Evaluation of phase-plane derivatives at singular points

Biief dbmmunication Evaluation of Phase-plane Derivatives at Singular Points by H. H. DENMAN

and A.

D. ANDERSON

Engineering Technology Ofice Ford Motor Company, Dearborn,

Michigan

Zntroduction We

consider nonlinear systems governed 2++f(x,li)

by equations of the form

= 0,

(1)

where 2 = dx/dt, etc. Singular points (S.P.) in the (x,v) phase-plane, v = 2, are defined by v, = 0, f(Xs, 0) = 0. If Eq. 1 is rewritten vu’ +f(x, v) = 0,

where (2) (3)

then v’ = dvldx as given by Eq. 3 is indeterminate at the S.P. A separatrix is a curve v(x) in the (x, v) phase plane which passes through a singular point and separates regions of qualitatively different motion. The nature of a separatrix in the neighborhood of a S.P. is specified if we can evaluate the derivatives vi, vJ1,. . . , where v: = dv/dx evaluated at (x,, 0). While some special results for v: are known, the general expression for vin) = dnv/dxn evaluated at (x,, 0) does not seem to have been determined. We find this expression and apply it to several examples.

Calculation

of Derivatives

at S.P.

It can easily be shown (e.g. by induction) (d/dx)^(vv’)

=

that

$; (” ; ‘) v(“+~-~) dk),

where r is t,he integer part of 4 (n+ l), (“k) is the binomial 2

coefficient,

and

indicates that for n odd, the term k = &(n + 1) is to be taken with weight 4. Applying the operator (d/ds)n to Eq. 3 yields vh+l-k)

523

V(k) +f(n,

=

0,

Brief Communication wheref’“)

= (d/dx)“f.

At a S.P., w(O)= us = 0, so that (5) becomes (at a S.P.)

(6) where the factor 1+ 6,, has been introduced to make Eq. 6 correct for n = 1. Several special cases are of interest. For n = 1, (4” + (ww,

4 + (qp4,

= 0,

(7)

which indicates the well-known result that two separatrices at a S.P. For n = 2, Eq. 6 gives

may intersect

v,” = - (1/3?4)f(2’ (cc,,0).

(8)

In the familiar case of linear friction, where f(x, 8) = 2YV+ g(x),

(9)

Eq. 6 yields, for n> 1, = - $

[(n+ l)ui+2y]vj”)

Example

1: Cubic

(” ; ‘) @+1-k) vfd _ gin)_

(10)

Oscillator

For the linearly damped soft cubic oscillator, dimensionless time variable, as vdv/dx+2yv+x-v2xs

Eq. 3 can be written, in a = 0,

(11)

which has singular points at 0, of:l/v. Then at X~ = l/v, w:2+2yu;-2

= 0,

(12a)

w; = 6( 3~; + 2y)-l v, ?I, v, = - 3(4vi + 2~)-l[(wz)~ - 2~~1, Note that the slopes of the separatrices that win) is proportional to v+l.

Example

2: Damped

Simple

(I2b) etc.

at (xs, 0) are independent

(I2c) of

V,

and

Pendulum

In this case, Eq. 3 can be written vdv/d8+ 2yv+sinB are at ( 4 nrr, 0). For x, = n-, we readily obtain

= 0. Singular points

(vi)” + 2y4 - 1 = 0, v,“=

(134

0,

v,“’ = -

(13b)

(4vi + 2y)-l,

etc.

(13c)

It is easily verified that all even derivatives are zero at x, = n. These results agree with those of Ku (l),obtained by equating coefficients after multiplication of two power series.

524

Journal

of The Franklin

Institute

Brief Communication .

’

I&arks Power series expansion of the separatrix in the neighborhood of a singular point can yield quite useful information. These coefficients in the past have been obtained by rather laborious methods. However, Eq. 6 gives the power series coefficient for any term of the series in terms of lower-order coefficients. Furthermore, Eq. 6 is linear in wl;“) (except for n = 1) which makes evaluation particularly simple. Thus the separatrix may easily be approximated to arbitrary accuracy within the circle of convergence of the power series. ---------POWER ---TRAJECTORIES

SEPARATRICES SERIES

iX

-7 --

FICA 1. Phase-plane

\

rxrves for the linearly damped cubic oscillator, y = O-125, v = 1.

The expression for v: is quadratic and therefore has two real roots (or none). However, a pair of distinct real roots may occur even though there is a single bona-fide separatrix through the singular point. This is the case for the damped cubic oscillator. The two slopes at the singular points, v’+ = - y + (~2 + 2)*, vi-- = - y - (72 + 2)*, correspond to motion away from a:d into the singular points, respectively. For this example, it is obvious that the phase-plane trajectories through the S.P. corresponding to motion away from the singular points are not separatrices, for they do not separate qualitatively different motions. Figure 1 illustrates that the neighborhoods of these trajectories contain

Vol.

287,

No. 6 June

1969

525

Brief Communication ’ . points on either side of the curve which result in the same qualitative motions. Thus we reject the positive slope as not belonging to a separatrix. Such trajectories are indicated in the figure; however, Eq. 6 can still be used to calculate these trajectories. Examination of Fig. 1 indicates that the trajectories with negative slopes at the singular points are the separatrices. (It may, in fact, be possible to extend this result to problems of a more general nature.) The dotted curve in the figure is the power series approximation for one separatrix using the first seven terms in the expansion of V(X). Also, note that Eq. 5 may be used to determine the phase-plane trajectory for any initial condition (2,~). Thus, both stable and unstable trajectories may be examined easily.

Reference (1) Y. H. Ku, “Analysis and Control of Non-Linear The Ronald Press Co., 1958.

526

Systems”,

pp. 95-97, New York,

Journal

of The Franklin

Institute