 1 / 3 3 .

2. a VS. 09 curve.

FIG.

3. BEHAVIOUR AT INCREASING FREQUENCY When we proceed through F,-curve* at an increasing frequency, starting from (r = 0, (a)= (Oo), it is possible when crossing ellipse El, that the representative point of the system goes to the zone

- °

Let us consider Fig. 3. I f FZ
.

"\ \x•

,~,32

\ ~,\

•

"

8/27

\\ !
I

x\

'!1 ;t 213

{ER) I I

0"5 [

i

FIG. 3. Behaviour of the system when the frequency is increasing. * Fl-curves are shown in mixed lines and F2-eurv~ in continuous and slight lines (Figs. 3 and 4),

- -

I

i

.~5

/

_

/.

h

I

,/

I

05

1/4

5___~_.__t

. . . . . .

I

i

o

FIG. 4. Behaviour of the system when the frequency is

decreasing.

Jump phenomenon in a Van der Pol oscillator This critical value is:

485

If we do not take into consideration the trivial solution Pl = 1/2, we have:

F~=0"05597

~+(p~-:~)~=~

and the corresponding value of 0- is 0-c= 0"2383. If F2 F~. F 2 determines then discontinuity in the frequency response curves. Moreover, this value, close to 15[268, is not included in Cartwright's interval.

which is the equation of a circle centered in (0, 3/4), with a radius ~/~4, and which passes through the intersection of two ellipses, with ordinate 3•4. We consider only this part of the circle corresponding to F 2 < 1/4, i.e. this part of this circle with an abscissa included between 0 and 1/2. If 8/81 < F 2 < 1/8, the system does not show any jump phenomenon and it is then impossible to join a Fl-Curve with frequency variation only: the free oscillation term stays in the signal. If 4]81 < F 2 <8/81, there is a jump phenomenon on a F2-curve but, as in the preceding ease, we cannot pass on a Ft-curve. If F 2 <4/81, we stay on a Fz-curve without any jump phenomenon. 5. CONCLUSION

This work is a quantitative study of the jump phenomenon in a frequency response of a system governed by a Van der Pol differential equation. Besides the jump phenomenon which appears when we go through a vertical tangent point, it has been sccn how we can go from one determination to another and thus create a complex hysteresis phenomenon as indicated in Fig. 5. More details have then been given on the work already done by P~

4. BEHAVIOUR AT DECREASING FREQUENCY

Let us consider Fig. 4. If F 2 > 1 / 4 , we go from F 2 to F 1 normally on line p = I/2. Moreover, if f 2 < 8/27, we observe a jump on F 1. If 1/8
1 ~ i

~

F¢.412"/'

0'~

0.2

If we substitute these values in (3), we have: 0

( p l - ½)[0-2+ (pl _ ~)2_ & ] =0

i

I

1

I

0"5 ~

f

|

!

Flo. 5. Hysteresis phenomenon.

o~

A. LE POURHIET and J.-G. PAQUET

486

CARTWRIGHT for complex oscillations, and a threshold amplitude ( F 2 = 1 / 8 ) , for which the system cannot loose the free oscillation term, has been given. This irreversibility p h e n o m e n o n can also be considered in the same way as the "dangerous stability frontier" for zero input systems [2]. Quantitative analysis o f these discontinuities reveals the behaviour o f Van der Pol and L o r d Rayleigh oscillators and opens a new field for experimental studies. REFERENCES [1] J. J. STOKER: Nonlinear Vibrations, pp. 147-187. Interscience, New York (1966). [2] J. C. GILLE,P. DECAULNEand M. P~LEGRIN:Mdthodes Modernes d'l~tude des Systdmes Asservis. Dunod, Paris (1967). [3] N. MINORSKY: Introduction to Non-linear Mechanics, pp. 341-354. J. W. Edwards, Ann Arbor, Mich. (1947). [4] W . J . CUNNINGHAM : Introduction to Nonlinear Analysis, pp. 213-220. McGraw-Hill, New York (1958). [5] L. StDEmADES: Les solutions forc~es de l'rquation de Van der Pol. L'Onde Electrique, tome 45, No. 3, Octobre 0965), pp. 1216-1224. [6] C. HAYASHI: Non-linear Oscillations in Physical Systems. McGraw-Hill, New York (1964). [7] G. CHAPPAZ: Aspects analytiques et analogiques des solutions de l'rquation de Van der Pol en r~.gime forc6 sinusoidal. Int. J. nonlinear Mech. 3, 245-269 (1968). [8] A. ANDRONOVand A. WI-rr: Zur Theorie des Mitnehmens yon Van der Pol. Arch fiir Elektrotech (1930). [9] R. CHAL~AT: Sur rrquation de Lord Rayleigh. Colloques internationaux du C.N.R.S., No. 148, Editions du C.N.R.S., p. 287 (1965). [10] A. LE POURHIETand J. F. LE MAITRE: Une mrthode grnrrale d'rtude de la stabilitY, d'un syst~me non linraire oscillant. Int. J. Control 12, 281-288 (1970). Ill] M. L. CARTWRIGHT: Forced oscillations in nearly sinusoidal systems. J. Inst. Elec. Eng. (London) 95(3), 88-96 (1948). [12] A. W. GILLIES: On the transformation of singularities and limit cycles of the variational equations of Van der Pol. Q.J. Mech. Appl. Math. 7, part 2 (1954). [13] M. PELEGRIN, J. C. GILLE and P. I)ECAULNE" Les Organes des Syst~mes Asservis, Dunod, Paris (1965).

with

B

3~ C

=__

~°°2 --

1 LC

eo--

I C

a='-; . C

This equation is identical to (1). APPENDIX 2 Physical system represented by Rayleigh equation Let us consider the control system given in Fig. 6

FIG. 6. Servomechanism with servomotor.

kv~ kv~

kv~ kv4

cm"

FIG. 7. Characteristics torque-velocity of the servomotor. which has a two-phases servomotor with t o r q u e velocity characteristics as in Fig. 7 [13]. In servomechanisms these motors are used generally at low speed and we have approximately

C,.= - a O - b O a + k V, where a, b and k are constants. I f J a n d f r e p r e s e n t s the inertia and friction for the servomotor we can write

JO+ f O = - a O - b O 3 + k V APPENDIX 1

Physical system represented by a Van der Pol equation Let us consider the circuit as shown on Fig. 1 [4] in which the current at the input o f the negative resistance is the following:

with

V= A( eo sin c o t - c 0 - dO), and therefore

i = - ]~x + O x 3 ,

0 + f + a + A k d 0 + A k c 0 + boa = Aeoksi n cot. J J J J

x being the voltage. The differential equation describing the circuit is:

This equation is of the Lord Rayleigh type with a negative friction if

C5c - yx + fix 3 + l f xdt = I sin cot, with the condition that resistance R be negligible. I f we take the derivative o f this equation, we have:

5¢- aSc + too2 x + B x 2 ~ = e0co cos cot

d< _f+a kA Then results o f this paper can be applied and lead to the knowledge o f discontinuities for frequency response curves. This could be the subject o f interesting experimental research.

Jump phenomenon

in a Van der Pol oscillator

Rq~um~--Le pr6sent travail traite d'une nouvelle presentation du phenom/me de saut darts un osciUateur de Van der Pol. II eat montr6 comment, dans le signal de sortie, un composant mathematique h la fr~luence naturelle de l'oscillateur, peut donner lieu ~ des discontinuit6s complexes dans la reponse frhtuencielle. En cons~luence, certains phenom~nes secondaires peuvent ~tre expliqu~s, par exemple l'hyst~r~is qui caract~rise le phenom~ne de saut lorsqu'on op~re avec une fr~luence croissante et d~croissante. D'autres d~tails qHantitatifs et qualitatifs sont aiors d o n n ~ sur r~sultats obtenus ant~rieurement par M. L. Cartwright et A. Gillies. Zusammenfassung--Die Arbeit behandelt eine neue Darstellung des Sprungph~nomens in einem Van der PolOszillator. Gezeigt wird nun, dab eine mathematische Komponente in dem Ausgangssignal bei der natiirlichen Frequenz des Oszillators komplexe Diskontinuit~lten in deren Frequenzgang verursachen kann. Daraus ktlnnen einige

487

sekund~re P h i n o m e n e gedeutet werden, z.B. die Hysteresis, die das Sprungphinomen charakterisiert, wenn bei anwachsender und fallender Frequenz gearbeitet wird. Weitere quantitative und qualitative Einzelheiten zu friiheren yon M. L. Cartwright und A. W. Gillies erhaltenen Resultaten werden angegeben. Pe3~oMe--Hacroamaa pa6oTa OTHOCHTC~i K HOBOMy npej1cTamlemmo $1B/IeHH~I cxam~a B aBToKoy[e6aTe•e BaH ~cp HoGs. I-IoKa3blBaeTc~l KaK, B BbIXO~HOM CHFHa.He, MaTeMaTHqeCKI~ 3JIeMeHTHa HaTypa.rlbHOl~tIaCTOTe aBTOKOgIC6aTedI~l, MO~KeTnpHl~CTH K KoMnJIeKCHHMHDepbIBHOCT$IM B qaCTOTHOl~ xapawrepgcTm~e. BBHay 3TOrO, HegOTOptae BTOpHu._h'me BBJ~eHH~ MOryT 6I,I'rl, O6"I~gCHeHH, Hanpm~ep FHCTODe3HC xapaKTepH3ylOmHlt SB.rleHHe ci