Nonlinear analysis in the phase plane

Nonlinear Analysh the Phase Plane

zn

b y THEODORE A. BICKART

Department of Electrical Engineering Syracuse University, Syracuse, N e w York

ABSTRACT: The study of a second order system often involves solution of the first order phase plane equation. The Lienard construction and its several extensions have been very popular as procedures for obtaining the solution graphically. Here-to-fore the classic Van der Pol equation and any others similar to it have not beeen amenable to solution in the phase plane by a Lienard-like construction. The extension to the Lienard construction given in this paper shows how to obtain a solution to such problems using a graphic construction.

Introduction A second order system described by the pair of first order equations: dy _ dt

dx

qJ (x, y)

d-T =

9C(x, y)

(1)

is characterized in the x - y plane by the first order equation: dy _

~ (x, y)

dx

9C(x, y)"

(2)

This paper is concerned with the graphical solution of Eq. 2.

Background The familiar Lienard construction (1, 2) is a technique for obtaining a solution to Eq. 2, when Eq. 2 has the particular form: dy = dx

x -4- X ( y ) y

dy d--x =

x y + Y(x)"

(3a)

or

(3b)

A generalization of the Lienard construction due to Clauser (3) is applicable when Eq. 2 has the form: dy _ dx

x + X (y) y + Y(x)"

256

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Nonlinear Analysis in the Phase Plane

A further generalization by Bickart (4) may be employed when Eq. 2 appears as : dy _ x + X (5) dx y + Y" In Eq. 5, X may be a function of either x or y and Y a function of either x or y. A very interesting generalization of the Lienard construction given by Pell (5) is useful when Eq. 2 appears in the form: dy _ dx

X l(x) + X2 (y) y

(6a)

Obviously, the technique is also applicable to: dy dx = -

x Y1(x) "4- Y2(y)"

(fib)

In a manner similar to that employed by Clauser in generalizing the Lienard construction, we can generalize PelFs generalization so as to have a construction for : dy _ dx

X l ( X ) A- X~ (y) Y1 (x) -4- Y2 (y)"

(7)

A useful extension of the Lienard construction given by LeCorbeiller (6) is applicable when Eq. 2 is of the form: dy _ dx

x + X (x, y) y

(8a)

x

(8b)

The procedure will also apply to: dy = dz

y -4- Y (x, y)"

LeCorbeiller's extension differs from those previously mentioned in that the construction process is more like that involved in the method of isoclines (7, 8). In each of the truly Lienard-like constructions, employed in graphically solving Eq. 3 through Eq. 7, there is one underlying theme: Given a solution point P, find a point A, using only graphical construction methods, such that the line normal to A P has the slope of the solution curve passing through P. With this accomplished, the point P', a short distance from P in the direction of the normal, is taken as an approximation to another point of the solution curve. Extension

of the Lienard

Construction

to a N e w C l a s s o f P h a s e P l a n e

Equations It is possible to further extend the Licnard construction to systems for which Eq. 2 has the particular form:

Yol. 278, No. 4, October 1964

257

Theodore A. Bickart d y -~. _

(9)

x -~- X I ( X ) X 2 ( y )

dx

y

This extension makes it possible to obtain a phase plane solution to the classic Van der Pol equation using a strictly graphical construction. Solution curves for the Van der Pol equation have not been obtained in the phase plane ( x - 2 plane) using a Lienard-like construction; the Lienard construction has been used to obtain a solution in the phase plane of Lienard. See for example (9). The procedure involved in graphically solving Eq. 9 may be described in the following way. First, starting from the point P, coordinates (xo, yo), the point B, coordinates (X~(yo), Xl(Xo)), is located. This is accomplished by plotting the curves x = X2(y) and y = X l ( x ) in the x - y plane and then

:'-~ (Xz(~',)BXH,,)) /

I

P ~ [x,,yol

!

*

I

x~

I

Xz(y)

X

~=XllX)

FIO. 1. Procedure for locating the point (X~(yo), Xx(xo)).

following the procedure illustrated in Fig. 1. Second, starting from point B, the point A, coordinates ( - X I ( x o ) X s ( y o ) , 0), is established. This is possible through a graphical implementation of the quarter square multiplication technique as shown in Fig. 2. Third, the point P' (which will serve as a new point P in the repetition of these steps) is located a short distance from P on the line through P and normal to A P . (Note that the slope of the normal must be that of the solution curve Eq. 9 at the point (xo, yo) by virtue of the established coordinates of A and P.) To illustrate this extension of the Lienard construction in the solving of a problem, consider tile Van (ter Pol equation: .~- ~(1-x2)2+x=0

258

(10)

Journal of The Franklin Institute

Nonlinear A nalysis in the Phase Plane X- -I/4y t

(o,XHx.I+ Xm(yo3)

•

.,,,~

i ........... 'l

--

'

(xtcy.), x,(x.))

(01XI(X.) -- XZ(y.))

. (-XHx")Xz~Y'l'O)~i' i ,, ,

(-~[x+-,x(..,.y.,] ,J ~)

-

(-v4[~,~.

~TRANSLATION OF/4iNIS INTERVAL ALONG THE X XIS UNTIL t THE POINT {-I Xl IX*)- Xl lY*I] ,O) LIES ON THE ORIGIN LOCATES THE POINT A WITH COOROINATES : z t

(-,~[x ...... x,,,.,] .,~[x,,..,mX,,,.,],o)

L

= (-Xdx.lXz(y.hO)

FIG. 2. Procedure for locating the point (-Xl(xo)Xz(yo), 0).

4

4

i i i

/"--~z---."

,~. ' h.

. . ,.

I

3'

4'

+(I-xZ)

FIG. 3. Phase plane trajectories of Van der Pol equation.

I. 278, No. 4, October 1964

259

Theodore A. Bickart

which upon setting 2 = y, yields the phase plane equation: dy _ dx

x -- e(1 -- x2)y y

(11)

The graphical solution for e = 1 and for two different initial conditions (one inside and one outside the limit cycle) is shown in Fig. 3. One of the steps in the step-by-step process has been emphasized in the illustration so as to highlight the previously discussed details. As indicated in Fig. 2, the construction which locates the point A from point B, requires the translation of a line segment. I t is possible to eliminate this translation at the expense of adding another curve, as a construction aid, to the x - y plane. A relatively simple construction, using two parabolas, to locate point A is illustrated in Fig. 4; no translation is involved. The construction procedure m a y be modified in a simple manner to treat problems described b y : dy _ dx

(12)

x Y + Y1 (x) Y2 (y)"

If the number of curves has not become too depressingly great, then it is a relatively simple m a t t e r to set down the construction steps to handle: dy _ dx

(13)

x + Xl(x)X2(y) Y -[- YI(X) Y2 (y)"

In effect, we combine the construction steps used in solving Eqs. 9 and 12. In the construction for Eq. 9, the point ( - X ~ ( x o ) X 2 ( y o ) , 0) is located and serves as point A. If the curve y = - Fl(x) is plotted on the x - y plane, it y-x = V2(y+ xl z

y+x=l/2(y-xl =

"~ ( X2iy, i, Xr:x.~)

,, i

x

A

Fro. 4. Alternate procedure for locating the point

260

( - X I (xo)X2 (yo), 0).

Journal of The Franklin Institute

Nonlinear Analysis in the Phase Plane

then becomes quite easy to locate the point (0, - F I E - - X 1 (xo)X2 (yo)J). I f this latter point serves as point A, then systems described b y : dy = _ x dx y -ff F I E - X I ( x ) X 2 ( y ) J

(14)

m a y be handled. I n a similar m a n n e r constructions m a y be established for: dy = _ x -4- F ~ E - Y l ( x ) Y 2 ( y ) ] dx y

(15)

dy = _ x q- F 2 [ - Y , ( x ) Y2(y)-] dx y -~- F I ~ - X I ( z ) X 2 ( y ) J "

(16)

and :

T h e constructions for solving Eqs. 12 through 16 should in no w a y be construed as the t o t a l i t y of extensions and generalizations on the Lienard-like construction for Eq. 9; they are only the more obvious extensions.

Concluding

Discussion

Equations 3 through 7 show t h a t a Lienard-like construction for obtaining a graphical solution to the x - y plane Eq. 2 has, in the past, been restricted to those equations for which ~3(x, y) and ~: (x, y) can be expressed as a sum of functions of x or y. Specifically, it has been d e m o n s t r a t e d in this paper, t h a t a construction exists which allows q3 (x, y) to be the sum of x and a function of a product of a function of x and of a function of y and 9C(x, y) to be the s u m of y and a function of a product of a function of x and of a function of y. Of greater importance is the fact t h a t with constructions available for obtaining sums and products there is little inherent limit on the allowed functional f o r m of q5 (x, y) and 9¢ (x, y ) - - p e r s e v e r a n c e m a y be expected to yield the specific construction steps for most x - y plane equations which we might encounter.

References (1) Nicholas Minorsky, "Nonlinear Oscillations," New York, D. Van Nostrand Company, pp. 109-110, 1962. (2) J. J. Stoker, "Nonlinear Vibrations," New York, Interseience Publishers, pp. 31-33, 1950. (3) Francis H. Clauser, "The Behavior of Nonlinear Systems," Jour. Aeronautical Sciences, Vol. 23, No. 5, pp. 411-434, May, 1956. (4) Theodore A. Bickart, "Extension of the Lienard Construction," IEEE Trans. Circuit Theory, Vol. CT-11, No. 2, June, 1964. (5) W. H. Pell, "Graphical Solution of the Single-Degree-of-Freedom Vibration Problem with Arbitrary Damping and Restoring Forces," Jour. App. Mech., Vol. 24, pp. 311-312, 1957. (6) P. LeCorbeiller, "Two-Stroke Oscillators," I R E Trans. Circuit Theory, Vol. CT-7, No. 4, pp. 387-398, December, 1960. (7) W. J. Cunningham, "Introduction to Nonlinear Analysis," New York, McGraw-Hill Book Company, pp. 28-36, 1958. (8) Y. H. Ku, "Analysis and Control of Nonlinear Systems," New York, Ronald Press Co., pp. 16-18, 1958. (9) Nicholas Minorsky, "Nonlinear Oscillations," New York, D. Van Nostrand Co., Chapter 4, 1962.

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