Some properties of the generalized Van der Pol equation

General investigation Let us consider the following generalized V an der Pol equation [1]: where CO, V, q are constants, and £ is a sm all param eter.. 1.1 we first transform it into th

S O M E PRO PER TIES OF T H E G EN ERALIZED VAN der PO L EQ U A TIO N

NGUYEN VAN DAO (HANOI)

1 General investigation Let us consider the following generalized V an der Pol equation [1]:

where CO, V, q are constants, and £ is a sm all param eter W hen q — 0 we obtain the well-

known Van der Pol equation, for which there exists only one stable stationary oscillation

o f th e t y p e

To investigate the vibrations described by Eq (1.1) we first transform it into the standard form by means of the formulae

The transformed equations are

~ = fứ[l - (ứ cos99 + <7cosw)2]sin2 9?, (1.4)

— £ ứ [l — ( ứ COS 99+ g COS V / ) 2] sin 9? COS 9?,

where ip = 99 — CO/.

By employing the averaging method of non-linear oscillations [2] we obtain in the first

approximation the following equations for a and rp:

= e a /iO , v ),

Eqs (1.5) have a zero solution a = a 4tặ =3 0 and the other non-trivial stationary solutions

a = ^ 0 detei mined by the system

The equation of variation for the zero solution is

d i a dt

Hence, the zero solution is stable if ( e > 0):

The equations o f variation for the non-trivial stationary solutions have the form

dòa w

(1.8)

d5y>

~dt

The characteristic equation o f this system is

X2

Therefore, the conditions o f stability for nontrivial solutions are

(1.9)

We shall consider n o w various resonant cases that correspond to the concrete values o f

2 The Resonant C ase V = 3co

In this case the averaged equations (1.5) tak e the fo im

da

(2.1)

= - e ^ < ? s i n 3 v ,

SO

/ ĩ ( í’ ,y>) = Y ' 8 ~ 4 " t' 4 y>’

(2.3)

Since

1 fl2

/ i (0,yO = ỳ - 4 9

then the zero solution a = 0 of the system (2.1) is stable if

q < Ý 2 , q > \ ' 2 ,

- | / 2 < q < ị / ĩ .

The non-trivial stationary solutions a+) o f the system (2.1) are determined from

the equations:

1 The first subcase

a ị - 2 q a + + 2 q 2 - 4 = 0 ,

h e n c e

As

t h e n t h e c o n d i t i o n s o f s t a b i l i t y ( 1 9 ) b e c o m e

We have

(4Ệ),

branch a+ = <7 + ^ 4 - # * is stable.

2 The second subcase

s in 3 y * = 0, cos3y>* = — 1,

hence

3v* = (2« + 1)tt, a* = - ạ ± v / 4 - ạ 2.

Consequently, when q < 0, the branch a+ = — q — Ị/ 4 — q 2 is unstable and the branch

a + — — <7+ y/4- 0 2 is stable.

is presented in Fig 1 The bold lines correspond to the stable regimes of vibrations With

the increase of the parameter q the amplitude o f vibrations changes along the lines

BCDEGHK. At the points c (q = — |' 2 ) and G (q = 2) there is a jump in the amplitude

An analogous phenomenon is also observed when decreasing the parameter q Thus, in

the case V = 3co the amplitude of stationary vibration o f the generalized system is bigger than that o f the initial system.

3 Vibrations in the case V = OJ

N o w , the averaged equations (1.5) are o f the fo rm

(3.1)

therefore

W e have

a dip

A (a, v) = T1

/ , (0, vO*:

I - 2 + - y + q 2 + a q c o s y - — c o s 2 ^ l ,

/ l ( 0 , v ) , = - ^ - | - 2 + g2- - y C o s 2 V, j ,

It will readily be seen that the zero solutio n a = 0 is stable if/ 1 ( 0, y>*) < 0, o r i f q < —2 and q > 2.

T he first stationary solution a *, yj* is determined by the equations

sinyj* = 0 , a ị + 2qa* + q 2- 4 = 0 ,

fr o m w hich w e obtain

= — <7±2, y>* = 2wr, where /1 is a n integer.

Since 1 - ^ - 1 = 1 1 = 0, then the co n d itio n s o f stability o f the stationary solu-

\ d y /* \ d a u

tions also take the form (2.4) We have

where the upper sign formula corresponds to the straight line a+ = - q + 2, while the lower

one to the straight line a+ = - q - 2 , from which it follows that for q > 0 the straight

line a+ = —q — 2 is unstable and the line a+ = —q + 2 is stable.

The s e c o n d stationary value a+, V i is determ ined as the solution o f the equations

w h ere t h e upper sign formula corresponds to t h e straight line a+ = <7 + 2 , while th e lo w e r

o n e to th e straight line a+ = q — 2.

F o l lo w in g the inequalities (2.4) w e con clu d e that the line Ơ* = q - 2 is unstable and

th e li n e = q + 2 is s t a b le w h ere q < 0.

and therefore the first condition of stability (1.9) is not satisfied and the stationary solution obtained is unstable.

The branches of the amplitude-curves that correspond to the stable stationary vibra­ tions in the case V = a> are presented in Fig 2.

siny* = 0, a \ - 2 q a + + q 2 — 4 = 0 ,

h e n c e

y* = (2n + 1)tt, a+ = q ±2.

F o r th e c a s e u n d e r c o n s id e r a tio n w e h a v e

In this case, we have

4 Stationary vibrations in the remaining cases (v = For these cases the averaged equations becom e quite simple

1

d a

dt

ea

T

% 0 '

dt

It is easy to sh o w that the zero solution a = 0 is stable if q2 ^ 2, and unstable if —1/2 <

< q < \ 2.

The nontrivial stationary solution

is stable for the values (ỹ2 < 2 Thus, for q2 < 2 the considered system vibrates with the

a ị —4 - 2q 2 . For q 2 > 2 the damping of vibrations takes place [1] (Fig 3).

5 Experimental Results

The influence of the parameters q and V upon the amplitudes of stationary vibrations has been investigated on the analog computer Meda 41TC The operating amplifiers work

in that regime where the tensions at the output and the input have to be changed within

the range o f - 1 0 to + 1 0 volt C onsequently, for th e normal work o f the computer it is necessary to transform the Eq (1.1) in to the «machinery» equations.

T o that end we introduce the notations

z = y 2 , y = X + q c o s v t ,

x - i - , i - ± , y - J L , z J L , q J L ,

-*■*> / * » z*> <7* > are the values o f X, z, q that correspond to 10 volt.

Eq (1.1) can now be rewritten in the form

- X = (’S k ^ X - e X + e k ^ X Z ,

where

k = i*- K11 —

Y = k 21X + k 22Q c o s v t , k 2l = , * 2 2 = — Thus, the «machinery» equations take the form

z = y 2,

K = k 2] X - \ - k 22Q ^ o s v t

.y* = 6, z* = 36, k n = 1, k 12 = 36, /c2i = 3 * * 2 2 = - y • The analog circuit used in this study is sh ow n in Fig 4 The results o f experim ents are presented in Figs 1, 2, 3 by round points.

F 4.

Ì J H a le , Oscillations in nonlinear systems, N ew York—London 1963.

2 N N B o g o liu b o v , Yu a M itr o p o lsk y , The asymptotic methods in the theory o f nonlinear oscillations,

Moskva 1963.

S t r e s z c z e n i e

P E W N E W L A S N O S C I U O G Ĩ L N IO N E G O R 0 W N A N IA

V A N d e r PO LA

N ieliniow e uogĩlnione ukfady zawsze posiadajạ bardzicj rốznorodne wlaắciwosci w porĩvvnaniu

z wyjsciowymi ukJadami Stw oizcnic nowych, bliskich do znanych, nieliniowych ukỉadĩvv i zbadanie ich wlasnosci ma wazne znaczenic w danym artykule zbadano drgania nieliniowych ukỉadỏvv, opisywanych uogolnionym rốwnanicm Van dtT Pola w postaci podancj przez J Halego [1J R ozpatrzono stacjonarne drgama w r6znych warunkach rezonansu i ich statecznosc.

Okazujc siẹ, ze drogạ odpow icdnicgo wyboru parametrow badanego ukladu m ozem y kierowac amplitu­

de stacjonarnych drgan od zcra do pcwncj okreslonej wartosci Teorctyczne wnioski sạ sprawdzane przez seriẹ doswiadczeri na maszynie analogowej ME D A 41TC Tcorctyczne i doốwiadczalne wyniki sạ w pdnej zgodnosci.

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Received July 25, 1975.

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to a shock load

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