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Global bifurcations in the forced vibration of a damped string
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c#zt%7.&2/93 1993 Perptm
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GLOBAL BIFURCATIONS IN THE FORCED VIBRATION OF A DAMPED STRING OLIVER M. O’REILLY* Institute of Mechanics, Swiss Federal institute of Technology (ETH). CH-8092 Ziirich, Switxerland (Received 5 July 1991) Abstract-Using a Shil’nikov-type model, some global bifurcations present in the averaged quations for the harmonic vibration of a damped pretensioned linearly elastic string are examined. In particular, we are interested in examining those bifurcations which allow the string to change its direction of whirling These bifurcations are shown to be directly related to the structural instability of a homoclinic connection. From symmetry and stasbility considerations, we are able to show that this instability produces gluing bifurcations and homoclinic explosions and provides an explanation for some of the chaotic motions observed in this system. The consequences of our analysis for related physical systems which exhibit similar modal coupling is discussed. 1. INTRODUCTION
This paper examines some of the global bifur~tions observed in the amplitude-m~ufated equations of motion for the vibrating string. The string considered is pretensioned and fixed at both ends. One of these ends is excited ha~oni~ffy in a direction transverse to the length of the string. When the excitation frequency is close to a transverse harmonic, the planar periodic vibration becomes unstable and a periodic ballooning or whirling motion is produced, provided the excitation amplitude is sufficiently large. As the excitation frequency is increased, the whirling motion increases in amplitude. Further increase in the frequency produces a hysteretic collapse of the response amplitude and a smaller-amplitude planar periodic motion results. This phenomenon was first reported by Harrison [l] and inspired a large body of analytical and experimental work (cf. O’Reiffy and Holmes [2] for the relevant references). The majority of this work focused on the case where the excitation frequency is close to the first transverse harmonic and the string is modeled as a linearly elastic continuum. Amplitude-modulated chaotic behavior for the string was first suspected when Miles [3] showed the presence of Hopf bifurcations in the averaged equations of motion. Their presence was first observed numerically by Johnson and Bajaj [4,5] and experimentally by Molten0 and Tufiffaro [6]. O’Reiffy and Holmes [2] confirmed these results and analyzed the non-periodic behavior. Owing to the degenerate nature of the integrable limit of the averaged equations, they found that it was not possible to employ the Mefnikov technique [73 and obtained information on the global bifurcations. They did show, however, that in the absence of damping the chaotic motions can be explained via a Shif’nikov-type model for a Hamiltonian saddle center (cf. Miefke et aI. [8] and Miefke [9]). For the damped case, they indicated that the chaotic behavior could be interpreted using a Shif’nikov-type model for a saddle focus. However, no thorough investigation of this case was performed. Shi~nikov-ty~ mechanisms were originally developed by Shif’nikov [lo] whife generafizing the work of Andronov and Leontovich [ 11J on the generation of limit cycles in planar dynamical systems with homocfinic orbits to higher dimensions. The model possesses a homocfinic orbit to a fixed point of the flow for given parameter values of the system. Due to the structural instability of the homocfinic orbit, it will not be preserved as these parameters are varied. Shif’nikov showed that, depending upon the spectrum of the fixed point, a single stable periodic orbit [lo] or a countable number of unstable periodic orbits [12] can be produced when the homocfinic connection is broken. Generalizations of these results were subsequentially obtained [13-U] and some of them depend crucially on the dimension of the unstable manifold (cf. [lS] in particular). Development of the ConfeyMoser conditions [16], which can be directly applied to these models to show the existence * Present address: ~~rtment 94720, U.S.A.
of Mechanical Engineering, Unive~ity of California at Berkeley, Berkeley, CA 337
0. M. O’REILLY
338
of a hyperbolic invariant horseshoe, and the success of the models in explaining some of the global bifurcations observed in the Lorenz equations (cf. Afraimovich et al. [17]) precipitated the considerable subsequent interest in Shil’nikov-type mechanisms (cf. Wiggins [ 18, 193 and the references therein). Prior to performing a Shil’nikov-type analysis, we examine the local bifurcation phenomena which occur at the fixed point of interest, which gives us explicit expressions for the eigenvalues of this fixed point. Numerical simulations show the existence of a homoclinic connection to this fixed point for certain parameter values. A three-dimensional map is then constructed which describes the dynamics in a neighborhood of this connection. In constructing this map, we employ symmetry considerations extensively. This allows us later to use several known results available for more restrictive versions of our map. We then examine the global bifurcation behavior for two ranges of the spectrum of the fixed point. In one of these ranges a global bifurcation known as (after Gambaudo et al. [20]) the gluing bifurcation occurs; in the second range a homoclinic explosion is observed. In the former case transient chaos is observed and in the latter stable chaotic attractors are present. Shil’nikov-type analyses for physical systems have, to our knowledge, rarely ever been performed. In the majority of applied research of this topic the researcher concludes that the presence of a homoclinic connection implies the existence of a (nearby) hyperbolic invariant set. This may be the case; however, a detailed analysis produces far more interesting results. In our case, it allows us of distinguish between chaotic behavior observed numerically by Johnson and Bajaj [4, 51, which occurs due to period-doubling and homoclilnic phenomena; it forewarns us of the presence of transient chaotic behavior and provides an explanation of the mechanism which produces a qualitative change in the modal behavior of the string. These results are also applicable to other systems and we discuss this at the conclusion of the paper. Finally, we remark that our exposition has been influenced by the works of Holmes [21], Glendinning and Sparrow [22], Glendinning [23], Tresser [24] and Kovacic and Wiggins [ZS]. 2. THE
AVERAGED
EQUATIONS
OF MOTION
The equations of motion for the string are obtained by modeling the string as a linearly elastic continuum, with displacements in the plane parallel to the direction of the external forcing (henceforth known as the vertical direction), in the plane perpendicular to the plane of the excitation (henceforth known as the horizontal direction) and along the string axis (henceforth known as the longitudinal direction). Coupling between these three displacements is present in the strain energy density function. The three partial differential equations which describe the vibration of the string are then derived using the EulerLagrange equations. A Taylor series expansion to third order in the strains is then performed, followed by a Galerkin projection of the fundamental harmonics in the vertical, horizontal and longitudinal directions. This procedure results in three cubically coupled ordinary differential equations (cf. [2] for explicit details). As the chaotic motions observed experimentally [2,6] occur on time scales significantly lower than that of the forcing, these equations may be averaged with respect to the period of the forcing to produce a further set of three coupled ordinary differential equations. The equations for the longitudinal motion can be eliminated; the resulting averaged equations obtained after a resealing are: d
$Pl
=
- YPI - (A + AEh,
+ BMpz
$PZ
=
-
-
YP~
-
(A
+
Ah2
BMPI
d 541 = - yqi + (A + AQpi + BMq2 + a d 5% = - yq2 + (A +
WP,
-
BMq,.
These are the equations derived originally by Miles [3] and examined by various researchers [2, 4, 51. In the above equations the subscripts 1, 2 denote the vertical and
Global bifurcations in the forced vibration of a damped string
339
horizontal directions, respectively, and the qis are the generalized modal coordinates and the PiS the corresponding momenta. The harmonic excitation is introduced by two parameters: A, the detuning parameter, which defines the proximity of the forcing frequency to the transverse harmonic frequency and a which defines the forcing amplitude. E and M are the energy (p: + P: + qf + qj) and angular momentum (pLqz - qlP2) of the generalized modal coordinates and momenta and A and B are constants, equal to -1.5 and 1.0, respectively, for the vibrating string model. The parameter y models the linear viscous damping present in the system. We note that the averaged equations are valid when the geomet~c non-linea~ties (cubic terms) are weak and the forcing and damping small. The system described by (1) may be viewed as an O(2) symmet~c system whose symmetry is broken by the harmonic forcing. This particular symmetry breaking produces a 1-l resonance phenomena between the mode in the plane of the forcing (the vertical mode) and the mode in the plane perpendicular to the forcing plane (the horizontal mode). These equations have been shown by Miles [33 to possess pitchfork, saddle-node and Hopf bifurcations (cf. [3], Fig. 1) and by Johnson and Bajaj [4, S] to possess sequences of period-doubling bifurcations (cf. [4], Figs 5 and 8) and chaotic solutions. In our analysis we shall confine our attention to the homoclinic phenomena and shall not address the period-doubling bifurcations. A key feature of (1) is the invariance under the flow of the pl-ql plane. This implies that the flow on this plane will always be integrable. The system of equations (I) also possesses the symmetry (P~,p~,q~,qz)~(p~, -P2t%-42h (21 We note that in the absence of this symmetry the construction of a single map would sufliee to describe the dynamics of interest. Its presence necessitates the construction of four maps.
3. BIFURCATION
IN THE INVARIANT
PLANE
An invariant plane for (1) is the ql-pi plane. Analysis of the phenomena occurring in this plane determines the behavior of the vertical mode, in particular its stability:
$1=- YPI -
(A + AE)q,
= - yqi + (A + AEfp, -f- a
(3)
where E denotes the p1anar energy p: + qf. When a = y = 0, the system has a fixed point at the origin and its phase space is foliated by periodic orbits. Addition of forcing (a) alters the phase plane depending upon the value of A and y. As A is increased from zero, the fixed point (qL,pc)which was originally at the origin increases in magnitude. Further increase in A results in a saddle-node bifurcation (cf. [33, Fig. 1). This bifurcation produces an additional two fixed points (qo, po). Subsequent increase in A results in a pitchfork bifurcation occurring at (q6, PC).This pitchfork birurcation does not affect the stability of (qe,pJ in the invariant plane; however, (qt,P.) is now unstable in the out-of-plane direction (cf. Fig. 1). In the experimental system this corresponds to the onset of the whirling motions. These two whirling motions, which correspond to the stable branches of the pitchfork bifurcation, satisfy the symmetry condition (2) and may be viewed as corresponding to clockwise (p2,q2 > 0) and counterclockwise Qt, q2 c 0) whirling motions of the string. The two other fixed points in the invariant plane are both stable in the out-of-plane direction, although one of them (PO)is a saddle in the invariant plane and the other (qo) is a sink. The aforementioned stability information can be ascertained by linearizing (1) about a planar equilibrium point (ql,pl,q2 = O,p, = 0) and determining the eigenvalues: i“1.2
=
-
R3.4
=
-v+
y f
+(A
+ AE)(A + 3AE)
(W
,/-(A
+ AE)(A + (A + B)E)e
WI
The eigenvalues c?I, t determine the stability of the planar equilib~um in the ql-Pr plane, whereas .&, 4 determines the out-of-plane stability. Numerical evidence, which we sha11
0. M. O'REILLY
340
bl 3
F:
-3
-3
I
q, Fig. 1. The p,-ql
qt
invariant plane: (a) a = 1.0, y = 0.0, A = 3.4, (b) u = 1.0, y = 0.5, A = 3.4.
discuss later, shows a homoclinic orbit which is biasymptotic to (qZ,ps); so, we now characterize the eigenvalues of this fixed point. We note that when A lies between the pitchfork bifurcation points, the eigenvalues of &pC) have the form 1 1.2 =
P f iw
/13 = v A‘$ = 1
(5)
where p = - y, v < 0 and 1 > 0. i.e. the fixed point is a saddle focus with a threedimensional stable manifold and a one-dimensional unstable manifold. The dimensions of these manifolds imply that the homociinic connection is a codimension-one bifurcation problem and only one parameter in our analysis needs to be varied in order to examine its structural instability. The parameter we choose to vary is the detuning parameter A; however, similar results will be obtained by varying a or y. For the remainder of the analysis we shall assume that -v, w > - p, 1. We remark that in the absence of damping (ql, pe) is a saddle center and the system (1) is Hamiltonian. As mentioned earlier, this system afso possesses a Shil’nikov-type mechanism. For more details on this system the interested reader is referred to [2, 8, 9J and the references therein.
4.
CONSTRUCTION
OF THE
SHIL’NIKOV
TYPE
MAP
We are interested in constructing a dynamical system which describes the dynamics of orbits which pass dose to the fixed point (qe,pc). From numerical evidence a homoclinic orbit, r, exists which is biasymptotic to this fixed point at a given value of A = A* (a, y are fixed). In keeping with the literature. we define a new parameter ,u = A - A* and, so, at ~1= 0 the homoclinic connection, r, exists. This a priori knowledge of the existence of r is required for the construction we are about to perform. Our purpose is to describe the behavior of orbits which remain close to r and for a range of values of p about I( = 0. For the case under consideration, a map was previously derived by Kovacic and Wiggins [25] (cf. [18], pp. 261-267) while analyzing the damped Sine-Gordon equation. Our construction mirrors theirs, with some slight modifications which we shall need later in our analysis. The first step in the procedure is to perform a coordinate change (PlrPzr4rr(lz)-+(Pt - &lP2191
and, using the Hartman-Grobman
- 4.1q2) = (&,X2*X3*X4)
(6)
linearization theorem (cf. [19]), obtain the vector field in
Global bifurcations in the forced vibration of a damped string
341
Fig. 2. Definitions of the Poincare sections used to define the map T,
a neighborhood of the origin:
dx3
dt = vx3
An implicit assumption here is that the linearization is valid only in the neighborhood of the origin and, so, we have omitted the higher-order terms in (7). Two maps are defined: a “local map”, To, which describes the dynamics in an E neighborhood of the origin and a ‘global map”, T,, which describes the dynamics of orbits which enter and leave this neighborhood of the origin (cf. Fig. 2): X:u,+= {(x~,x~,x~,x~)ER*Ix~ = O,cez*p'ru s x1 s E,O 5 x4 5 E,O s x3 I;E}
(W
CYO-={(X,,~~,X~,X~)ER~~X~=O,~~~~~'~~X~I;~,O~X~~E,OI;
(W
-X~ZSE}
X'o+= {(xI,x~,x~,x~)ER*~x~ = 0, ce2=p'm 5 xl s E,O 5 -x4 s E,O s x3 s E}
(8~)
X:b-= {(xI,x~,xJ,x~)E R* Ix2 = 0, seZ*P’m5 x1 s .s, 0 s - x4 s C,0 s - x3 s E}
(gd)
and c :+ = {(x,,x~,x~,x~)ER~IO
{(x~,x~,x~,x~)E
R410 < -x3,x4
= E}
(W
Xi+ ={(x,,x3,x3,x4)~R4~O
(80
The bound sezxp’* s xl is calculated by considering an orbit which intersects any of the four X;“*s at x1 = E and establishing the conditions for the size of Z$‘*, so that the only intersection of an orbit in the neighborhood of the origin Ze (= ZY,’ u Rx‘ u Zb’ u EL-) is the initial one at x1 = E.
342
0. M. O’REILLY
For a complete definition of the map which accounts for the symmetry conditions, it is necessary to construct separate maps, for each of the four Zg:‘* sections. We shall first construct the map for Ilf’ and use the invariance of the flow under (_x,,x2,x3,x4) + -x4) to construct the remaining maps. Consider an orbit which enters (XI, x2, -x3, a neighborhood of the origin through XI’ at (x1,,, xzo, xJo, xao) and leaves through Cl’ at (.x~~,xzI,x~~, ~41 = .s).The time of flight, r, for such an orbit can be calculated by directly integrating (7d): r+og
0$ .
Substitution of the time of flight into the integrations of (7a-c) yields the following map, which describes the behavior of some of the orbits in a neighborhood of the origin: U+ To : x:“o+ -+x:1+ +-)“Acos(~log(~))
Xl x2 = 0 x3
[
x4
Il-
-
xr(~>“^sin(~fog(~))
(10)
& v/i x3 0 F 4 &
Furthermore, we restrict the size of Ey’ so that PO’ will map Z$’ on to X:rl+.This can be achieved by restricting the size of the small parameter E.The “global map”Ty’ is defined by considering the definitions of the Poincare sections which it maps:
x:
[!L1 Xl x2
-pr,
0
+
-P2
x3
x4
(11)
*
-P
Because the homoclinic connection, I-, is structurally unstable, this connection will be present only when p = 0. This instability is accounted for by the introduction of the additional parameters p, jtr, and ,ur2 in (11). The parameter x* is the intersection of r with CZ+: x* =
;(I + e2np’o),0,0,0 , (
The full map T"+ = Ty+0Tl;'can now be defined: T”+:x:“,+-+x;
>
(12)
Global
bifurcations
in the forced vibration
of a damped string
343
We note that because x z = 0 on any of the four subsets of Co, it is eliminated from our analysis. This enables us to reduce the dynamics (of interest) of (1) from a four-dimensional system of ordinary differential equations to a three-dimensional map. To make the analysis more tractable, several resealings are performed:
tan@,) = i,
h
tan(@) = i,
tan(Q3) = s
XI [I [
for T”+:
which result in the resealed representation
x3
+
x4
axi(x~)-~COs(O10g(X,$) + 0,) + cxJ(x‘+)-” - jA-i + x: dx,(x,)-“cos(olog(x4) + 02) +fx3(x~)-” - $2 gx1(x4)-Pcos(olog(x4) + (IQ + ix3(x4)-” - p
.1 (144
It now remains to define the remaining three maps. These are obtained by examining the symmetry condition (4) and their effects on the map ci+:
T”-: Cl;-- -2:b
=
T”+b4
+
1x41,~
-,
-
P)
Wb)
T”:~b+~~cl;=T”‘~-_,-~i-,-i,c-r-c)
T’-:Zb-+Zb
W)
=IY+(x4+Ix41,f-+--1;i+-i,c+-c,p+-p).
(144
The definition of the map T: X0 + X0 for a given point is, therefore, dependent on the subset of & in which it lies: T(xIJ,,x~)E
%+
=
T”+(xI,
~3,
x4)
(W
T(xI,xJ,x~)E%-
=
T”-(XI,
~3,
x4)
(W
T(x,,x~,x,)EX~+
= T’+(x~, x3, x4)
WC)
T(x,,x3,x4)~X~-
z T’-(xl,
(W
x3,
x4).
In the following sections of this paper we shall examine the fixed points of the map T. These fixed points correspond to periodic orbits of the original equations (1). Of particular interest will be the stability of these fixed points, which is obtained from the Jacobian of the map evaluated at the fixed point of interest:
(16) where DTi I = a(x,)-“cos(wlog(x,) DT,z =
c(x4)_”
DTi3 = +
%(X4)
- l -pp
cos(w
UXl(X4)
- ’ -Pi
sin(o log(x4) + @i) - vcx3x4 l-”
DT2, = d(x4)-“cos(olog(x4) DTz
+ CD,)
log(x4)
+
0,)
+ 02)
=f(x4)-’
DT,, = - dx,W
- l -pp
cos(w
log(x4) + 02)
+ dx1(x4)- l -p 0 sin(w 108(x4) + a2) - vfx3x; ’ -”
0. M.
344
O’REILLY
DT+,, = g(~.+)-~ cos(o log(xe) + Q3) DTJz = i(x+)-’ DTa3 = - gx,(xJ)-
l -pp cos(0 log(x.+) + @a)
+ gxr(xq)_ l -p w sin(w log(x4) + Qa) - vixax; 1-“. The Jacobians for the other three representations of the map T can be obtained using the transformations (14b-d). Description of the full (steady-state and transient) dynamics for any of the attractors we will discuss may require all of the four submaps of T. However, our analysis is simplified considerably by noting that the steady-state dynamics of some of the attractors can be described using only some of the submaps of the map 7’. Of particular interest will be iterations of the map whose steady-state behavior can be. completeIy described using only T”+ or T’- and those which require T”- and T ‘+. The former orbits (following Conley [26-j) are known as non-transit orbits and the latter, transit orbits. For the analysis of the two symmetric non-transit attractors, it suffices to consider the behavior of the map 7’“+ and, for the transit attractor, by using the symmetry transformation s:
(X1,X2,X,rX~)~(X1,X2,
--x3,
-x.4)
(17)
it suffices to examine the behavior of the map T’+. Before analyzing the global bifurcations of interest, we wish to return briefly to the string. Interpreting the non-transit orbits in terms of their experimental counterparts, one observes that such orbits, which may be periodic or chaotic, preserve the direction of whirling of the string. This preservation occurs even though the orbit (whether chaotic or periodic) will appear in an experiment performed on the string as a precessing or amplitude-modulated ellipse (cf. [Z, 27-j). On the other hand, for the transit orbits the whirling direction of the string will vary, as shown in Fig. 3. Our later analysis will show the variation of orbits from transit to non-transit and uice versa. If this is interpreted in terms of the whirling direction of the string, then the Shil’nikov mechanism can be viewed as describing the coupling mechanism between the two whirling modes.
Fig. 3. A transit orbit showing how the direction of whirling of the string changes as the orbit evolves. The vertical and horizontal axes correspond to the vertical and horizontal directions of the string-s motion, respectively.
Global bifurcations 5. THE
in the forced vibration of a damped string GLUING
345
BIFURCATION
We now consider a special case of the map T where -p > 1. In the unscaled parameters this corresponds to the case where -p > 1. Numerical evidence (cf. Fig. 4) shows that for a = 1.0, y = 0.5, a homoclinic orbit, r, exists when A = A* = 3.405 01 = 0). The eigenvalues of the fixed point (q. pJ are 1= 0.405, p = -0.5, v = - 1.405 and w = 1.743. As A is varied, two possible scenarios are encountered. When A < A* b < 0) two stable, symmetric, periodic non-transit orbits appear and when A > A* @> 0) a single asymmetric, periodic transit orbit is present. This is a classic example of a gluing bifurcation (cf. [20] and the references therein for more information on this bifurcation). As ~1 -+ O-, the two symmetric periodic orbits “glue” together and form the “figure eight” homoclinic loop. Because of the symmetry, the homoclinic loop will break and form an asymmetric orbit which grows in size as p increases from 0. As the eigenvalues varied in value by less than 2% for the p values considered, we shall consider them as constant in the analysis that follows. Similarly, we shall assume that p and x4 are small. We note that similar behavior has been observed at other system parameter values. These orbits similarily satisfied -p > 1. Theorem 1. Assuming that at p= 0, homoclinic connection, r, to the fixed point (qr,pg) exists and that the eigenvalues of the fixed point have the form stated above, then for p < 0 two stable non-transit periodic orbits exists. For ~1> 0 a single stable transit periodic orbit exists. Furthermore, a gluing bifurcation occurs at p = 0. Proof. The proof of the existence of the non-transit orbits is obtained by applying theorem 1 of Neimark and Shil’nikov 1133 to both of the maps 7’“+ and T’-. Their theorem proves that, as p+ O-, each of the maps has two stable fixed points which approach homoclinicity. For the original system (1) these stable fixed points correspond to stable periodic orbits. The proof of the existence of the transit orbit follows identically by applying the theorem to 7”‘- or T’+. Asp+O+, the transit orbit approaches homoclinicity and, so, atp=Oa gluing bifurcation occurs.
I
(a
, (b)
-I I-
Fig. 4. The gluing bifurcation: (a) 01= 1.0, y = 0.5, A = 3.40 (II < Ok(b) a = 1.0, y = 0.5, A = 3.405 (J = 0); (c) a = 1.0, y = 0.5. A = 3.415 01 > 0).
346
O.M.
O'REILLY
The proof of this theorem can be obtained in a more intuitive manner by using a heurestic approach developed by Glendinning and Sparrow [22] and Glendinning [23]. For our analysis we need only consider the maps 7’“+ (for the behavior of the non-transit orbit) and T’+ (for the behavior of the transit orbit). For each of these maps we solve for the x4 coordinate of a fixed point. If the x4 coordinate of the fixed point approaches 0 as p approaches 0, its period tends to infinity and the fixed point is stable, then the fixed point corresponds to the periodic orbit which is approaching homoclinicity. Solving for x4 for both maps and taking a lowest order approximation gives F II+ =x4 +
P+ =
+/A-
i(x4)-"[pI-l +/d-2
g(x4)-Pcos(wlog(x4)
x4 + p +
i(x4)-“[c(rt
-XT]
+ @3)bl-l +
prz
+g(x4)-~~0~(d0g(x4)+uq[~rl
-x:-J
(lga)
-x:3.
(1W
- XT]
For a given set of system parameters, the x4 coordinate of the fixed point is given by the zeros of F. Both of these functions are shown in Fig. 5. In the case of F”+, F’+ this graphical analysis shows the presence of a single fixed point as P-P O- and p + O+, respectively. The stability of these fixed points can be obtained from the respective Jacobians (16) by noticing that at --p > 1 all of the terms in the Jacobian will be small, assuming x4 is small; so, the fixed points of the maps are stable. A similar results can be obtained by applying the bilinear transformation to the characteristic equation of the Jacobian and then using the Routh-Hurwitz stability criterion. The period of these orbits is ascertained by examining how the roots of F vary as p is varied. When x4 is small, the coefficients of i and g in (lSa, b) will be negligible and lx41z 1~1.Furthermore, an orbit which passes close to the saddle focus
(a)
(b) F
Y.
F”
(d)
(c)
F”
F”.
ILL X.
(e) F
(f Y.
1 F”
rc X.
FL K
X4
,._/ / L x.
Fig. 5. Fixed points of the map for -p > I: (a) non-transit orbit, p = - 0.1, (b) transit orbit, p = - 0.1; (c) non-transit orbit, p = 0.0; (d) transit orbit, p = 0.0; (e) non-transit orbit, P = 0.1; (f) transit orbit, p = 0.1.
Global bifurcations in the forced vibration of a damped string
347
Fig. 6. A transient chaotic orbit which eventually becomes periodic. The’ system parameter values are 01= 1.0, y = 0.45, A = 3.4 and the initial conditions arc (ql,pI,q2,p2) = (0.982214, - 0.342663. - 1.05, - 0.285).
will spend the majority of its period in the neighborhood of this fixed point and a lower bound period of such an orbit is given by the time of flight (9): 119) From (19) we see that as p tends to zero period, T tends to infinity and the periodic orbit approaches homoclinicity. The implication of the gluing bifurcation for the system behavior when the parameter ranges are close to those for this bifurcation is the presence of transient chaos. This was first observed by Holmes [21 J while analyzing a two-dimensional Shil’nikov-type map for a saddle focus and has been subsequently studied by Gambaudo and co-workers (cf. [ZOJ and the references therein). These transit orbits require more than two of the four maps we have derived to describe their behavior. However, as their steady-state behavior is that of a fixed point, it can be described using only one of the maps. Figure 6 illustrates one such example for the system under consideration. To date, none of the experimental results published [2, 6-J report the presence of the gluing bifurcation. This is probably due to the fact that it occurs in a very small region of the parameter space. The implication of the gluing bifurcation for the experimentalist is the likelihood of an increased presence of transient chaos, Because the chaotic motions are amplitude-modulated, they occur on time scales which are much larger than that of the external forcing and may, thus, be incorrectly interpreted. However, if one assumes that Miles’ equations are valid for the experimental system, then the eigenvalue spectrum from (4a, b) may be calculated as a check for transient chaos phenomena. In the parameter range we have discussed (a = 1.0, y = OS), Johnson and Bajaj [4, S] have found transit and non-transit chaotic motions for ~40 and ~1~0. These orbits occur after a sequence of ~riod-doubling bifurcations. As p is large for these orbits, our analysis is invalid for their interpretation. They may, however, occur as a result of the interaction of the unstable manifolds of (4.,pt) and the ~~od~oubling sequence. A similar behavior in a Riissler-type attractor of the three-dimensional Volterra equations has been discussed by Arnedo et al. [28). However, for their system one could prove the presence of a hyperbolic invariant set in a neighborhood of the homoclinic connection. This is not the case here and the interaction of these orbits with the unstable manifolds remains an open question.
6. THE HOMOCLINIC
EXPLOSION
A typical example of a homoctinic explosion is shown in Fig. 7. For a = 1.0, y = 0.45, a homoclinic orbit, I-, exists when A = A* = 3.078 & = 0). The eigenvalues of the fixed
us
0. M. O’REILLY
I
(0)
(b)
-1.5
1.5
Fig. 7. The homoclinic explosion: (a) 01.= 1.0, y = 0.45. A = 3.06 @ c 0); (b) 01= 1.0, y = 0.45, A = 3.078 @ = 0); (c) a = 1.0, y = 0.45, A = 3.20 b > 0).
point &,pL) are 1 = 0.493, p = - 0.45, v = - 1.39 and w = 1.88 and satisfy the condition - p < 1. A similar behavior that satisfies this condition is observed elsewhere in the parameter space. As ,u is varied, two possible scenarios are encountered. When p < O., two symmetric, chaotic non-transit attractors appear and when p ) 0, a single chaotic transit attractor is present. Kovacic and Wiggins [ZS] have analyzed a similar situation and proved the existence of a single hyperbolic set. Our case is considerably more subtle. Here several chaotic attractors exist and the region of their delineation is given by a homoclinic connection. Theorem 2. Assuming that at ~1= 0, a homoclinic connection, r, to the fixed point (qz, p,) exists and that the eigenvalues of the fixed point have the form stated above, then for p < 0 two hyperbolic invariants set exist, corresponding to the two non-transit attractors. For p > 0 a single hy~rbolic invariant set exists which corresponds to the transit attractor. For 7’“+, Wiggins (Cl83, pp. 261-267) has proven the existence of a hyperbolic invariant set @male horseshoe [29]) using a generalization of the Conley-Moser conditions. The existence of the second such set (for the map T’-) can be proven by using the symmetry condition (18). By using the symmetry transformation S, the requisite horizontal and vertical slabs can be constructed for the analysis of the T’+ map (cf. Fig. 8). The stretching and contracting conditions on T’+ are trivially satisfied because of the relation of its Jacobian to that of T”+. Finally, by using the transformation (x3 + - x3 on the horseshoe construction for T’+, Wiggins’ analysis for that of T”+ can be directly applied and the existence of a hyperbofic invariant set proven. OnIy one such attractor exists as we have used the symmetry condition (18) to account for the behavior of T”-.
Global bifurcations
in the forced vibration
of a damped string
349
(b)
(a)
Fig. 8. Projections of the vertical (V) and horizontal (H) slabs required for the construction of the horseshoe map: (a) non-transit orbit case; (b) transit orbit case.
(a)
(b)
G".
G’*
X.
(c 1 G”
(e)
6"
(f) G”*
G”
.
Fig. 9. Fixed points of the map for -p < 1: (a) non-transit orbit, p = - 0.1; (b) transit orbit, p = - 0.1; (c) non-transit orbit, p = 0.0; (d) transit orbit, p = O.@(e) non-transit orbit, p = 0.1; (f) transit orbit, p = 0.1.
We have just proven the existence ofchaotic attractors for p ~0; a homoclinic connection exists at p = 0 and it is of interest to examine the behavior of the submaps of T to examine the transition from one attractor to another as ~1goes through zero. To examine this transformation, we focus our attention on the fixed points of two of our four maps: T”+ and T’+ and assume that x4 and p are small. Solving for two of the coordinates (x, and x3) of the fixed points in terms of x4 coordinate, we obtain a transcendental equation for the x4 coordinate of the fixed point: G”+ = 0 for T”’ and G’+ = 0 for T’*. The values of x,.which
0. M. O’REILLY
350
satisfy these equations are the .x4 coordinates of the fixed points. From a graphical analysis (cf. Fig. 9), we see that the map T”+ has a finite number of fixed points when p < 0. The number of such fixed points grows unboundedly as p approaches 0.0 and, at /J = 0.0, one has a countable infinity of periodic orbits. When p increases from 0.0 the number of these periodic orbits decrease rapidly to 0. For the map T’+, the situation is, as expected, reversed. Determination of the stability of these fixed points is non-trivial. However, because a hyperbolic invariant set exists in a neighborhood of p= 0, the most accessible method of determining their stability is to infer their instability from the proof of the existence of the hyperbolic invariant set. This graphical analysis shows us that, as p tends to zero, an increasing number of periodic orbits are generated. Following the work of Yorke and Alligood [30] (see also [22, 23]), these periodic orbits transform through period-doubling and saddle-node bifurcations and eventually result in a chaotic attractor. What is unusual in our case is that examining how these periodic orbits are created and destroyed allows us to explain the existence and non-existence of the chaotic attractors as P is varied.
7.
IMPLICATION
FOR
OTHER
SYSTEMS
Because the Shil’nikov phenomena discussed here provide a mechanism for a transition from one direction of whirling to another (p > 0) or the preservation of the whirling direction (p < 0), they may be viewed as a mode-coupling mechanism. Such a modecoupling behavior has also been observed in other systems: the spherical pendulum (cf. Tritton [31]), surface waves in a cylindrical tank that is forced horizontally (cf. Funakoshi and Inoue [32]) and parametric vibrations of plates (cf. Yang and Sethna [33]). The amplitude-modulated equations for the first of these two systems are also given by (l), but the values of the constants A and B differ from those for the string. Both these systems have also been shown to possess a local bifurcation behavior similar to that of the string (Miles [34, 353). However, our numerical integrations of these two systems, while showing the presence of homoclinic connections, does not appear applicable. We conjecture that this is probably due to the presence of global bifurcations. In conclusion, our analysis, while providing a clearer picturer of the global bifurcations and mode-coupling mechanisms, needs to be applied on a case to case basis. Acknowledgements-This
work
was supported
by a post-doctoral
appointment
under Professor J. Dual at the
Institute of Mechanics of the Swiss Federal Institute of Technology in Ziirich. The author acknowledges the many useful conversations with and encouragement from Joseph Cusumano, Philip Holmes, Alexander Mielke and Francis C. Moon. Finally, the simulations shown in Figs 1.4.6 and 7 were performed using the package “Kaos” developed by J. Guckenheimer and S. Kim.
REFERENCES I. H. Harrison, Plane and circular motions of a string. J. Acoust. Sot. Am. 20, 874-875 (1948). 2. 0. O’Reilly and P. Holmes, Nonlinear, nonplanar and nonperiodic vibrations of a string. J. Sound Vih. 153. 413-435 (1992). 3. J. W. Miles, Resonant, nonplanar motion of a stretched string. J. Acoust. Sot. Am. 75. 855-861 (1984). 4. J. M. Johnson and A. K. Bajaj, Amplitude modulated and chaotic dynamics in resonant motion of strings. J. Sound Vib. 1uI. 87-107 (1989). 5. A. K. Bajaj and J. M. Johnson, Asymptotic techniques and complex dynamics in weakly non-linear forced mechanical systems. Inf. J. Non-Lineor Me& 25. 21 l-226 (1990). 6. T. C. A. Molten0 and N. B. Tutillaro, Torus doubling and chaotic string vibrations: experimental results. J. Sound Vih. 137, 327-330 (1990). 7. V. K. Melnikov, On the stability of the center for time periodic perturbations. Trans. Moscow Math Sot. 12, 153-204 (1963). 8. A. Mielke, P. Holmes and 0. O’Reilly, Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center. J. Dyn. Difl Eqns. 4. 95-126 (1992). 9. A. Mielke. Topological methods for discrete dynamical systems. GAMM-Mitteilungen. 2 (1990). 10. L. P. Shil’nikov. Some cases of generation of periodic motions in an n-dimensional space. Sou. Moth. Dokl. 3. 394-397 (1962). Il. A. A. Andronov and E. A. Leontovich, Generation of limit cycles from a separatrix forming a loop and from the separatrix of an equilibrium state of saddle-node type. Am. Math. Sot. Transl. (Series 2) 33, 189-231(1963). 12. L. P. Shil’nikov. A case of the existence of a countable number of periodic motions. Son Math. Dokl. 6.163-166 (1965).
Global bifurcations in the forced vibration of a damped string
351
13. Ju. I. Neimark and L. P. Shil’nikov. A case of generation of periodic motions. Sou. Math. Do&l. 6, 305-309 (1965). 14. L. P. Shil’nikov, On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of a saddle type. Moth. USSR-Sb. 6,427-438 (1968). IS. L. P. Shfnikov, A cont~bution to the probIem of the structure of an extended neigh~rh~ of a rough equilibrium state of saddle-focus type. Moth. LrSSR-Sb. IO, 91-102 (1970). 16. J. Moser. Stub/e and Random Motions in Dynumicul Sys&rms.Princeton University Press. Princeton (1973) 17. V. A. Afraimovich V. V. Bykov and L. P. Shil’nikov, On the origin and structure of the Lorenz attractor. Sou. Phys. Dokl. 22. 253-255 (1975). 18. S. Wiggins, Global B$wcations and Chaos. Springer, New York (1988). 19. S. Wiggins, introduction to Applied Nonlinear Dynamical Sysrems and Chaos. Springer, New York (1990). 20. J. M. Gambaudo, P. A. Glendinning and C. Tresser, Stable cycles with compficated structure. J. Physique L&r. 46.653-657 (1985). 21. P. Holmes, A strange family of thr~-dimensional vector fields near a degenerate singularity. J. Oi@ Eqns. 37, 38243 (1980). 22. P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits. J. Star. Phys. 35,645-696 (1984). 23. P. Glendinning. Bifurcations near homoclinic orbits with symmetry. Phys. Letr. 103A. 163-166 (1984). 24. C. Tresser, About some theorems by L. P. Shil’nikov. Ann. fttsr. H. Poincare 40.441461 (1984). 25. G. Kovacic and S. Wiggins. Orbits homoclinic to resonances:chaos in a model of the forced Sine-Gordon equation, Cattech. preprint (1989). 26. C. C. Conley, On the ultimate behavior of orbits with respect to an unstable critical point: 1. oscillation. asymptotic and capture orbits. J. Dig Eqn 5, 136-158 (1969). 27. 0. M. O’Reilly, The Chaotic Vibration of a String, Cornell University Ph.D. dissertation (1990). 28. A. Arnedo. P. Coullet and C. Tresser. Occurrence of strange attractors in threedimensional Volterra equations. Phys. Lert. 79A. 259-263 (1980). 29. S. Smale, Differentiable dynamical systems. Buff. Am. Mark. Sot. 73, 747-817 (1967). 30. J. A. Yorke and K. T. Alligood, Period doubIing cascadesof attractors: a prerequisite for horseshoes.Commun. Math. Phys. 101, 305-321 (1985). 31. D. J. Tritton. Ordered and chaotic motion of a forced spherical pendulum. Eur. J. Phys. 7, 162-169 (1986). 32. M. Funakoshi and M. Inoue. Surface waves due to resonant horizontal oscillation. J. Fluid Mech. 192.219-247 (1988). 33. X. L. Yang and P. R. Sethna, Local and global bifurcations in parametrically excited vibratioins of nearly square plates. Inr. J. Non-Linear Me&. 26, 199-220 (1991). 34. J. W. Miles, Resonant motion of a sphericaf pendulum. Phpicn flD, 309-333 (1984). 35. J. W. Miles, Resonantly forced surface waves in a circular cylinder. J. Fluid Meek. 149, 15-31 (1984).
NLH 28:3-F
c#zt%7.&2/93 1993 Perptm
s6.00 + .a2 Rat Ltd
GLOBAL BIFURCATIONS IN THE FORCED VIBRATION OF A DAMPED STRING OLIVER M. O’REILLY* Institute of Mechanics, Swiss Federal institute of Technology (ETH). CH-8092 Ziirich, Switxerland (Received 5 July 1991) Abstract-Using a Shil’nikov-type model, some global bifurcations present in the averaged quations for the harmonic vibration of a damped pretensioned linearly elastic string are examined. In particular, we are interested in examining those bifurcations which allow the string to change its direction of whirling These bifurcations are shown to be directly related to the structural instability of a homoclinic connection. From symmetry and stasbility considerations, we are able to show that this instability produces gluing bifurcations and homoclinic explosions and provides an explanation for some of the chaotic motions observed in this system. The consequences of our analysis for related physical systems which exhibit similar modal coupling is discussed. 1. INTRODUCTION
This paper examines some of the global bifur~tions observed in the amplitude-m~ufated equations of motion for the vibrating string. The string considered is pretensioned and fixed at both ends. One of these ends is excited ha~oni~ffy in a direction transverse to the length of the string. When the excitation frequency is close to a transverse harmonic, the planar periodic vibration becomes unstable and a periodic ballooning or whirling motion is produced, provided the excitation amplitude is sufficiently large. As the excitation frequency is increased, the whirling motion increases in amplitude. Further increase in the frequency produces a hysteretic collapse of the response amplitude and a smaller-amplitude planar periodic motion results. This phenomenon was first reported by Harrison [l] and inspired a large body of analytical and experimental work (cf. O’Reiffy and Holmes [2] for the relevant references). The majority of this work focused on the case where the excitation frequency is close to the first transverse harmonic and the string is modeled as a linearly elastic continuum. Amplitude-modulated chaotic behavior for the string was first suspected when Miles [3] showed the presence of Hopf bifurcations in the averaged equations of motion. Their presence was first observed numerically by Johnson and Bajaj [4,5] and experimentally by Molten0 and Tufiffaro [6]. O’Reiffy and Holmes [2] confirmed these results and analyzed the non-periodic behavior. Owing to the degenerate nature of the integrable limit of the averaged equations, they found that it was not possible to employ the Mefnikov technique [73 and obtained information on the global bifurcations. They did show, however, that in the absence of damping the chaotic motions can be explained via a Shif’nikov-type model for a Hamiltonian saddle center (cf. Miefke et aI. [8] and Miefke [9]). For the damped case, they indicated that the chaotic behavior could be interpreted using a Shif’nikov-type model for a saddle focus. However, no thorough investigation of this case was performed. Shi~nikov-ty~ mechanisms were originally developed by Shif’nikov [lo] whife generafizing the work of Andronov and Leontovich [ 11J on the generation of limit cycles in planar dynamical systems with homocfinic orbits to higher dimensions. The model possesses a homocfinic orbit to a fixed point of the flow for given parameter values of the system. Due to the structural instability of the homocfinic orbit, it will not be preserved as these parameters are varied. Shif’nikov showed that, depending upon the spectrum of the fixed point, a single stable periodic orbit [lo] or a countable number of unstable periodic orbits [12] can be produced when the homocfinic connection is broken. Generalizations of these results were subsequentially obtained [13-U] and some of them depend crucially on the dimension of the unstable manifold (cf. [lS] in particular). Development of the ConfeyMoser conditions [16], which can be directly applied to these models to show the existence * Present address: ~~rtment 94720, U.S.A.
of Mechanical Engineering, Unive~ity of California at Berkeley, Berkeley, CA 337
0. M. O’REILLY
338
of a hyperbolic invariant horseshoe, and the success of the models in explaining some of the global bifurcations observed in the Lorenz equations (cf. Afraimovich et al. [17]) precipitated the considerable subsequent interest in Shil’nikov-type mechanisms (cf. Wiggins [ 18, 193 and the references therein). Prior to performing a Shil’nikov-type analysis, we examine the local bifurcation phenomena which occur at the fixed point of interest, which gives us explicit expressions for the eigenvalues of this fixed point. Numerical simulations show the existence of a homoclinic connection to this fixed point for certain parameter values. A three-dimensional map is then constructed which describes the dynamics in a neighborhood of this connection. In constructing this map, we employ symmetry considerations extensively. This allows us later to use several known results available for more restrictive versions of our map. We then examine the global bifurcation behavior for two ranges of the spectrum of the fixed point. In one of these ranges a global bifurcation known as (after Gambaudo et al. [20]) the gluing bifurcation occurs; in the second range a homoclinic explosion is observed. In the former case transient chaos is observed and in the latter stable chaotic attractors are present. Shil’nikov-type analyses for physical systems have, to our knowledge, rarely ever been performed. In the majority of applied research of this topic the researcher concludes that the presence of a homoclinic connection implies the existence of a (nearby) hyperbolic invariant set. This may be the case; however, a detailed analysis produces far more interesting results. In our case, it allows us of distinguish between chaotic behavior observed numerically by Johnson and Bajaj [4, 51, which occurs due to period-doubling and homoclilnic phenomena; it forewarns us of the presence of transient chaotic behavior and provides an explanation of the mechanism which produces a qualitative change in the modal behavior of the string. These results are also applicable to other systems and we discuss this at the conclusion of the paper. Finally, we remark that our exposition has been influenced by the works of Holmes [21], Glendinning and Sparrow [22], Glendinning [23], Tresser [24] and Kovacic and Wiggins [ZS]. 2. THE
AVERAGED
EQUATIONS
OF MOTION
The equations of motion for the string are obtained by modeling the string as a linearly elastic continuum, with displacements in the plane parallel to the direction of the external forcing (henceforth known as the vertical direction), in the plane perpendicular to the plane of the excitation (henceforth known as the horizontal direction) and along the string axis (henceforth known as the longitudinal direction). Coupling between these three displacements is present in the strain energy density function. The three partial differential equations which describe the vibration of the string are then derived using the EulerLagrange equations. A Taylor series expansion to third order in the strains is then performed, followed by a Galerkin projection of the fundamental harmonics in the vertical, horizontal and longitudinal directions. This procedure results in three cubically coupled ordinary differential equations (cf. [2] for explicit details). As the chaotic motions observed experimentally [2,6] occur on time scales significantly lower than that of the forcing, these equations may be averaged with respect to the period of the forcing to produce a further set of three coupled ordinary differential equations. The equations for the longitudinal motion can be eliminated; the resulting averaged equations obtained after a resealing are: d
$Pl
=
- YPI - (A + AEh,
+ BMpz
$PZ
=
-
-
YP~
-
(A
+
Ah2
BMPI
d 541 = - yqi + (A + AQpi + BMq2 + a d 5% = - yq2 + (A +
WP,
-
BMq,.
These are the equations derived originally by Miles [3] and examined by various researchers [2, 4, 51. In the above equations the subscripts 1, 2 denote the vertical and
Global bifurcations in the forced vibration of a damped string
339
horizontal directions, respectively, and the qis are the generalized modal coordinates and the PiS the corresponding momenta. The harmonic excitation is introduced by two parameters: A, the detuning parameter, which defines the proximity of the forcing frequency to the transverse harmonic frequency and a which defines the forcing amplitude. E and M are the energy (p: + P: + qf + qj) and angular momentum (pLqz - qlP2) of the generalized modal coordinates and momenta and A and B are constants, equal to -1.5 and 1.0, respectively, for the vibrating string model. The parameter y models the linear viscous damping present in the system. We note that the averaged equations are valid when the geomet~c non-linea~ties (cubic terms) are weak and the forcing and damping small. The system described by (1) may be viewed as an O(2) symmet~c system whose symmetry is broken by the harmonic forcing. This particular symmetry breaking produces a 1-l resonance phenomena between the mode in the plane of the forcing (the vertical mode) and the mode in the plane perpendicular to the forcing plane (the horizontal mode). These equations have been shown by Miles [33 to possess pitchfork, saddle-node and Hopf bifurcations (cf. [3], Fig. 1) and by Johnson and Bajaj [4, S] to possess sequences of period-doubling bifurcations (cf. [4], Figs 5 and 8) and chaotic solutions. In our analysis we shall confine our attention to the homoclinic phenomena and shall not address the period-doubling bifurcations. A key feature of (1) is the invariance under the flow of the pl-ql plane. This implies that the flow on this plane will always be integrable. The system of equations (I) also possesses the symmetry (P~,p~,q~,qz)~(p~, -P2t%-42h (21 We note that in the absence of this symmetry the construction of a single map would sufliee to describe the dynamics of interest. Its presence necessitates the construction of four maps.
3. BIFURCATION
IN THE INVARIANT
PLANE
An invariant plane for (1) is the ql-pi plane. Analysis of the phenomena occurring in this plane determines the behavior of the vertical mode, in particular its stability:
$1=- YPI -
(A + AE)q,
= - yqi + (A + AEfp, -f- a
(3)
where E denotes the p1anar energy p: + qf. When a = y = 0, the system has a fixed point at the origin and its phase space is foliated by periodic orbits. Addition of forcing (a) alters the phase plane depending upon the value of A and y. As A is increased from zero, the fixed point (qL,pc)which was originally at the origin increases in magnitude. Further increase in A results in a saddle-node bifurcation (cf. [33, Fig. 1). This bifurcation produces an additional two fixed points (qo, po). Subsequent increase in A results in a pitchfork bifurcation occurring at (q6, PC).This pitchfork birurcation does not affect the stability of (qe,pJ in the invariant plane; however, (qt,P.) is now unstable in the out-of-plane direction (cf. Fig. 1). In the experimental system this corresponds to the onset of the whirling motions. These two whirling motions, which correspond to the stable branches of the pitchfork bifurcation, satisfy the symmetry condition (2) and may be viewed as corresponding to clockwise (p2,q2 > 0) and counterclockwise Qt, q2 c 0) whirling motions of the string. The two other fixed points in the invariant plane are both stable in the out-of-plane direction, although one of them (PO)is a saddle in the invariant plane and the other (qo) is a sink. The aforementioned stability information can be ascertained by linearizing (1) about a planar equilibrium point (ql,pl,q2 = O,p, = 0) and determining the eigenvalues: i“1.2
=
-
R3.4
=
-v+
y f
+(A
+ AE)(A + 3AE)
(W
,/-(A
+ AE)(A + (A + B)E)e
WI
The eigenvalues c?I, t determine the stability of the planar equilib~um in the ql-Pr plane, whereas .&, 4 determines the out-of-plane stability. Numerical evidence, which we sha11
0. M. O'REILLY
340
bl 3
F:
-3
-3
I
q, Fig. 1. The p,-ql
qt
invariant plane: (a) a = 1.0, y = 0.0, A = 3.4, (b) u = 1.0, y = 0.5, A = 3.4.
discuss later, shows a homoclinic orbit which is biasymptotic to (qZ,ps); so, we now characterize the eigenvalues of this fixed point. We note that when A lies between the pitchfork bifurcation points, the eigenvalues of &pC) have the form 1 1.2 =
P f iw
/13 = v A‘$ = 1
(5)
where p = - y, v < 0 and 1 > 0. i.e. the fixed point is a saddle focus with a threedimensional stable manifold and a one-dimensional unstable manifold. The dimensions of these manifolds imply that the homociinic connection is a codimension-one bifurcation problem and only one parameter in our analysis needs to be varied in order to examine its structural instability. The parameter we choose to vary is the detuning parameter A; however, similar results will be obtained by varying a or y. For the remainder of the analysis we shall assume that -v, w > - p, 1. We remark that in the absence of damping (ql, pe) is a saddle center and the system (1) is Hamiltonian. As mentioned earlier, this system afso possesses a Shil’nikov-type mechanism. For more details on this system the interested reader is referred to [2, 8, 9J and the references therein.
4.
CONSTRUCTION
OF THE
SHIL’NIKOV
TYPE
MAP
We are interested in constructing a dynamical system which describes the dynamics of orbits which pass dose to the fixed point (qe,pc). From numerical evidence a homoclinic orbit, r, exists which is biasymptotic to this fixed point at a given value of A = A* (a, y are fixed). In keeping with the literature. we define a new parameter ,u = A - A* and, so, at ~1= 0 the homoclinic connection, r, exists. This a priori knowledge of the existence of r is required for the construction we are about to perform. Our purpose is to describe the behavior of orbits which remain close to r and for a range of values of p about I( = 0. For the case under consideration, a map was previously derived by Kovacic and Wiggins [25] (cf. [18], pp. 261-267) while analyzing the damped Sine-Gordon equation. Our construction mirrors theirs, with some slight modifications which we shall need later in our analysis. The first step in the procedure is to perform a coordinate change (PlrPzr4rr(lz)-+(Pt - &lP2191
and, using the Hartman-Grobman
- 4.1q2) = (&,X2*X3*X4)
(6)
linearization theorem (cf. [19]), obtain the vector field in
Global bifurcations in the forced vibration of a damped string
341
Fig. 2. Definitions of the Poincare sections used to define the map T,
a neighborhood of the origin:
dx3
dt = vx3
An implicit assumption here is that the linearization is valid only in the neighborhood of the origin and, so, we have omitted the higher-order terms in (7). Two maps are defined: a “local map”, To, which describes the dynamics in an E neighborhood of the origin and a ‘global map”, T,, which describes the dynamics of orbits which enter and leave this neighborhood of the origin (cf. Fig. 2): X:u,+= {(x~,x~,x~,x~)ER*Ix~ = O,cez*p'ru s x1 s E,O 5 x4 5 E,O s x3 I;E}
(W
CYO-={(X,,~~,X~,X~)ER~~X~=O,~~~~~'~~X~I;~,O~X~~E,OI;
(W
-X~ZSE}
X'o+= {(xI,x~,x~,x~)ER*~x~ = 0, ce2=p'm 5 xl s E,O 5 -x4 s E,O s x3 s E}
(8~)
X:b-= {(xI,x~,xJ,x~)E R* Ix2 = 0, seZ*P’m5 x1 s .s, 0 s - x4 s C,0 s - x3 s E}
(gd)
and c :+ = {(x,,x~,x~,x~)ER~IO
{(x~,x~,x~,x~)E
R410 < -x3,x4
= E}
(W
Xi+ ={(x,,x3,x3,x4)~R4~O
(80
The bound sezxp’* s xl is calculated by considering an orbit which intersects any of the four X;“*s at x1 = E and establishing the conditions for the size of Z$‘*, so that the only intersection of an orbit in the neighborhood of the origin Ze (= ZY,’ u Rx‘ u Zb’ u EL-) is the initial one at x1 = E.
342
0. M. O’REILLY
For a complete definition of the map which accounts for the symmetry conditions, it is necessary to construct separate maps, for each of the four Zg:‘* sections. We shall first construct the map for Ilf’ and use the invariance of the flow under (_x,,x2,x3,x4) + -x4) to construct the remaining maps. Consider an orbit which enters (XI, x2, -x3, a neighborhood of the origin through XI’ at (x1,,, xzo, xJo, xao) and leaves through Cl’ at (.x~~,xzI,x~~, ~41 = .s).The time of flight, r, for such an orbit can be calculated by directly integrating (7d): r+og
0$ .
Substitution of the time of flight into the integrations of (7a-c) yields the following map, which describes the behavior of some of the orbits in a neighborhood of the origin: U+ To : x:“o+ -+x:1+ +-)“Acos(~log(~))
Xl x2 = 0 x3
[
x4
Il-
-
xr(~>“^sin(~fog(~))
(10)
& v/i x3 0 F 4 &
Furthermore, we restrict the size of Ey’ so that PO’ will map Z$’ on to X:rl+.This can be achieved by restricting the size of the small parameter E.The “global map”Ty’ is defined by considering the definitions of the Poincare sections which it maps:
x:
[!L1 Xl x2
-pr,
0
+
-P2
x3
x4
(11)
*
-P
Because the homoclinic connection, I-, is structurally unstable, this connection will be present only when p = 0. This instability is accounted for by the introduction of the additional parameters p, jtr, and ,ur2 in (11). The parameter x* is the intersection of r with CZ+: x* =
;(I + e2np’o),0,0,0 , (
The full map T"+ = Ty+0Tl;'can now be defined: T”+:x:“,+-+x;
>
(12)
Global
bifurcations
in the forced vibration
of a damped string
343
We note that because x z = 0 on any of the four subsets of Co, it is eliminated from our analysis. This enables us to reduce the dynamics (of interest) of (1) from a four-dimensional system of ordinary differential equations to a three-dimensional map. To make the analysis more tractable, several resealings are performed:
tan@,) = i,
h
tan(@) = i,
tan(Q3) = s
XI [I [
for T”+:
which result in the resealed representation
x3
+
x4
axi(x~)-~COs(O10g(X,$) + 0,) + cxJ(x‘+)-” - jA-i + x: dx,(x,)-“cos(olog(x4) + 02) +fx3(x~)-” - $2 gx1(x4)-Pcos(olog(x4) + (IQ + ix3(x4)-” - p
.1 (144
It now remains to define the remaining three maps. These are obtained by examining the symmetry condition (4) and their effects on the map ci+:
T”-: Cl;-- -2:b
=
T”+b4
+
1x41,~
-,
-
P)
Wb)
T”:~b+~~cl;=T”‘~-_,-~i-,-i,c-r-c)
T’-:Zb-+Zb
W)
=IY+(x4+Ix41,f-+--1;i+-i,c+-c,p+-p).
(144
The definition of the map T: X0 + X0 for a given point is, therefore, dependent on the subset of & in which it lies: T(xIJ,,x~)E
%+
=
T”+(xI,
~3,
x4)
(W
T(xI,xJ,x~)E%-
=
T”-(XI,
~3,
x4)
(W
T(x,,x~,x,)EX~+
= T’+(x~, x3, x4)
WC)
T(x,,x3,x4)~X~-
z T’-(xl,
(W
x3,
x4).
In the following sections of this paper we shall examine the fixed points of the map T. These fixed points correspond to periodic orbits of the original equations (1). Of particular interest will be the stability of these fixed points, which is obtained from the Jacobian of the map evaluated at the fixed point of interest:
(16) where DTi I = a(x,)-“cos(wlog(x,) DT,z =
c(x4)_”
DTi3 = +
%(X4)
- l -pp
cos(w
UXl(X4)
- ’ -Pi
sin(o log(x4) + @i) - vcx3x4 l-”
DT2, = d(x4)-“cos(olog(x4) DTz
+ CD,)
log(x4)
+
0,)
+ 02)
=f(x4)-’
DT,, = - dx,W
- l -pp
cos(w
log(x4) + 02)
+ dx1(x4)- l -p 0 sin(w 108(x4) + a2) - vfx3x; ’ -”
0. M.
344
O’REILLY
DT+,, = g(~.+)-~ cos(o log(xe) + Q3) DTJz = i(x+)-’ DTa3 = - gx,(xJ)-
l -pp cos(0 log(x.+) + @a)
+ gxr(xq)_ l -p w sin(w log(x4) + Qa) - vixax; 1-“. The Jacobians for the other three representations of the map T can be obtained using the transformations (14b-d). Description of the full (steady-state and transient) dynamics for any of the attractors we will discuss may require all of the four submaps of T. However, our analysis is simplified considerably by noting that the steady-state dynamics of some of the attractors can be described using only some of the submaps of the map 7’. Of particular interest will be iterations of the map whose steady-state behavior can be. completeIy described using only T”+ or T’- and those which require T”- and T ‘+. The former orbits (following Conley [26-j) are known as non-transit orbits and the latter, transit orbits. For the analysis of the two symmetric non-transit attractors, it suffices to consider the behavior of the map 7’“+ and, for the transit attractor, by using the symmetry transformation s:
(X1,X2,X,rX~)~(X1,X2,
--x3,
-x.4)
(17)
it suffices to examine the behavior of the map T’+. Before analyzing the global bifurcations of interest, we wish to return briefly to the string. Interpreting the non-transit orbits in terms of their experimental counterparts, one observes that such orbits, which may be periodic or chaotic, preserve the direction of whirling of the string. This preservation occurs even though the orbit (whether chaotic or periodic) will appear in an experiment performed on the string as a precessing or amplitude-modulated ellipse (cf. [Z, 27-j). On the other hand, for the transit orbits the whirling direction of the string will vary, as shown in Fig. 3. Our later analysis will show the variation of orbits from transit to non-transit and uice versa. If this is interpreted in terms of the whirling direction of the string, then the Shil’nikov mechanism can be viewed as describing the coupling mechanism between the two whirling modes.
Fig. 3. A transit orbit showing how the direction of whirling of the string changes as the orbit evolves. The vertical and horizontal axes correspond to the vertical and horizontal directions of the string-s motion, respectively.
Global bifurcations 5. THE
in the forced vibration of a damped string GLUING
345
BIFURCATION
We now consider a special case of the map T where -p > 1. In the unscaled parameters this corresponds to the case where -p > 1. Numerical evidence (cf. Fig. 4) shows that for a = 1.0, y = 0.5, a homoclinic orbit, r, exists when A = A* = 3.405 01 = 0). The eigenvalues of the fixed point (q. pJ are 1= 0.405, p = -0.5, v = - 1.405 and w = 1.743. As A is varied, two possible scenarios are encountered. When A < A* b < 0) two stable, symmetric, periodic non-transit orbits appear and when A > A* @> 0) a single asymmetric, periodic transit orbit is present. This is a classic example of a gluing bifurcation (cf. [20] and the references therein for more information on this bifurcation). As ~1 -+ O-, the two symmetric periodic orbits “glue” together and form the “figure eight” homoclinic loop. Because of the symmetry, the homoclinic loop will break and form an asymmetric orbit which grows in size as p increases from 0. As the eigenvalues varied in value by less than 2% for the p values considered, we shall consider them as constant in the analysis that follows. Similarly, we shall assume that p and x4 are small. We note that similar behavior has been observed at other system parameter values. These orbits similarily satisfied -p > 1. Theorem 1. Assuming that at p= 0, homoclinic connection, r, to the fixed point (qr,pg) exists and that the eigenvalues of the fixed point have the form stated above, then for p < 0 two stable non-transit periodic orbits exists. For ~1> 0 a single stable transit periodic orbit exists. Furthermore, a gluing bifurcation occurs at p = 0. Proof. The proof of the existence of the non-transit orbits is obtained by applying theorem 1 of Neimark and Shil’nikov 1133 to both of the maps 7’“+ and T’-. Their theorem proves that, as p+ O-, each of the maps has two stable fixed points which approach homoclinicity. For the original system (1) these stable fixed points correspond to stable periodic orbits. The proof of the existence of the transit orbit follows identically by applying the theorem to 7”‘- or T’+. Asp+O+, the transit orbit approaches homoclinicity and, so, atp=Oa gluing bifurcation occurs.
I
(a
, (b)
-I I-
Fig. 4. The gluing bifurcation: (a) 01= 1.0, y = 0.5, A = 3.40 (II < Ok(b) a = 1.0, y = 0.5, A = 3.405 (J = 0); (c) a = 1.0, y = 0.5. A = 3.415 01 > 0).
346
O.M.
O'REILLY
The proof of this theorem can be obtained in a more intuitive manner by using a heurestic approach developed by Glendinning and Sparrow [22] and Glendinning [23]. For our analysis we need only consider the maps 7’“+ (for the behavior of the non-transit orbit) and T’+ (for the behavior of the transit orbit). For each of these maps we solve for the x4 coordinate of a fixed point. If the x4 coordinate of the fixed point approaches 0 as p approaches 0, its period tends to infinity and the fixed point is stable, then the fixed point corresponds to the periodic orbit which is approaching homoclinicity. Solving for x4 for both maps and taking a lowest order approximation gives F II+ =x4 +
P+ =
+/A-
i(x4)-"[pI-l +/d-2
g(x4)-Pcos(wlog(x4)
x4 + p +
i(x4)-“[c(rt
-XT]
+ @3)bl-l +
prz
+g(x4)-~~0~(d0g(x4)+uq[~rl
-x:-J
(lga)
-x:3.
(1W
- XT]
For a given set of system parameters, the x4 coordinate of the fixed point is given by the zeros of F. Both of these functions are shown in Fig. 5. In the case of F”+, F’+ this graphical analysis shows the presence of a single fixed point as P-P O- and p + O+, respectively. The stability of these fixed points can be obtained from the respective Jacobians (16) by noticing that at --p > 1 all of the terms in the Jacobian will be small, assuming x4 is small; so, the fixed points of the maps are stable. A similar results can be obtained by applying the bilinear transformation to the characteristic equation of the Jacobian and then using the Routh-Hurwitz stability criterion. The period of these orbits is ascertained by examining how the roots of F vary as p is varied. When x4 is small, the coefficients of i and g in (lSa, b) will be negligible and lx41z 1~1.Furthermore, an orbit which passes close to the saddle focus
(a)
(b) F
Y.
F”
(d)
(c)
F”
F”.
ILL X.
(e) F
(f Y.
1 F”
rc X.
FL K
X4
,._/ / L x.
Fig. 5. Fixed points of the map for -p > I: (a) non-transit orbit, p = - 0.1, (b) transit orbit, p = - 0.1; (c) non-transit orbit, p = 0.0; (d) transit orbit, p = 0.0; (e) non-transit orbit, P = 0.1; (f) transit orbit, p = 0.1.
Global bifurcations in the forced vibration of a damped string
347
Fig. 6. A transient chaotic orbit which eventually becomes periodic. The’ system parameter values are 01= 1.0, y = 0.45, A = 3.4 and the initial conditions arc (ql,pI,q2,p2) = (0.982214, - 0.342663. - 1.05, - 0.285).
will spend the majority of its period in the neighborhood of this fixed point and a lower bound period of such an orbit is given by the time of flight (9): 119) From (19) we see that as p tends to zero period, T tends to infinity and the periodic orbit approaches homoclinicity. The implication of the gluing bifurcation for the system behavior when the parameter ranges are close to those for this bifurcation is the presence of transient chaos. This was first observed by Holmes [21 J while analyzing a two-dimensional Shil’nikov-type map for a saddle focus and has been subsequently studied by Gambaudo and co-workers (cf. [ZOJ and the references therein). These transit orbits require more than two of the four maps we have derived to describe their behavior. However, as their steady-state behavior is that of a fixed point, it can be described using only one of the maps. Figure 6 illustrates one such example for the system under consideration. To date, none of the experimental results published [2, 6-J report the presence of the gluing bifurcation. This is probably due to the fact that it occurs in a very small region of the parameter space. The implication of the gluing bifurcation for the experimentalist is the likelihood of an increased presence of transient chaos, Because the chaotic motions are amplitude-modulated, they occur on time scales which are much larger than that of the external forcing and may, thus, be incorrectly interpreted. However, if one assumes that Miles’ equations are valid for the experimental system, then the eigenvalue spectrum from (4a, b) may be calculated as a check for transient chaos phenomena. In the parameter range we have discussed (a = 1.0, y = OS), Johnson and Bajaj [4, S] have found transit and non-transit chaotic motions for ~40 and ~1~0. These orbits occur after a sequence of ~riod-doubling bifurcations. As p is large for these orbits, our analysis is invalid for their interpretation. They may, however, occur as a result of the interaction of the unstable manifolds of (4.,pt) and the ~~od~oubling sequence. A similar behavior in a Riissler-type attractor of the three-dimensional Volterra equations has been discussed by Arnedo et al. [28). However, for their system one could prove the presence of a hyperbolic invariant set in a neighborhood of the homoclinic connection. This is not the case here and the interaction of these orbits with the unstable manifolds remains an open question.
6. THE HOMOCLINIC
EXPLOSION
A typical example of a homoctinic explosion is shown in Fig. 7. For a = 1.0, y = 0.45, a homoclinic orbit, I-, exists when A = A* = 3.078 & = 0). The eigenvalues of the fixed
us
0. M. O’REILLY
I
(0)
(b)
-1.5
1.5
Fig. 7. The homoclinic explosion: (a) 01.= 1.0, y = 0.45. A = 3.06 @ c 0); (b) 01= 1.0, y = 0.45, A = 3.078 @ = 0); (c) a = 1.0, y = 0.45, A = 3.20 b > 0).
point &,pL) are 1 = 0.493, p = - 0.45, v = - 1.39 and w = 1.88 and satisfy the condition - p < 1. A similar behavior that satisfies this condition is observed elsewhere in the parameter space. As ,u is varied, two possible scenarios are encountered. When p < O., two symmetric, chaotic non-transit attractors appear and when p ) 0, a single chaotic transit attractor is present. Kovacic and Wiggins [ZS] have analyzed a similar situation and proved the existence of a single hyperbolic set. Our case is considerably more subtle. Here several chaotic attractors exist and the region of their delineation is given by a homoclinic connection. Theorem 2. Assuming that at ~1= 0, a homoclinic connection, r, to the fixed point (qz, p,) exists and that the eigenvalues of the fixed point have the form stated above, then for p < 0 two hyperbolic invariants set exist, corresponding to the two non-transit attractors. For p > 0 a single hy~rbolic invariant set exists which corresponds to the transit attractor. For 7’“+, Wiggins (Cl83, pp. 261-267) has proven the existence of a hyperbolic invariant set @male horseshoe [29]) using a generalization of the Conley-Moser conditions. The existence of the second such set (for the map T’-) can be proven by using the symmetry condition (18). By using the symmetry transformation S, the requisite horizontal and vertical slabs can be constructed for the analysis of the T’+ map (cf. Fig. 8). The stretching and contracting conditions on T’+ are trivially satisfied because of the relation of its Jacobian to that of T”+. Finally, by using the transformation (x3 + - x3 on the horseshoe construction for T’+, Wiggins’ analysis for that of T”+ can be directly applied and the existence of a hyperbofic invariant set proven. OnIy one such attractor exists as we have used the symmetry condition (18) to account for the behavior of T”-.
Global bifurcations
in the forced vibration
of a damped string
349
(b)
(a)
Fig. 8. Projections of the vertical (V) and horizontal (H) slabs required for the construction of the horseshoe map: (a) non-transit orbit case; (b) transit orbit case.
(a)
(b)
G".
G’*
X.
(c 1 G”
(e)
6"
(f) G”*
G”
.
Fig. 9. Fixed points of the map for -p < 1: (a) non-transit orbit, p = - 0.1; (b) transit orbit, p = - 0.1; (c) non-transit orbit, p = 0.0; (d) transit orbit, p = O.@(e) non-transit orbit, p = 0.1; (f) transit orbit, p = 0.1.
We have just proven the existence ofchaotic attractors for p ~0; a homoclinic connection exists at p = 0 and it is of interest to examine the behavior of the submaps of T to examine the transition from one attractor to another as ~1goes through zero. To examine this transformation, we focus our attention on the fixed points of two of our four maps: T”+ and T’+ and assume that x4 and p are small. Solving for two of the coordinates (x, and x3) of the fixed points in terms of x4 coordinate, we obtain a transcendental equation for the x4 coordinate of the fixed point: G”+ = 0 for T”’ and G’+ = 0 for T’*. The values of x,.which
0. M. O’REILLY
350
satisfy these equations are the .x4 coordinates of the fixed points. From a graphical analysis (cf. Fig. 9), we see that the map T”+ has a finite number of fixed points when p < 0. The number of such fixed points grows unboundedly as p approaches 0.0 and, at /J = 0.0, one has a countable infinity of periodic orbits. When p increases from 0.0 the number of these periodic orbits decrease rapidly to 0. For the map T’+, the situation is, as expected, reversed. Determination of the stability of these fixed points is non-trivial. However, because a hyperbolic invariant set exists in a neighborhood of p= 0, the most accessible method of determining their stability is to infer their instability from the proof of the existence of the hyperbolic invariant set. This graphical analysis shows us that, as p tends to zero, an increasing number of periodic orbits are generated. Following the work of Yorke and Alligood [30] (see also [22, 23]), these periodic orbits transform through period-doubling and saddle-node bifurcations and eventually result in a chaotic attractor. What is unusual in our case is that examining how these periodic orbits are created and destroyed allows us to explain the existence and non-existence of the chaotic attractors as P is varied.
7.
IMPLICATION
FOR
OTHER
SYSTEMS
Because the Shil’nikov phenomena discussed here provide a mechanism for a transition from one direction of whirling to another (p > 0) or the preservation of the whirling direction (p < 0), they may be viewed as a mode-coupling mechanism. Such a modecoupling behavior has also been observed in other systems: the spherical pendulum (cf. Tritton [31]), surface waves in a cylindrical tank that is forced horizontally (cf. Funakoshi and Inoue [32]) and parametric vibrations of plates (cf. Yang and Sethna [33]). The amplitude-modulated equations for the first of these two systems are also given by (l), but the values of the constants A and B differ from those for the string. Both these systems have also been shown to possess a local bifurcation behavior similar to that of the string (Miles [34, 353). However, our numerical integrations of these two systems, while showing the presence of homoclinic connections, does not appear applicable. We conjecture that this is probably due to the presence of global bifurcations. In conclusion, our analysis, while providing a clearer picturer of the global bifurcations and mode-coupling mechanisms, needs to be applied on a case to case basis. Acknowledgements-This
work
was supported
by a post-doctoral
appointment
under Professor J. Dual at the
Institute of Mechanics of the Swiss Federal Institute of Technology in Ziirich. The author acknowledges the many useful conversations with and encouragement from Joseph Cusumano, Philip Holmes, Alexander Mielke and Francis C. Moon. Finally, the simulations shown in Figs 1.4.6 and 7 were performed using the package “Kaos” developed by J. Guckenheimer and S. Kim.
REFERENCES I. H. Harrison, Plane and circular motions of a string. J. Acoust. Sot. Am. 20, 874-875 (1948). 2. 0. O’Reilly and P. Holmes, Nonlinear, nonplanar and nonperiodic vibrations of a string. J. Sound Vih. 153. 413-435 (1992). 3. J. W. Miles, Resonant, nonplanar motion of a stretched string. J. Acoust. Sot. Am. 75. 855-861 (1984). 4. J. M. Johnson and A. K. Bajaj, Amplitude modulated and chaotic dynamics in resonant motion of strings. J. Sound Vib. 1uI. 87-107 (1989). 5. A. K. Bajaj and J. M. Johnson, Asymptotic techniques and complex dynamics in weakly non-linear forced mechanical systems. Inf. J. Non-Lineor Me& 25. 21 l-226 (1990). 6. T. C. A. Molten0 and N. B. Tutillaro, Torus doubling and chaotic string vibrations: experimental results. J. Sound Vih. 137, 327-330 (1990). 7. V. K. Melnikov, On the stability of the center for time periodic perturbations. Trans. Moscow Math Sot. 12, 153-204 (1963). 8. A. Mielke, P. Holmes and 0. O’Reilly, Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center. J. Dyn. Difl Eqns. 4. 95-126 (1992). 9. A. Mielke. Topological methods for discrete dynamical systems. GAMM-Mitteilungen. 2 (1990). 10. L. P. Shil’nikov. Some cases of generation of periodic motions in an n-dimensional space. Sou. Moth. Dokl. 3. 394-397 (1962). Il. A. A. Andronov and E. A. Leontovich, Generation of limit cycles from a separatrix forming a loop and from the separatrix of an equilibrium state of saddle-node type. Am. Math. Sot. Transl. (Series 2) 33, 189-231(1963). 12. L. P. Shil’nikov. A case of the existence of a countable number of periodic motions. Son Math. Dokl. 6.163-166 (1965).
Global bifurcations in the forced vibration of a damped string
351
13. Ju. I. Neimark and L. P. Shil’nikov. A case of generation of periodic motions. Sou. Math. Do&l. 6, 305-309 (1965). 14. L. P. Shil’nikov, On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of a saddle type. Moth. USSR-Sb. 6,427-438 (1968). IS. L. P. Shfnikov, A cont~bution to the probIem of the structure of an extended neigh~rh~ of a rough equilibrium state of saddle-focus type. Moth. LrSSR-Sb. IO, 91-102 (1970). 16. J. Moser. Stub/e and Random Motions in Dynumicul Sys&rms.Princeton University Press. Princeton (1973) 17. V. A. Afraimovich V. V. Bykov and L. P. Shil’nikov, On the origin and structure of the Lorenz attractor. Sou. Phys. Dokl. 22. 253-255 (1975). 18. S. Wiggins, Global B$wcations and Chaos. Springer, New York (1988). 19. S. Wiggins, introduction to Applied Nonlinear Dynamical Sysrems and Chaos. Springer, New York (1990). 20. J. M. Gambaudo, P. A. Glendinning and C. Tresser, Stable cycles with compficated structure. J. Physique L&r. 46.653-657 (1985). 21. P. Holmes, A strange family of thr~-dimensional vector fields near a degenerate singularity. J. Oi@ Eqns. 37, 38243 (1980). 22. P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits. J. Star. Phys. 35,645-696 (1984). 23. P. Glendinning. Bifurcations near homoclinic orbits with symmetry. Phys. Letr. 103A. 163-166 (1984). 24. C. Tresser, About some theorems by L. P. Shil’nikov. Ann. fttsr. H. Poincare 40.441461 (1984). 25. G. Kovacic and S. Wiggins. Orbits homoclinic to resonances:chaos in a model of the forced Sine-Gordon equation, Cattech. preprint (1989). 26. C. C. Conley, On the ultimate behavior of orbits with respect to an unstable critical point: 1. oscillation. asymptotic and capture orbits. J. Dig Eqn 5, 136-158 (1969). 27. 0. M. O’Reilly, The Chaotic Vibration of a String, Cornell University Ph.D. dissertation (1990). 28. A. Arnedo. P. Coullet and C. Tresser. Occurrence of strange attractors in threedimensional Volterra equations. Phys. Lert. 79A. 259-263 (1980). 29. S. Smale, Differentiable dynamical systems. Buff. Am. Mark. Sot. 73, 747-817 (1967). 30. J. A. Yorke and K. T. Alligood, Period doubIing cascadesof attractors: a prerequisite for horseshoes.Commun. Math. Phys. 101, 305-321 (1985). 31. D. J. Tritton. Ordered and chaotic motion of a forced spherical pendulum. Eur. J. Phys. 7, 162-169 (1986). 32. M. Funakoshi and M. Inoue. Surface waves due to resonant horizontal oscillation. J. Fluid Mech. 192.219-247 (1988). 33. X. L. Yang and P. R. Sethna, Local and global bifurcations in parametrically excited vibratioins of nearly square plates. Inr. J. Non-Linear Me&. 26, 199-220 (1991). 34. J. W. Miles, Resonant motion of a sphericaf pendulum. Phpicn flD, 309-333 (1984). 35. J. W. Miles, Resonantly forced surface waves in a circular cylinder. J. Fluid Meek. 149, 15-31 (1984).
NLH 28:3-F