On the analysis of hopf bifurcations

002fk7225/83/030247-16$03.WO 6 1983 Pergamon Press Ltd.

hr. I EngnR Sci Vol. 21. No. 3, pp. 141-162, 1983 Printed Bntam

inGreat

ON THE ANALYSIS OF HOPF BIFURCATIONSi K. HUSEYIN and A. S. ATADAN Department of Systems Design, University of Waterloo, Waterloo, Ontario, Canada

Abstract-Theoscillatory instability and the family of limit cycles associated with a general autonomous dynamical system described by n nonlinear first order differential equations and an independently assignable scalar parameter are examined via an intrinsic method of harmonic analysis. The method is essentially a variation of the classical method of ,“harmonic balancing”, and is designed to eliminate the drawbacks and shortcomings associated with the latter. Inileed, the new approach yields consistent approximations for the nonlinear dynamical bifurcation problem under consideration through a systematic perturbation procedure. It has thus been possible to derive explicit, formula-type expressions for the post-critical family of the periodic solutions, frequency of oscillations and the path which represents the family bifurcating from a flutter-critical point on an initially stable equilibrium path. The results are available to be used directly in the analysis of specific problems which fall within the scope of the formulation, without actually performing much analysis. Two illustrative examples are provided. 1. INTRODUCTION

theories concerning discrete autonomous systems have developed in two main directions. In one line of research only potential systems are considered; the second direction involves non-potential systems as well, and is more comprehensive. The behaviour of a potential system can be described by an all-embracing potential function (e.g. the total potential energy of conservative systems) which provides a convenient basis for nonlinear analyses, particularly for the development of general theories. A relatively recent development, for example, concerning the equilibrium and stability behaviour of the multiple-parameter potential systems is available as a monograph[l]. Catastrophe theory basically parallels this quantitative approach 121. If a system involves non-potential forces, however, a dynamical formulation may be essential. As a matter of fact, such systems can exhibit dynamical instabilities as well as static befurcations while conservative systems are capable of the latter only. One way of exploring such more genral response characteristics of discrete systems is to consider the governing first order ordinary differential equations in a state-space formulation. If such a one-parameter family of nonlinear autonomous equations forms the basis of a general study, one first notes that the salient features of such a system are the equilibria and limit cycles (or it was tacitly assumed so for a long time) so that as time tends to infinity the asymptotic behaviour of the system approaches one of these steady states. More recently, however, a third possibility has emerged; it has been discovered[3] that even innocent looking equations (e.g. of order 3) can describe a complicated motion which appear to be random. Such “chaotic motions” are associated with “strange attractors” and are receiving increasing attention both at experimental as well as theoretical levels [4,5]. Our understanding of the behaviour of dynamical systems will certainly be greatly enhanced with further studies concerning this rather puzzling phenomenon. On the other hand, the equilibrium configurations of an autonomous system and their stability have been explored rather fully with regard to one-parameter as well as multiple-parameter situations. It has been demonstrated, for instance, that a divergence point on an equilibrium path of a one-parameter system, where the Jacobian becomes singular, can generally be identified either as a point of bifurcation or a limit point[6]. The former is classified further as an asymmetric, stable symmetric or unstable symmetric point of bifurcation[7,8], in analogy with conservation systems. At compound divergence points, interactions between modes occurs, resulting in significant imperfection sensitivities [8]. For a system under the influence of several independent parameters, the concepts of general and special critical points, which were first introduced in the analysis of potential systems, are extended to autonomous systems[9] to identify and characterize critical divergence conditions. In fact, this classification has been instrumental in obtaining distinct forms of the equilibrium surface explicitly[lO]. It has thus NONLINEAR bifurcation

tA part of this paper was presented in CANCAM 1981,Moncton,N.B., Canada. 247

248

K.HUSEYINandA.S. ATADAN

been shown analytically that the phenomena like Fold, Cusp, etc. catastrophes are exhibited by these more general systems as well. This development represents the first direct generalization of elementary catastrophes to non-potential systems. If the critical point on an equilibrium path is a flutter point at which the real part of a pair of complex conjugate eigenvalues of the Jacobian crosses the imaginary axis with non-zero velocity, while the rest of the eigenvalues continue to have negative real parts, an oscillatory type instability occurs which leads to a family of limit cycles. This phenomenon is known as Hopf bifurcation, and following Hopf [ll] has been studied by several authors. A book[l2,13] devoted to various aspects of Hopf bifurcation deals with both discrete and continuous systems, and contains many references. Recent publications include a comprehensive article on factorization theorems and repeated branching of solutions [ 141, applications of multiple time scaling[l5] and method of averaging[l6], bifurcations in flow-induced oscillations in terms of a finite-dimensional [ 171as well as an infinite-dimensional analysis [ 181,generic bifurcations [ 191,a survey of bifurcations of dynamical systems [20], and bifurcation formulae derived from center manifold theory[21]. A recent book by Hassard, Hazarinoff and Wan deals with the theory and applications of Hopf bifurcation where certain formulae are produced on the basis of Poincare normal form and the numerical evaluation of these formulae are discussed[22]. Essentially, the phenomenon of Hopf bifurcation is related to non-linear oscillations, and the methods available in that context such as averaging, multiple time scaling, etc. can be adopted to analyze this type of bifurcations. Another approach is described as the method of harmonic balancing which is often criticized for not yielding consistent results unless some information about the behaviour of the system is a priori known[23,24]. Allwright[25] studied the Hopf bifurcation phenomenon in a feed-back control system through “a second order harmonic balance technique” in which he assumes solutions involving the first and second harmonics and applies the traditional balancing technique. Harmonic balancing is a relatively simple method, and in this paper a modified version of it is introduced to overcome the observed inconsistencies. The variation can best be described as an intrinsic application of harmonic balancing which yields consistent results systematically without requiring advance information about the solution. The formulation and analysis parallel the multiple-parameter perturbation technique[l] applied to the analysis of the equilibria, and may, therefore, be considered as a step in the direction of unification of the theories underlying dynamical and statical bifurcation analyses. Another important feature of the present paper is that, in addition to describing a conceptually simple method of analysis, it also leads to explicit results in general terms via the same method. These results are readily available to be used directly in the solution of specific problems in mechanics, network analysis, chemical processes or in any other discipline without much analysis. Indeed, if the governing equations of a specific problem fall within the scope of the formulation here, or if they are brought within that scope through appropriate transformations, then the family of periodic solutions, bifurcating path from the critical point and frequency-parameter relationships can all be determined by evaluating certain derivatives and substituting into formula-type results given here. This is illustrated with the aid of two examples. 2.ANALYSIS Consideran autonomous system represented by a one-parameter family of nonlinear ordinary differential equations

where z is the n-dimensional state vector and 77is an independent real scalar parameter. The nonlinear functions Zi(zj, q)(i, j = 1,2,. . ., n) are assumed to be real analytic in the state variables zi and 7, which span an (n + 1) dimensional Euclidean space (En+,), in a region (R) of interest. It is further assumed that the system (1) has a single valued equilibrium solution f(T) in R and all the eigenvalues of the Jacobian,

(2)

On the analysis of Hopf bifurcations

249

evaluated on this path, have negative real parts for n < 7, or n > nlcwhere 7, is a critical value of 7. Throughout the theory here, it is assumed without loss of generality, that the eigenvalueshave negative real parts only for 11< 77,. As n is increased from an initial point, a critical point on the path may be reached where at least one real eigenvalue of the Jacobian (2) vanishes and, with a further increase of n, becomes positive. This phenomenon is known as divergence instability and it has been explored in [6-81. Another type of instability occurs at an essentially different critical point on f(n) when the real part of at least one pair of complex conjugate eigenvalues vanishes and then becomes positive as n increases. At this critical point, a family of periodic orbits bifurcate from the equilibrium path. This particular phenomenon is known as Hopf bifurcation and it will be studied in the following analysis. To facilitate the analysis, a new coordinate system is attached to the equilibrium solution

f(v) by z=f(d+y.

(3)

Introducing the transformation (3) into the system (l), one obtains $Y

= Y(Y,

9)

(4)

such that Y(T)

=

0

(5)

is the trivial solution. It then follows that Y(0, 7) = 0 which yields Y(0, q) = Y&O, q) = Y&O, q) = . . * = 0

(6)

where the subscript 0 denotes differentiation with respect to 7. By means of a nonsingular transformation y = TX,

(7)

(4) can be transformed into

ix=

X(x, 77)

(8)

which can also be expressed in the scalar form as ~Xi

=

Xi(X’, 77)

in the region R. The transformation matrix T is chosen such that the Jacobian [aX(x, n)/ax] evaluated at the critical point C has the block-diagonal form of $X(x,

n)/, = block diag. [D, C,, C,, . . ., R,, RZ, . . .]

(10)

with real elements. The blocks 0, C, and R, are given by

D=[-“w, ;] c*=[-“:,

3(4=1,2,...)

(11) (12)

250

K. HUSEYIN and A. S. ATADAN

and R, = [a,](~ = 1,2,. . .)

(13)

where (Ye< 0 and (Y,< 0 for n < 7,. In order to obtain the block-diagonal form of (lo), the transformation matrix T can be chosen as T = [dR d’; . . ., c;, c;, . . . / . . ., r,y,. . . .]

where superscript R and I denote the real and imaginary parts of the eigenvectors d and c,. The eigenvectors d, cq and r, correspond to the imaginary, complex conjugate and real eigenvalues which are associated with the blocks D, C, and R,, respectively. It is easy to see that, under the transformation (7), the properties (6) are carried over the vector function X as x(o,q)=x~(o,

(1%

q)=x~(o,q)=...=o.

The system of differential eqns (9) can be written as

-$x=J(q)x

+X(x, q)

(16)

where J(n) is the Jacobian along the equilibrium path f(n) and ~i(X’, 7) represents the rest of the terms. Expanding the Jacobian J(n) into Taylor series at the critical point C one obtains

where Xij is given by (10) and Xii”= [a2Xi/JXia~]1,~etc. Let h&q) = ~(7) 7 io(q) be the pair of eigenvalues of 5(v) that crosses the imaginary axis at 77= Q, with non-zero velocity (Fig. 1) as 77is increased. Then for 77= 77,one has A,,~(?~) = T

iw(q,) = T iw,,

(18)

and, in addition, the real parts of A,,*(r))should satisfy (19)

w

w

-1\

t

WC

I

Y

WC

I

(a)

(b) Fig. 1.

On the analysis of Hopf bifurcations

under the basic assumptions the Appendix)

251

of Hopf. It can be shown that this condition is equivalent to (see

XI 10 + x220+ 0

(20)

which will be employed later in the analysis. It is then concluded that if the relation (20) is satisfied, a family of periodic solutions bifurcates from the equilibrium path

x’( 7) = 0,

(21)

at the critical point C(X’ = 0, n = 7,). The bifurcated periodic solutions of (9) are assumed to be in the parametric form given by

xi=x’(t; E) 17 =

(22)

de)

(23)

such that x’(t; E) = x’(t + T; E)

(24)

where T is the period. The solutions given by (22) and (23) are substituted back into the differential eqns (9) leading to

$X’(t;E)

E

Xi[X'(t;

E), T(E)]

(25)

where 6 is the perturbation parameter. The periodic solution x’(t; E) may be represented by a Fourier series as X’(t; E)= 2 [pi,(E) COSmU(e)t+ rim(E)sin mti(rZ)t] m=O

(26)

where I;.O( c) G 0.

(27)

The coefficients pi,(t) and rim(E)are the ith entries of the amplitude vectors corresponding to the mth harmonic. Without loss of generality it is assumed that r,,(c) = 0

(28)

since the system of differential equations given by (25) are autonomous. Due to the same property the solutions (26) can be transformed into 2P-periodic series M

Xi(7; l) = x [pi,(e) COSm7 + rim,(c)sin m7],

m=O

(2%

by means of the time scaling 7 =

wt

(30)

which also transforms (25) into o(&i(r; IJESVol.21.No. 3-E

E) = xJxj(7; E), T(E)].

(31)

K.HUSEYINandA.S. ATADAN

2.v

One can now generate perturbation to the parameter l, as

equations,

by successive

differentiation

of (31) with respect

&xi t wp; = x,x’,

(32)

OYx,i+ 2w’xl t 0,x: = Xij,,Xi’X”’ + 2XijyiioXi’77’ t xijxy,

(33)

Wf’rx;+ 3of’x; + 30’xy + W,xr = Xijk,Xj’xk’xl’ t 3Xjjk”jk0Xi’Xh’77’ t 3XijkXi”Xk’+ 3Xi,““x”(v’)2 + 3Xij”X’“77’ + 3xij,x”?7” t XijX””. . . )

(34)

where w”’= (d3w/dc3)/,, xi = (dx’/dr)l,., X: = [a(dx’/dr)/&]l,, Xijo= [a’X;/a~j&7](,, etc. It is noted that the coefficients Xi09 Xjoo, Xi000do not appear in the perturbation eqns (32)-(34) due to (15). The coefficients Xi,, Xijk, Xijo etc. will be called system coejicients. The derivatives of Pim(e), r;,,,(E), W(E) and n(e) are obtained by solving first, second, etc. order perturbation equations where a perturbation equation is called kth order if (31) is differentiated k times with respect to E. These derivatives are then used to construct ordered approximations for the coefficients

pi,(~) =

Pimp

+

~p’:me’+. ,.

P:,E +

rim(e) = rimet rjm.c +irbc2

(35)

(36)

t ....

$E‘ +...

(37)

1 O(E) = w, t W’E+ -6J”c2 t . . . 2

(38)

q(e) =

77, +

$6 +

and

where prime denotes differentiation

with respect to E and pimc = rim, = 0

(39)

for all i and m since, at the critical point C, ~~(7; 0) = 0. The solution of the kth order perturbation equation can be written as M

x”k)( 7) = m?O[pi: cos m-r + r$) sin mT]

(40)

where xick)(r) is the kth term of the Taylor expansion:

+;xiq7)e2 t .....

xi(7; E) = Xi’(T)6

(41)

In order to obtain the solutions given by (40) for (k = l), the periodic substituted into the first order perturbation eqn (32) leading to

rPinl M sin mr t w, x m m=O

11_ Pl+l,m Pm

4, 4,

1 1 r&

rb+l,m Cm

cos mr

solution

(29) is

253

On the analysis of Hopf bifurcations

-0

=

I I

wc

1

O

- WC 0 I _--____+_-_-~-; I I ,

2

% -id,

aq 1

P;,

rim

’

Pm I

Pq+l,m

I_______!___ 0

rim

i1

m=O

I

P;,

1 asJ

cos mr + i Y’qm m=O , r9+lJfl

sin m7

I

rsm -

P&l

(42)

where the subscripts q and s correspond to the complex conjugate and real eiganvalues, respectively. The first derivatives of the Taylor expansions of (35) and (36) are determined from (42), by setting the constant term (m = 0 case) and the coefficients of cos rn~ and sin mr (m 2 1 case) to zero for each m, as follows ph = pi, = r:,= 0 (m 2 1)

(43)

p;, = - r;t

(44)

p;, = r;, = Pi71= P&l., = p:1 = 0

(45)

where (28) is employed. At this stage, it is noted from (43)-(45) that, pll or y21is suitable candidate for E. Indeed, replacing E by pll and introducing (43)-(45) into (40) for (k = I), the solution of (32) is expressed as

sin 7

(46)

where dot denotes differentiation with respect to p ,, which replaces E as the perturbation parameter. The solution xi(‘)(~), for an arbitrary k# 1, is obtained in the same way as the solution ii(~) except that the solutions of the former (k- 1) perturbation equations and the condition given by (20) have to be taken into account. With this in mind the solutions of (33) and (34) can be determined as .. r12 ., r22

fh2

:I [I lj22

+

&2

Pq+l,2 tis?

sin 7 +

cos

2rt

?@ .,

sin 27,

(47)

rq+t,2 .. Tr2

sin37

(48)

254

K.HUSEY1NandA.S.ATADAN

where

612

=&x,2r&,,+4&2) c

On the analysis of Hopf bifurcations

1 $13 = - sJ(3 c

VL +

Y2,)

&,=&q-34 c - (9w:+ a’,- ofJe, + 2a,w,e2 ... pq3 = (90;+ (I’,- w;)2 + (2qpJJ2 (9& + a”, - 4)e2 + 2ff,o,e, ... pq+,,3= -(9w2, ... ps3=-

(Y,Y~,+ 3~~2 9&)2,+4

...

r 13 = $3

VI -

c

y22)

(9&+ a’,- 6$e3 - 2cY,o,e4 rq3 = (94 + a’, - CiJy + (2qpJ2 ...

(90:+ a’,- wfJe4+ 2ff,o,e3 ... rq+,,3 = (90; t c-i’,- cog2 + (2q4J2 ... rs3

=

3&V,, - asvs2 SW’,t (Y’,

x,,

c$lt (

,,,,

-

x4+,,**

-cWC 1 %ffq )

2

-%xq,2 4

t

~xq+,,,2 UC

)

K. HUSEYIN and A. S. ATADAN

2%

During the same procedure

and

are also obtained. Hence the parameter-amplitude (37) and (38), are

and parameter-frequency

relations, given by

(52)

77(Pll) = % +;i(P,,)’

1 ..

O(P,d = WC+-@(Pld2. 2 where +j and L;i,which are assumed to be nonzero, Introducing now

(53)

are given by (50) and (51), respectively.

77=77,.+/J

(54)

so that the critical point C becomes the origin of the x’ - F space, one has

(55) where

The periodic

solutions

are then constructed

E = PII. It should be noted that the imaginary

(46)-(48) into (41) and setting

parts of the eigenvalues

d -w(n) dq in addition to Hopf’s condition

by substituting

I c

~,,~(n) may satisfy

= w” = 0

(19). It is shown in the Appendix

that w0 can be obtained

in

On the

analysis

251

of Hopf bifurcations

terms of the system coefficients given by (A4) which upon introducing

into (56) yields

Xl20- x*,0 = 0

(57)

and (51) takes the form cli = -

$a,,+cl*,).

(58)

At this stage it is interesting to examine the slope (dwlda), which can equivalently be written as do dodT -=-da dvda’ at the critical point C. Hopf’s condition (19) together with (56) imply that the paths that the eigenvalues follow have zero slope at C as, for example, in Fig. l(b). On the other hand, if (56) is not satisfied, the eigenvalues h,,*(v) cross the imaginary axis in (o - (w)plane with nonzero slope. If g and/or 3 vanish, one has to proceed with higher order perturbations in order to obtain first order approximations for the parameter-amplitude and parameter-frequency relations. It can be shown through this procedure that the third derivatives of p and w are necessarily zero, and the fourth derivatives are required for a first order estimate. This, however, becomes rather cumbersome in this generality although the method remains systematic and readily applicable to specific problems. It is noted that the post-flutter path (55) bifurcating from the fundamental equilibrium path x = 0 at the critical point C is often called supercritical (subcritical) if @> 0 (fi < 0). It can be shown that the periodic motions represented by the supercritical (subcritical) path (55) are stable (unstable), and all the transient solutions approach (leave) these limit cycles as t tends to infinity. In Fig. 2 illustrating these phenomena, solid (dashed) lines represent stable (unstable) states. 3. EXAMPLES 1 An electrical network

Consider the electrical network given by Hirsch and Smale[26] which is shown in Fig. 3. It is assumed that the capacitor C and the inductor L are linear elements (of unit values) whereas the resistor R is a nonlinear element which is described by hdk, 17)= (id3 - &

(59)

where vR, iR and 17 represent the voltage, the current and a variable parameter (i.e., temperature) of the resistor, respectively. P

I

:I i f

C

C

P41

(a)

(b)

Fig. 2.

I

‘\

PII \

\

\

\ I

K. HUSEYIN and A. S. ATADAN

Fig. 3.

The state equations can be obtained as

where iL = z’, u, = z2 and zi = dz’/dt(i = 1,2). The differential eqn (60) is recognized as a special case of Lienard’s equation. Since the Jacobian of (60) is in the canonical form and zi = 0 yields zf = 0 for all 7, one can replace zi by xi, and x’(q) = 0 defines the initial equilibrium path. One also can apply the time scaling T = wb,as in (30), and write the state equations in the following form

(61) where X: = dx’/dT and o is the frequency. The eigenvalues of the Jacobian

162) can be determined as

(63)

where ]qj< 2. Clearly Q = 0, and r) = O-t k. It is observed that h,,2(p) have negative (positive) real parts for p < 0 (CL> 0), indicating stability (instability) of the equilibrium path for p < 0 (P ’ 0). The nonzero system coefficients associated with (61) are readily determined as

XII0= 1 &ii=-6

(65) (66)

where (65) implies that the Hopf’s condition (20) is satisfied. Then, the roots A,,*(p) cross the imaginary axis with nonzero velocity and in fact, since (57) is also satisfied, they cross it with zero slope (see Fig. 4). It follows that a family of 27r-periodic solutions of (61) bifurcates from the fundamental equilibrium path at the critical point C.

On the analysis of

259

Hopfbifurcations

w

Fig. 4,

From (50) one immediately obtains

which indicates that the bifurcating periodic solutions are stable (Fig, 2(a)). The equations of these limit cyctes can be constructed readily on the basis of (29) as 3 cos r-“jZ@“)%n3?

*

,“-I ==PiI

‘I

[

-sin~-3!$piJzcos3~

3

The curvature of the amplitude-frequency relation is given by (51) in the theory which yields j=()

(691

in this particular case. Using the results (67) and (69) in (55) and (53), respectively, one obtains

(7% and 0= 1

(71)

since w, is given by (64). The supercritical (post-flutter) path (70) represents the relation between the amplitude of the first harmonic of the inductor current and the parameter (i.e. tempera~re). 2 A mathematical

model

This model is presented as an additional illustration of the applicability of the results obtained in the theory directly. Consider the system of differential equations 1 xt= x2+ apx’ - a(x’)” - ax’(x*)*

2 Xf = - x’ + a/Ax*- (Y(x’)2x*- *(X*)3 x: = -2x3 + 3cuJA.x’ - &‘)-l

(72) (73) (74)

260

K.HUSEYINandA.S. ATADAN

where (Y< 0. Since the system given by (72)-(74) is autonomous, its periodic solutions can be made 2n periodic by scaling the time as 7 =

wt.

(75)

Introducing (75) into (72)-(74) and writing the resulting differential equations in the matrix form one has

Note that the system of differential equations given by (76) are already in the standard form as far as the general theory is concerned. Therefore the transformations introduced in the general analysis are not required. It is also observed that in this example xi = 0 implies xi = 0 for all y; therefore, xi(p) = 0 is the equilibrium path. The eigenvafues of the Jacobian (77) I are given by

h&L) = - 2 + 3(Y/.l

(79)

and A,,*have negative (positive) real parts for CL> 0 (II < 0) which indicates a reversed stability property for the equiiibrium path. The nonzero coefficients of (76) are

x,, = -x2,

=

0,

=

1

x33=ff3=-2

(81)

x,Io=x220=~

@2)

x,,, = 3a X,,,,

=

X2222

=X,,,,

(80)

(83) =

-6a

x,,,, = x,,,, = - 2CY.

(84) (85)

It is observed from (82) that the Hopf’s condition given by (20) is satisfied, indicating that the periodic solutions bifurcate from the equilibrium path x’(p) = 0 at p = 0 which can be obtained from the derivations of Section 2 as

(86)

Following the theory (see (49)) &=((j=O

(87)

261

On the analysis of Hopf bifurcations since the Hopf’s condition

is satisfied. From (50) and (51) one obtains the second derivatives

c;i=O

which are used in (55) and (53) to construct relation, respectively, as follows

the post-flutter

P =

as

(89) path and the amplitude-frequency

(PIJ2

w= 1. Since the curvature of (90) is positive and the equilibrium bifurcating periodic solutions are unstable [ll].

(90) (91) path is stable for p > 0, the

REFERENCES [1] K. HUSEYIN, Nonlinear Theory ofE/astic Stability. Noordhoff, The Netherlands (1975). [2] K. HUSEYIN, The Multiple-parameter Stability Theory and its Relation to Catastrophe Theory, Problem Analysis in Science and Engineering (Edited by Branin and Huseyin), pp. 229-255. Academic Press (1977). [3] E. LORENZ, J. Atmospheric Sci. 20, 130-141 (1963). [4] F. C. MQON and P. H. HOLMES, J. Sound and Vib. 65,275-296 (1979). [5] 0. E. ROSSLER, Chaotic Oscillations: An Example of Hyperchaos, Lectures in Appl. Math., Nonlinear Oscillations in Biology (Edited by Hoppensteadt), Vol. 17, pp. 141-156(1979). [6] V. MANDADI and K. HUSEYIN, Hadronic J. 2, 657-681 (1979). [7] K. HUSEYIN, ASME J. Appl. Mech. 103, 183-187(1981). [8] V. MANDADI and K. HUSEYIN, Int. J. Nonlinear Mech. 15, 159-172(1980). [9] V. MANDADI and K. HUSEYIN, Mech. Res. Comm. 4, 179-183(1977). [lo] K. HUSEYIN and V. MANDADI, Proc. 15th Int. Cong. Theo. Appl. Mechanics, Toronto (Edited by Rimrott and Tabarrok), pp. 281-294. North Holland, Amsterdam (1980). [ll] E. HOPF, Berichten der Mathematisch-Physikalischen Klasse der Sdchsischen Academic der Wissenschaften, Vol. XCIV, pp. 1-22 (1942). [12] J. E. MARSDEN and J. MCCRACKEN(Editors), The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, Vol. 19, Springer-Verlag, New York (1976). [I31 P. R. SETHNA and Cl. R. SELL, ASME J. Appl. Mech. 45, 234-235 (1978). [I41 D. D. JOSEPH, Bifurcation Theory and Applications in Scientific Disciplines (Edited by 0. Gurel and 0. E. Rossler), Vol. 316, pp. 150-167(1979). [IS] L. S. SMITH and L. Morino, AIAA J 14,333-341 (1976). [16] P. R. SETHNA and S. M. SHAPIRO, ASME .I Appl. Mech. 44,755-762 (1977). 1171 P. J. HOLMES, J. Sound Vib. 53.471-503 (1977). [18] P. J. HOLMES and J. E. MARSDEN, Automatica 14, 367-384 (1978). [19] J. K. HALE, Heriot- Watt Symposium, Vol. I, pp. 57-157, Res. Notes in Math 17, Pitman, London (1977). [20] P. J. HOLMES and J. E. MARSDEN, Nonlinear Partial Diferential Equations and Applications (Edited by J. M. Chadam), pp. 163-208. Lecture Notes in Mathematics, 648. Springer-Verlag (1978). [21] B. D. HASSARD and Y. H. WAN, J. Math. Anal. Appl. 63(l), 297-312 (1978). [22] B. D. HASSARD, N. D. KAZARINOFF and Y. H. WAN, Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Notes Series, 41. Cambridge Uhiversity Press, London (1980). [23] A. H. NAYFEH and D. T. MOOK, Nonlinear Oscillations, Wiley, New York (1979). [24] A. I. MESS and L. 0. CHUA, IEEE Trans. Circuits Sys. CAS-26, 235-254 (1979). [25] P. J. ALLWRIGHT, Proc. Cam&. Phil. Sot. 82,453-467 (1977). ]26] M. W. HIRSCH and S. SMALE, Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, New York (1974).

(Recehed

13 December 1981)

APPENDIX In this Appendix it will be shown that the derivatives of the real and imaginary parts of the eigenvalues, evaluated at C, can be written in terms of the system coefficients. In order to obtain these relations the Jacobian J(q) is expanded into Taylor series around the critical point C as in (17) and then introduced into the characteristic equation

lJG.4 - h(P)Il = 0

262

K.HUSEYINand A.S. ATADAN

which yields

wherep=v-qcandh=A,,,= critical point are

a(p)? i&L). It is obvious from (Al) that the nonzero terms of the Jacobian J(T) at the x,1 = x2, = 0,

x34

= -

x,3

= loq

which are in fact given by (10)-(13). Differentiating (Al) with respect to q and evaluating at the critical point one has X,,,-A” X210

w, ; io, 1

0;

0

0

0 01 ____----_~_---___--I----X 310 0 ' a4- i0, @q. I 1 -o X 410 0 q-1%( --------~__'-__-o__l_;T; X (1" 01 0 -

iw, x I20 ; 0 X?20-A0, 0 - % ____-___-~__---_--_C---

I -

0

;

0

IO

X 320 I a,-io, oq 0 0 X 420 , - %I aq -w ________-,_-_-_----.,_---_ 0 0 0 Xm 1 where A"= (dA/dq& =

/ ,

0 0 0 P

t

I 0,

0

0 0

=o

(A2)

I a,- iw,

CT’+io” and A(Q) = io,. Note that, without loss of generality, (A2) is written only for A = io,. After

some algebra, (A2) yields fro= $x1,, + X220)

643)

w”

(‘44)

= 4~,2,

2

-

&I)

Clearly, (A3) demonstrates the equivalence of (19) and (20). and 644) leads to (57) for 0’ = 0.