On integral manifolds for autonomous systems with slowly modulated motions∗

ON INTEGRAL MANIFOLDS FOR AUTONOMOUS SYSTEMS WITH SLOWLY MODULATED MOTIONS* P. R. SETHNA Department of Aerospace Engineering and Mechanics. University of Minnesota. MN, U.S.A. (Received

28 February

1980)

Abstract-Autonomous systems having slow amplitude modulated motions are studied. A theorem is proved which gives the existence and estimates of size of integral manifolds for such systems. The mathematical results are then applied to study a nonlinear system with follower forces.

1. INTRODUCTION

Nonautonomous systems with periodic excitations are known to have stable amplitude modulated motions that are approximately of the same period as that of the excitation with a modulation period which is much larger than that of the excitation. Furthermore, the period of the modulation does not appear to have any simple relationship to the excitation period. Such motions have been known to occur in systems having internal resonance [l, 21, and can also occur in periodically forced regenerative systems such as the van der Pol oscillator [3]. These phenomena are similar to beat phenomena of linear systems. In the case of systems with internal resonance, the beating seems to occur between a major harmonic of one motion and the fundamental of another. In the case of the van der Pol oscillator it is due to the beating between the natural limit cycle motion and the periodic force. What distinguishes these nonlinear phenomena from beat phenomena in linear systems is that the motions in the nonlinear case are asymptotically stable and can occur in basically dissipative systems. Such phenomena were studied in [4] and the theorem on which this work is based is given in [5]. In the present work we generalize the results in [5] for the case when the system is autonomous. The theorem given here is similar to the one given by Hale [6]; the difference lies in the fact that the one given here can be used directly in specific applications. What is notable in this study is that we predict slowly modulated motions, similar to those in the nonautonomous case, but in this case the motions do not necessarily arise from the beating between two motions. We will demonstrate, this with the aid of an example. 2. PRELIMINARIES

Quite general weakly nonlinear autonomous

systems of the form

i= Bz+sZ(z, E, a)

(2.1)

where z E R”, where E and a are real parameters, and where Z is a function into R”, can, by the method of variation of constants, be transformed to a desired form, as given below. Next we give this form along with several conditions that we impose on the system. Consider [6], t = &X(6,x, y, is,a)

(2.2)

b=d+eO(O,

(2.3)

x, y, E, a)

f = Ay + .sY(O,x, y, E, a)

(2.4)

*This work is supported by funds from the National Science Foundation under Grant NSF-ENG-7640030. 401

402

P. R. SETHSA

where E E R, a E R, 0
0, 0,4

(2.6)

and suppose Co(a) is a constant solution of (2.6) so that X,({‘(a), 0, 0, a)=0 matrix C(t’(a), a)=dXo({‘(a), 40, a)/2x has eigenvalues with non-zero real there exist an E, and vector functions f(6, E, a), g(0, E, a) of dimensions n and which for 0
parametrically

f(e,E,a),

Y = de,

of 8.

for all a; if the parts, then [7] m respectively, period f& and

E, a)

represent an integral manifold of (2.2H2.4). Furthermore: fi_mo If(e,E,a) -to(a)\ = 0 Fz lg(& s, a)1=0

for all 8 E Rk,

lal
The stability of the manifold is locally the same as that of the solution y =O, b =0 of the system i =C({O(a), ~b,

s’=A6.

(2.7)

A standard application of this theorem is to the autonomous case of the van der Pol equation in which case x and 0 are both of dimension one and y is absent. Our interest in systems of the form (2.2H2.4) lies in the case when (2.6) has a nonconstant solution periodic in ET.We first note that for (2.6) to have a nonconstant periodic solution the dimension of x has to be greater than one. We will discuss the case when the averaged equation (2.6) has a periodic solution through a Hopf bifurcation. Suppose that C(e’(a), a) has a pair of complex conjugate eigenvalues i,(a) and I(a), such that L(0) = ivo, Ii(O)= - ivo, vo#O and d;i(O)/da=u >O, and all the remaining n -2 eigenvalues of C have real parts bounded away from zero for all Ial
C(CO(a),a)z + h(z, a)

(2.8)

where Ih(=o([z(~) for each a, and z ~0 is a solution of (2.8). Then the following theorem [8] on Hopf bifurcation for system (2.8) holds. Theorem A. System (2.8) has a unique periodic solution z*(r, a), bifurcating from the zero solution. This solution depends on a constant K which is determined by the nonlinear function h(z, a). This unique periodic solution occurs for a > 0 when K O. Remark. An examination of the proof of Theorem A shows that the linear variational

On integral manifolds for autonomous systems with slowly modulated motions

403

equation of (2.8) for the periodic solution z* has (n - 1) characteristic exponents with nonzero real parts. Thus, based on Theorem A, we can conclude that the averaged system (2.6) has, for a sufficiently small, a # 0, a periodic solution t: = to + Z:(sv(a)t), with v(O)= vo, of period 2n, i.e. Z:(s + 2n) = Z:(s) for all s, and the variational equation of (2.6Xfor the periodic solution has (n - 1) characteristic exponents with non-zero real parts. We now appeal to a standard theorem on integral manifolds of (2.2H2.4) [6], to get our main result. 3. THE

MAIN

THEOREM

If system (2.2)-(2.4) satisfies all the conditions stated above and if (2.8) has a Hopf bifurcation when the parameter a goes through zero, then there exists a constant E, which depends on a, such that for all E, 0 <&SE, system (2.2H2.4) has a unique integral manifold M, which has a parametric representation. x=f(& 4,~ aA

y=g(R 4, E, a)

where f and g are continuous functions of 0,4, E and a, for &JE R’, 0 < sd E, Ial G ao, and are multiply periodic in 8 of vector period Q and periodic in 4 of period 2n. Furthermore, lim If@, 4, s, a) - <:@)I = 0

Cd0

lim 8--O

Igut

4,

E, alI=

0

uniformly in e E Rk, 4 E R. If the cylinder x= e:(0), y = 0, which is an integral manifold of t = &X0(x,0, 0, a),

y=Ay

is stable (unstable), then M is stable (unstable). Proof. The proof of the theorem follows immediately from Theorem 18.2 in [7] and the discussion preceeding the statement of the theorem. 4.

APPLICATION

We briefly indicate below how the above theorem may be applied to systems of equations that arise in applications. Consider a damped two-degrees-of-freedom dynamical system with follower forces: Aij+ M+Cq=g(q, q, IO (4.1) where q E R2, A, B and C are 2 x 2 constant matrices where A and B are symmetric and positive definite, and C = K + oF, K = Kr and F, which represents follower forces, is not in general symmetric. The vector function g will be assumed to be a polynomial with the lowest order terms homogeneous cubic polynomials in q and Q. The parameter u is related to follower forces and p is some other parameter. Suppose for g=O, system (4.1) has two pairs of complex conjugate eigenvalues A,.,= 6, kiwi, &= 6 2 f iw, where Sj and wri= 1,2 are function of 6. Furthermore, we assume that for u = uc,, S/a,,) = O,i = 1,2 and w,(cr,,)= wjo # 0 and dd,(a,,)/da > 0. Let .s=e-cr,,, then system (4.1) can be transformed, by using real Jordan canonical variables y, into the form

dy ,,=Ly+W,

/4

where L,=diag

(L(l), L’2))+O(s2)

P. R. SETHNA

404

and dwj j= 1, 2. wj=,a o=ocI

&, ds, J = da ,_’

The analysis is valid for y small. To make this explicit, let y=sl”x s~‘~~(x, e, p), then (4.2) takes the form: * =

L,x +q(X, E, /cl

and let k(c1’2x, p) (4.3)

which is of the form (2.1). To reduce (4.3) to the structure af system (2.2)-(2.4), we introduce the variables a,, a,; I,+~,ti2 by the transformation x=(uI sin +r, a, cos JIr, a2 sin J12,a2 cos ti2)r, and then using a variation of constant procedure we get a system of equations of the form u2~$1,+2))

ijii4Ecs;al+Aju19 dj=WjO+E($+B,(u19

023 $13 $2))

i=l,

2

(4.4)

where Ai and Bj are periodic in $i and ti2 with period 2x in each variable. If we introduce new variables f3,= JldwjO,j= 1,2, then in the new variables a,, u2, 8, and 8,, (4.4) take the form (2.2)-(2.4) with the functions corresponding to X and 0 periodic in tI1, 8, with periods 27r/w,, and 27r/w,, respectively and the variable y=O. Let the ratio w,,-,/w~~be any number not near 1, l/3 or 3. Then averaging over 6, as in (lS), and using p1 and p2 for the variable representing the average of a, and a2, respectively, we have the averaged equations: (4.5) where Cjk are functions of the parameter ~1.The constant solutions of (4.5) for p1 #O, p2 #O, if they are represented by r1 and r2, satisfy S;+ i

Cj&=O

k=l

(4.6)

where lj=r,b),

j-

1, 2.

The autonomous averaged system (4.5) will have a Hopf bifurcation when the variational equation corresponding to the constant solution r,, r2 has eigenvalues with pure imaginary parts for some value of p = ~1,. The conditions for this to occur can easily be shown to be a=0 G,c22-c12c21>0

(4.7)

where a= C, lr: + C22r3 and where C, and rp i, j= 1, 2 are evaluated at /A=po. The parameter a is the parameter occurring in the theorem. If Ial>0 and c,,c22-c12c21~0~

(4.8)

for small values of a a periodic solution of (4.5) occurs and the theorem given in Section 3 is applicable. The physical and geometric interpretations of phenomena associated with this example, when interpreted in the scaled Jordan form coordinates X, are as follows. Suppose the periodic solution of (4.5) can be represented as follows: P1 =rl +YAst) P2 = r2 + Y2(st)

On integral manifolds for autonomous systems with slowly modulated motions

405

where rr and r2 are the solutions of (4.6) with p = p,. and where yr and y2 represent the nonconstant parts of the periodic solution. Then the vector variable x has the following parametric representation: Cr, +rk#~)l sindf, x1 CT1

+v1wl~o~$1 +0(E)

x3

[r2

+y2M)J

sin

[1x4

cr2

+

cos

x2

=

Y2Wl

(4.9)

Ifi $2

withO<4<2rt, 0<@,<2n, and 0~1//~<27r. Geometrically, the theorem asserts that the system has an integral manifold in the neighborhood of the surface (4.9) parameter&d by the three angular variables 4, rjl, J12,and if the periodic solution is stable the manifold is stable. Physically, the system has “modal” motions with amplitudes having average value rl and r2 which are the solution of (4.6) and slow modulations with a time scale ELThe amplitudes of this modulation are determined by y1 and y2, which quantities can be computed explicitly. As mentioned in the introduction, the phenomena associated with such systems, although beat-like in character, are not necessarily due to beating between two periodic motions since the basic modal frequencies w,,, and wzOin this example, can be any numbers, so long as the ratio w10/w20 is not near 1, l/3 or 3. Ackno&dgemenrs-The author would like to acknowledge the assistance of Mr. A. K. Bajaj in the calculations associated with the example in Section 4.

REFERENCES 1. P. R. Sethna, Vibrations of dynamical systems with quadratic nonlinearities ASME, J. appl. Me&. 32 576-582 (1965). 2. T. Yamamoto and K. Yasuda, On the internal resonance in a nonlinear two-degree-of-freedom system, Bull. JSME 20,140,168-175 (Feb. 1977). 3. J. J. Stoker, Nonlinear Vibrations, Chapter V. Interscience Publishers, New York (1950). 4. P. R. Sethna and A. K. Bajaj, Bifurcations in dynamical systems with internal resonance, ASME, J. appt. Me& 45.4.895-902 (Dec. 1978). 5. P. R. Sethna and A. K. Bajaj, Bifurcations in Non-linear Oscillatory Systems, Proceedings ofrhe MI1 Inrernarional Confirence on Non-linear Oscillarions. pp. 621-626. Prague (1978). 6. 1. K. Hale. Oscillations in Non-lineor Systems, p. 161. McGraw Hill, New York (1963). 7. Reference [6], page 160. 8. S. N. Chow and J. Mallet-Paret, Integral averaging and bifurcations, J. Df@ Eqnr. 26, 1, 112-159 (1977). Resume On etudie des systemes autonomes ayant des mouvements lents d’ampl i tude modulee. On demont re un theoreme qu i donne l’existence et des estimations de la taille des reproductions On applique ensuite les integrales de tels systemes. resultats mathematiques pour etudier un systeme non lineaire avec des forces transmises. Zusanunenfassunq: amplitudenmodulierten BewegAutonome Systeme mit langsamen, ungen werden untersucht. Ein Theorem, das die Existenr und GrCssenabschgtzungen fiir integrale Mehrfachheiten solcher Systeme angibt, wird beuiesen. Die mathematischen Ergebnisse werden dann verwendet, urn ein nichtlineares System mit Folgekre’ften zu untersuchen.