Bifurcation analysis of double pendulum with a follower force

Journal of Sound and Vibration (1992) W(2),

191-204

BIFURCATION ANALYSIS OF DOUBLE PENDULUM WITH A FOLLOWER FORCE J.-D. JIN Department of Aeronautical Engineering, Shenyang Institute of Aeronautical Engineering, Shenyang, People’s Republic of China AND

Y. MATSUZAKI Department of Aeronautical Engineering, Nagoya University, Chikusaku, Nagoya, Japan (Received 3 1 July 1990, and in revised form 10 January 199 1)

The stability and bifurcations of a double The focus is placed on a doubly degenerate a pair of pure imaginary eigenvalues: that The local qualitative behavior of the system

pendulum with a follower force are considered. system which possesses a zero eigenvalue and is, coupled flutter and divergence bifurcation. is examined in the neighborhood of the degen-

erate system. The four-dimensional equations of motion are reduced to the two-dimensional ones by using some qualitative reduction theory for dynamical systems, and it is shown that two distinct bifurcations can occur in the system according to the ratio of two damping coefficients of the double pendulum. Numerical simulations have been performed using the original four-dimensional equations to confirm the analytical results. Some global forms of behavior of the solutions have also been investigated numerically.

1. INTRODUCTION

Since Ziegler [I] discovered the destabilizing effect of linear viscous damping in a nonconservative elastic system, several investigators (see, for example, references 12, 31) have further proved that the stability characteristics of a two-degree-of-freedom elastic system subjected to a follower force are strongly dependent on the ratio of the two damping coefficients. Recently, the authors [4] studied such a system qualitatively from a viewpoint of bifurcation of solutions, with the aid of dynamical system theory. The study was concentrated on a local codimension-two bifurcation problem of the system near a degeneracy with double zero eigenvalues, and we showed that three kinds of bifurcations with various complicated motions can occur in the system according to the ratio of the two damping coefficients. This paper deals with the stability and bifurcations of the same twodegree-of-freedom system with a follower force. However, our focus is now on a different type of doubly degenerate case which possesses a zero and a pair of pure imaginary eigenvalues: that is, coupled flutter and divergence bifurcation. The local behavior of the system about the center equilibrium is studied by using dynamical system theory in the neighborhood of the double degeneracy. The center manifold theory [5] is used to reduce the four-dimensional equations of motion to three-dimensional ones near the degenerate system, and then the reduced system is transformed to a simpler form by use of Birkhoff’s normal form theory [6,7]. Because one of the normal form equations decouples from the remaining two, and can be ignored, we finally obtain two-dimensional equations. Hence, possible local flow behavior or motion of the original four-dimensional system can be conjectured from the reduced two-dimensional equations. 191 0022-460X/92/080191 + 14 .%03.00/O

0 1992 Academic Press Limited

192

J.-D. JIN AND Y. MATSUZAKI

Using a similar method Scheidl, Troger and Zeman [8] examined such a codimensiontwo bifurcation problem of a double pendulum (which may be considered to be a special case of the system treated in this paper) with an end elastic support. Taking into account symmetry breaking, Sethna and Shaw [9] studied a codimension-three bifurcation of two articulated tubes conveying fluid near the double degeneracy. Classification and unfoldings of such bifurcation problem were studied by Holmes [IO], Iooss and Langford [ 111, Langford and Iooss [7] and Guckenheimer and Holmes [ 121. Here, we will make use of some of their unfolding results in our study. Numerical simulations of the original full equations are performed to confirm the analytical prediction. Some global forms of behavior of the system are also investigated by using numerical methods.

2. DIFFERENTIAL EQUATIONS OF MOTION AND DOUBLY DEGENERATE SYSTEM We consider a double pendulum subjected to a follower force as shown in Figure 1. It consists of two rigid massless bars with equal length I, concentrated masses ml =2m and m2 = m, two springs with a spring constant c, and two viscous dampers with coefficients b, and b2. The generalized co-ordinates 9, and @ are taken to be small, and a load p is applied at the free end at an angle a~. The restoring moments induced at the hinges are cq, + b,@, and c(@- 9,) + b2(& - $I,). The damping coefficients b, and b2 are taken as positive and small, and the gravitational effect is not included. Then we obtain the equations of motion of the system using the Lagrange equation, and transform them to the four-dimensional first order equations [4],

dX/dr=OW+fW, where r denotes a non-dimensional x,=V),t

r ‘(‘)=

time defined by r = ,/mt,

x2, x3, x4)TER4,

and d@/dr,

x4 =

x?=dp,/dr,

x2=q2,

x=(x,,

(1)

~1,

p=(Ko,,~2)TE~4,

0

0

1

0 1

0

0

0

I

(F- 3)/2

(2 - F)/2

(5 -F)/2

(3F- 2Fir - 4)/2

F= PI/c,

Bi=b,/(l&),

-(B, +2&)/2 (B, + 4&)/2

B2 ’ -2B2 1

i= 1, 2,

f(X, P) = (0, 0, $J(K P), f4(X, P))‘, h(X, P) = f(x, -

-f(x,

X2)‘{

(d-

F)X, + [F(2 - Cl) -

31x2 + (B,

-x~)(x:+X:)+~ [(I -a)“xg-(x,

+$(X,-X,)(3X:+X:)+$[f(x,-ax&(1

+ 3B2)X3 -

-ax$]+

-a)‘x;]+O(I@).

3B2X4)

O(pcf),

(2)

BIFURCATION

ANALYSIS

I

d

OF DOUBLE

193

PENDULUM

m,=2m

1

QI

CQ, +b,d, I

Figure 1. A double pendulum system.

The trigonometrical terms in equations (2) have been expanded in Taylor series, and note that f is an odd function with respect to X: f(X, p) = -f(-X, p) for all p. The eigenvalue problem of A(p) yields a quartic characteristic equation (3) and the coefficients pi are dependent on the parameter ,u. We analyze in this study the local behavior of the system for the parameter p lying in the neighborhood of p =po for which A(po) has a zero eigenvalue and a pure imaginary pair. In addition, we assume that the remaining eigenvalue of A(po) is negative since we are interested in the bifurcation of stable phenomena. The above-mentioned assumptions require that three conditions must be satisfied: that is, p4=0,

P3'PIP2,

(4)

P3'0.

Fixing Bi and B2 for the time being, we obtain from the above conditions F =-B+[B2+16Bz(B,+6B~)]“2+3

1 3

0 ~(BI

+

ao=

Fo(Fo-3)

6B2)

where B=iB,-2B,-$B:Bz-3B,B:.

+1,

Equations (5) give a map N:

R2-,R21(BI,B2)w(F0,

ao).

The domain and image of N are D={(B,,

B2)~lR21BI, B2>0,

1={(F,

a)eR2(F=Fo,

B,, B2<& for some small E}, a=ao

for (B,, B*)ED},

(5)

194

J.-D.

JIN

AND

Y.

MATSUZAKI

F

Figure2. A part of set I.

respectively, and the local analysis will always be carried out near the set I. A part of set Z limited in the area of (3 < F<4, 1
R,=-;(B,

f23=0,

+6B,).

(6)

Here o=

B,+Bz-Fo(l--ao)(B,+2B2) BI +6B2

1. Ii2

We introduce a new parameter A=(F-F,,,

a-m,)~R*.

(7)

Then p =po+ (a, 0,0),and equations (1) become dX/dr=A(y)X+f(X,

p)=D(a)X+g(X,

A).

(8)

Note that the case of a= 0 corresponds to the degenerate system and we consider here the case of a + 0 only. Expanding the non-linear terms of equations (8) in power series with respect to a, we obtain dx/dr

=

o(a)x+g(x, 0) + a (apa)g(x, 0) + o(la1*1xf+ ~xls).

(9)

We will use the theory of center manifold to reduce equations (9) to three-dimensional ones which retain all information about the qualitative behavior of the original system.

3. REDUCTION TO THREE-DIMENSIONAL SYSTEM ON THE CENTER MANIFOLD The matrix A(po) can be put into Jordan normal form by a transformation matrix V which is composed of the eigenvectors of A(po). Now we introduce a transformation X= VY and substitute it into equations (9) to obtain dY/dz=[J+~(il)]Y+F(Y)+F,(Y)rZ+O(Ia121Y13+IY15),

(10)

BIFURCATION

ANALYSIS

OF DOUBLE

PENDULUM

195

where

J=[;

;j,

Jc=[i

!],

Jd=-;(B,+6Bz),

v- ‘g(VY, 0) = (r;, , Fd)‘,

F( Y) =

F,(y)=

-;

v-’&?(vr, o)]/a~=(Fk,

y= (Yc,ydjT,

YEER3,

fidjT,

(11)

YdER.

According to the center manifold theory, there exists the center manifold yd=h(y,, A) E IRS, and the local behavior of the system is governed by the following equation on the center manifold [4, 51: dycldr = (Jc+A,d~c+&4~c, + FdYc,

4 +Fc(yc, NY,, 4)

MYc, n)v+

o(1421Yc13+ lYc15).

Because the system considered has a symmetry, i.e., Fc( Y) = -F,(equations (12) can be reduced to d.v,ldr = (Jc+ &)Y,+

Y), Fd( Y) = -&(-

FAY,, 0) + O(lA121~cl + Ial lyc13+ lyc15h

(12)

Y),

(13)

where F, contains terms of order ]y,13 only, as in reference [4]. We will further reduce equations (13) to a two-dimensional system using Birkhoff’s normal form theory.

4. DERIVATION OF NORMAL FORM

Next, we will find the “simplest” form for the non-linear terms of the equations (13). This can be done by several reduction methods, such as those of Takens [ 131 and of Lyapunov-Schmidt [14], or the Birkhoff normal form theorem [6,7], and we use the last one here. As the Birkhoff theory appears in some mathematical papers, here we try to present the essence of the theory without describing technical details. For simplicity, we define r(& r) =,:(k

r),

c(k) = J,(k) + A,, (4,

M(r(A; T)) = F,.(r(A.; T), 0)

Then equations (13) become dr/dr = Cr + M(r).

(14)

Since we consider the case of d + 0 (that is, the system near the doubly degenerate one) the matrix C possesses eigenvalues, fir, 62 and fi3, given by

(15)

J.-D. JIN AND Y. MATSUZAKI

196

Let u, W, @ and IJ*, G*, w* denote the normalized eigenvectors and their adjoint eigenvectors corresponding to 6,) 6, and &, respectively. Now we introduce a transformation r(a.; r)=Z(r)u(a)+S(r)W(a)+i(r)k(a),

SEC.

ZER,

(16)

Then equations (14) become udz/dz+~vdr/dr+~ddS/dr=z6,v+~~~~+Sij;~+N(~,~,S),

(17)

where N(z, s, S)=M(zu+sw+fG).

(18)

Next we form the inner products of equations (17) with IJ* and w* to obtain ds/dr = 62s + d2,

dz/dr=fi,z+d,,

(19920)

where

s

2n

2rr

d, =(N,

,*)=;

Q=(N,

NV* dt,

w*)$

s0

0

NW* dr.

(21922)

Since N contains terms of I$ only, we may write N(z, s, i) = C z”YWz,,,,,,,,

(23)

where m, n and p are non-negative integers, and the summation C is always performed for m + n +p = 3. Then equations (21) and (22) are expressed as dl = C z’nsn.Fp(a,,np,v*) s 1 z”‘J’Z*dj~$,

d(‘ ) ER’9 mnp

d2=~z”‘s”~p(a,nnp, w*)~~z”Y~*dj;:;,

d”’ ,nnpEC .

(24)

The next step is to perform a non-linear change of variables so that we may remove from equations (19) and (20) some non-linear terms which have no influence on the topological structure of the solutions. Define new variables zI and s, by zI = z + 1 a,,,npz’n$~P,

a,nnpe R,

s1 = s + c P,n”pZ’n.fc

jL*E Q=,

(25)

where the coefficients arltnpand /&,“*depend on ,u. Substituting equations (25) into equations (19) and (20) we obtain dz,/dr=&z,+&(z,,

s,, S,),

ds,/dr=

c52s,+~2h

SI,

S,),

(26)

where

d=x z;"s;~fa::~p, a,=I z;"s;sfa:;;p,

(27)

and a9:p= [(,,, - 1Jr2 + (n +P%

a!2p=bc2+(n+p-

WI

+ (n

+k-P-

-p)i~&h,,np + d:i!p, l)io&L,+d!$!p.

Pa) (28b)

If the expressions in the brackets of equations (28) are non-zero, we can make the new equal to zero by appropriate choices of a,,* and P,nr,P.The expressions coefficients a:;;, bracketed are non-zero if n #p in equation (28a), and if n fp + 1 in equation (28b), for ,n,,pidentically zero in this case. But all A near zero, and we can make the corresponding aCi)

BIFURCATION

ANALYSIS

OF DOUBLE

197

PENDULUM

the bracketed expressions are zero when n=p in equation (28a), and when n =p+ 1 in equation (28b) at I. = 0. Therefore, we set (I,,,, = 0 and /?,”,,+ I n = 0 to obtain a(I) =#I) ,““I# mnn,

d’2’ a(2) mn+l n= mn+l n.

(29)

This reduces equations (26) to the normal form dz/dr = 6,~ + zlsl'd$:'l + z’d$,!,,

ds/dr = ~5~s+ Isj2sdh;), + z2sd%,

(30)

where we have rewritten zI as z and s1 as s for simplicity. An alternative normal form can be obtained in cylindrical co-ordinates s = p eiw ( p # 0) : that is, dp/dr=5,p+SRp3+ZRz2p,

dyl/dr=w,+S,p2+Z,z2,

dz/dr=6,z+zp2d#+z3d#,

(31)

where SR= Re (d${),

S,= Im (d$,%),

ZR= Re (d%),

Z,= Im (d$b).

(32)

The second of equations (31), which involves the variable representing the phase component of the solution, decouples from the remaining equations and will be ignored here. Then we can reduce the system finally to a two-dimensional one and rescale it by the linear transformations p’= mp and z’= @$$z to obtain dpldr = p(Cl + YP’ + v2),

dz/dr =z(c2 + 6p2 + .zz2),

(33)

where the prime for p and z has been omitted, and y=*l,

n =&/l&&l,

&= fl,

8 = d!:‘l/lS,l,

(34)

which are all dependent on BI and B2.

5. BIFURCATIONS

OF SOLUTIONS

We now investigate the bifurcations of solutions of the system through analyzing the two-dimensional equations (33) which possess all information about local behavior of the original system except the phase effect. For B,>O and B2>0, we have &=--I, 8~0, q23*585 and y = - 1 for B2/B, < 23.585. As shown in Figure 3, we can divide the B,-B2 plane into two regions A and B (which will be called the cases A and B, respectively) and distinct bifurcations occur in each region. Because of the symmetry with respect to both axes p

p= E,/B, = 23.505

Figure 3. The 81-82

plane. The plane is divided into two regions according

to the signs of y

198

J.-D. JIN AND Y. MATSUZAKI

and z in equations (33), the phase portraits are illustrated only in the first quadrant. The case A, in fact, corresponds to the case VIII of Table 7.5.2 in reference [12], and the unfolding is shown in Figure 4. The {,-cZ plane is divided into six regions by the bifurcation sets. Primary pitchfork bifurcations occur from the trivia1 solution (0,O) on the lines cl=0 and O and (m, 0) for cl < 0. Secondary pitchfork bifurcations can also occur from (0, a) on the line ci + &z=O, and from (G, 0) on the line cz= 0{,, and we obtain an equilibrium (J-G

+ &)/(I

+ M), JCL-

@,)/(I

+ M))

for 0. The bifurcation set and the corresponding phase portraits are given in Figures 4(a) and 4(b), respectively.

(b)

Figure 4. Unfolding of case A (BI =O.Ol, BZ= 0.3) : (a) bifurcation set; (b) phase portraits

For case B, we have y = -1. Introducing the change of variables r + -z, cl + -c, and 62 + -cz, we obtain, from equations (33) dpldr = P(C, + p2 - t7z2),

dz/dr = z(c2 - 8p2 + z*).

(35)

Equations (35) correspond to the case lb of reference [ 121, and we obtain the bifurcation sets and phase portraits of equations (33) for the unfolding as shown in Figures 5(a) and 5(b), respectively. The ~,-CZ plane is also divided into six regions by the bifurcation sets. The primary pitchfork bifurcations occur from the trivial equilibrium on the lines J, =0 and c2 = 0 and the secondary pitchfork bifurcations occur from non-trivial equilibria on the lines 13c,+ c2 =0 and cl + r7c2=0, behavior similar to that in case A. 6. DISCUSSION

AND NUMERICAL

SIMULATIONS

In order to provide a physical interpretation of the results shown in Figures 4 and 5, we present a brief discussion regarding the relationship between the dynamics of the

BIFURCATION

ANALYSIS OF DOUBLE PENDULUM

199

(b) Figure 5. Unfolding of case B (B, =0.2, &=0.2):

(a) bifurcation set; (b) phase portraits.

reduced plane autonomous system, equations (33), and the original four-dimensional physical system, equations (1). The relevant correspondence between small solutions of equations (33) and of equations (31) is easy to see if we take into account the rotational effect neglected at a stage in the analysis. In general, a stationary solution of equations (33), (p, z) for p#O, corresponds to a periodic orbit of equations (31), and equilibria of equations (33) which correspond under reflections in p represent the same periodic solutions, but reflections in z give distinct periodic solutions because of the symmetry of the system. According to the center manifold and normal form theory, solution curves given by equations (31) have the same topological structure as those given by equations (1), so that the stable equilibrium of equations (33) on the vertical axis in the phase portraits of Figures 4(b) and 5(b) represents a buckled state of the pendulum, whereas that on the horizontal axis represents a limit cycle motion (flutter) encircling the center equilibrium (the vertical trivial position) of the pendulum. As an example of flutter phenomenon, the evolution of two solutionst moving towards the limit cycle in the three-dimensional center manifold are sketched in Figure 6, which corresponds to the phase portrait in region 2 of Figure S(b). The stable equilibrium on the (p, z) plane, not just on the p or z axis, represents a flutter encircling a buckled position, but this phenomenon, although it appeared in our previous study [4] of this system, does not occur in the present problem. According to the preceding discussion it is easy to imagine the local behavior of the original physical system from the results shown in Figures 4 and 5 obtained by using the reduced two-dimensional equations. For case A, a buckled (regions 3-5) or stable vertical (region 6) state of the pendulum may be observed physically if the motions for the parameter values in regions 3-6 are started from some appropriate positions in the local state space. However, for the other initial conditions in these regions as well as in regions 1 and 2 t The “evolution of a solution” here means that a representative point (p(r), z(r), y(r)) moves on an integral curve in the direction of the arrow as r increases.

200

J.-D.

JIN AND Y. MATSUZAKI

Figure 6. Solution evolution on the center manifold corresponding Figure 5(b).

to the phase portrait in region 2 of

the behavior of the system cannot generally be determined from this theory because the configurations of the pendulum will ultimately go out of the local state space. In addition, there is no flutter phenomenon in this local case. For case B, all the motions started from any position in the local state space always stay in it; moreover, stable limit cycle motion (flutter) is possible as well. Thus, one of the three states of the pendulum: vertical position (region l), buckling (regions 4-6) and flutter (regions 2-4) may be observed ultimately according to the different parameter regions and initial conditions. To examine if the original system has the same qualitative features predicted, we have also solved the full equations numerically with the aid of the fourth order Runge-Kutta method. Some numerical results projected onto the x2-x4 plane for case B are shown in Figures 7(a)-7(c), which correspond to the phase portraits of the regions 4-6 of Figure 5(b), respectively. In Figure 7(a), solutions 1 and 2 evolve towards a limit cycle, and solution 3 evolves towards a static equilibrium, i.e., a buckled state of the pendulum, which corresponds to the phase paths 1-3, respectively, sketched schematically by dotted lines in region 4 of Figure 5(b). A part of the evolution of solution 3 is redrawn in Figure 7(d) to show some detail. The evolving solutions shown in Figures 7(b) and 7(c) both approach a static equilibrium, but they have slightly different shapes from one another, which correspond to the phase paths expressed by dotted lines in regions 5 and 6 of Figure 5(b), respectively.

7.

SOME GLOBAL BEHAVIOR

The results obtained above are local: that is, the predicted bifurcations and phase portraits of the system are effective only for a small range of the state variable x and the values of the parameter p near the specified value po. In general, we could therefore draw no information about global features of the solution from this theory. However, a numerical analysis of the local solutions in the global space allows us to determine some of their global nature, as done in reference 143. The flow behavior was numerically examined in a “large” state space by using directly the original full equations of motion for the local

BIFURCATION

ANALYSIS

OF

DOUBLE

201

PENDULUM

L

A----l\

I

202

J.-D.

JIN AND Y. MATSUZAKI

values of parameter p : i.e., p lies in the neighborhood of po. From the results obtained we conjecture that if the local solution set is regarded as a whole, we may look upon it as a “saddle” in case A and a “sink” in case B, respectively, in the global state space, as illustrated in Figure 8. Some of the results of numerical simulations for these cases are shown in Figure 9. For case A, three evolving solutions are shown in Figures 9(a) and 9(b), which correspond to the phase paths sketched by dotted lines in the phase portraits of region 5 in Figure 4(b). Solution 1 evolves away from the “saddle” towards the large limit cycle, and solutions 2 and 3 evolve towards the static equilibrium in the “saddle”. In Figure 9(b), a part of the evolution of solution 2 in Figure 9(a) is magnified to show detail. The solution evolving towards the local solutions, for case B, in the global space is shown in Figure 9(c), where the structure of the local solution set corresponds to the phase portraits of region 5 in Figure 5(b) or Figure 7(b). In case A, it would seem that there exists a “large” limit cycle in every region of Figure 4(b), as shown in Figure 8, and the solutions which come out of the “saddle” all evolve towards the limit cycle. This implies physically that a large amplitude flutter of the pendulum encircling the local domain will be observed ultimately in the area of those initial conditions for which one was not able to determine the behavior of the system by the local theory. However, in case B, the local solution set has attractiveness in a considerably large area in the global state space. Thus, all the motions that start from this global area will get into the local domain, and one of the three states, as mentioned earlier, will ultimately be observed.

(b)

(a)

Figure 8. Phase portraits of global solutions around the local solution zone. Shaded zones represent the area for the local solutions: (a) “saddle” and limit cycle; (b) “sink”.

8. CONCLUSIONS

In this paper we have analyzed a double pendulum subjected to a follower force. Attention was concentrated on the local behavior of the system near the doubly degenerate system with eigenvalues: (0, Gw). Using the center manifold theory and Birkhoff’s normal form theory we obtained two-dimensional equations of motion which retain all information about the qualitative behavior of the system in the local domain. Analyzing the twodimensional reduced system, we found that two different bifurcation phenomena can occur in the original system according to the ratio of two damping coefficients. The B,-B2 plane

BIFURCATION

ANALYSIS

OF DOUBLE

203

PENDULUM

1.67

-1.67

-3.33 I

-1.67

0.00

I.67

3.33

I

I

I

I (b)

0.17

*o

0.00

-0.17

Figure 9. Numerical simulation by equations (I) for phase portraits in Figure 7: (a) case A of B, =O.Ol, &=0.3 and (F, a)=(3.99.5, 1.255); (b) magnification of evolution of solution 2 in Figure 9(a) and solution 3; (c) case B of B, =0.2, &=0.2 and (F, u)=(3.92, 1.292).

and distinct bifurcations into two regions by the line defined by B2/Blx 23.585, occur in each region. The numerical simulation of the original full equations confirms analytical results in the local domain. Some global behavior of the local solutions was investigated by use of a numerical method. From the results obtained we may conjecture

is divided

204

J.-D. JIN AND Y. MATSUZAKI

that the local solution set as a whole looks like a “saddle” for case A and a “sink” for case B in the global state space. It seems that there exist the “large” limit cycles in case A around the local solution set at least for the values of parameter p limited in the neighborhood of po.

REFERENCES 1. H. ZIEGLER 1952 Ingenieur-Archio 20, 49-56. Die Stabilitatskriterien der Elastomechanik. 2. V. V. BOLOTIN 1963 Nonconservative Problems of the Theory of Elastic Stability. New York: Pergamon Press. 3. G. HERRMANN and I. C. JONG 1966 Journal of Applied Mechanics 33,125-l 33. On nonconservative stability problems of elastic system with slight damping. 4. J.-D. JIN and Y. MATSUZAKI 1988 Journal of Sound and Vibration 126, 265-277. Bifurcations in a two-degree-of-freedom elastic system with follower forces. 5. J. CARR 1981 Applications of Centre Manifold Theory. New York: Springer-Verlag. 6. G. Iooss and D. D. JOSEPH 1980 Elementary Stability and Bifurcation Theory. New York: Springer-Verlag. 7. W. F. LANGFORD and G. Iooss 1980 in Bifurcation Problems and Their Numerical Solution, 103-l 34. Interactions of Hopf and pitchfork bifurcations. Base1 : Birkhauser Verlag. 8. R. SCHEIDL, H. TROGER and K. ZEMAN 1983 International Journal of Non-linear Mechanics 19, 163-176. Coupled divergence and flutter of a double pendulum. 9. P. R. SETHNA and S. W. SHAW 1987 Physica 24D, 305-327. On codimension three bifurcations in the motion of articulated tubes conveying a fluid. 10. P. J. HOLMES 1980 in Nonlinear Dynamics (R. H. G. Helleman, editor), 473-488. New York: New York Academy of Sciences. Unfolding a degenerate nonlinear oscillator: a codimension two bifurcation. 11. G. Iooss and W. F. LANGFORD 1980 in Nonlinear Dynamics (R. H. G. Helleman, editor), 489-505. New York : New York Academy of Sciences. Conjectures on the routes to turbulence via bifurcations. 12. J. GUCKENHEIMER and P. 5. HOLMES 1983 Nonlinear Oscillations, Dynamical Systems and Bzfurcation of Vector Fields. New York : Springer-Verlag. 13. F. TAKENS 1974 Publications of the Institut des Haute Etudes ScientiJiques, Bures-Yvette, Paris 43, 47-100. Singularities of vector fields. 14. W. F. LANGFORD 1979 SIAM Journal of Applied Mathematics 37, 22-48. Periodic and steadystate mode interactions lead to tori.