The bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator

Physica D 34 (1989) 169-182 North-Holland, Amsterdam

THE BIFURCATION TO HOMOCLINIC TORI IN THE QUASIPERIODICALLY FORCED DUFFING OSCHJ~TOR Kayo IDE l and Stephen WIGGINS 2 t Graduate Aeronautical Laboratories, Caltech, Pasadena, CA 91125, USA 2Applied Mechanics, Caitech, Pasadena, CA 91125, USA Received 15 April 1988 Revised manuscript received 8 August 1988 Communicated by J.E. Marsden

Using recently developed techniques of Wiggins [1988] we study the bifurcation to homoclinic tori in the quasiperiodicaUy forced Dulling oscillator. We show how homoclinic tori give rise to chaotic dynamics for single-degree-of-freedom quasiperiodically forced oscillators in much the same way as Smale horseshoes in the periodically forced case. We give a complete description of the bifurcation set in the five-dimensional parameter space for the two-frequency forced Dufling oscillator and then show how the two-frequency results can be used to understand the effects of an arbitrary (but finite) number of frequencies.

I. Introduction Chaotic dynamics of single-degree-of-freedom periodically forced oscillators have received much study over the past ten years. Such systems admit a canonical reduction to a two-dimensional Poincar6 map (see Guckenheimer and Holmes [71 or Wiggins [17]) for which a variety of "'chaos diagnostics" are available. For example, the Smale-Birkhoff homoelinic theorem provides a mechanism for the chaos in terms of the transverse intersection of the stable and unstable manifolds of a hyperbolic, periodic point of the Poirear6 map. Such intersections can be detected in a large class of systems via Melnikov's method, see Guckenheimer and Holmes [7] or Wiggins [17] for a discussion of these ideas. Once chaos has been identified in the two-dimensional Poincar6 map, many aspects of it can be quantified by Lyapunov exponents and entropy (Katok [9]), fractal basin boundaries (Grebogi et al. [61), dimension (Farmer et al. [4]), etc. For examples arising in physical

systems see Guckenheimer and Holmes [7] and Wiggins [17]. However, there has been very little work done on chaos in quasiperiodically forced, singledegree-of-freedom oscillators. One reason is that there is no canonical reduction to a two-dimensional Poincar~ map as in the periodica'.ly forced case, so that results from the relatively well-developed theory of diffeomorphisms of the plane cannot be carried over. Such systems arise frequently in applications since often systems are externally excited by more than one frequency at varying amplitudes. In this paper we t, 'ecently developed techniques by Wiggins [17] to study chaotic dynamics in the quasiperiodically forced Duffing oscillator.. The mechanism for chaos that we study is orbits homoclinic to normally hyperbolic invariant toil. We will ~gue that orbits homoclinic to toil occur in quasiperiodical!y forced systems in much the same way that orbits homoclinic to periodic orbits occur in periodically forced systems. A theorem is

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K. Ide and S. Wiggins / Bifurcation to homoclinic tori in the Duj~ng oscillator

170

stated which characterizes what we mean by the term "chaos". We use a generalization of Melnikov's method in order to determine when orbits homoclinic to toil occur in the quasiperiodicaUy forced Duffing oscillator. We concentrate mainly on the Duffing oscillator forced with ;wo frequencies since this case still involves five variable parameters (i.e., damping, two amplitudes and two frequencies). We give a complete analysis of the bifurcation to homoclinic toil in the fivedimensional parameter space. We also comment on the case of an arbitrary, but finite, number of forcing frequencies. The first work done on the quasiperiodically forced Duffing oscillator was by Moon and Holmes [11] and Wiggins [16]. Romeiras and Ott [13] have studied chaos in the damped pendulum subject to quasiperiodic forcing. Theoretical work concerning chaos in quasiperiodically forced singledegree-of-freedom systems similar in spirit to Wiggins [17] has been done by Meyer and Sell [10] and Scheurle [14].

2. Analytical techniques We now summarize the analytical techniques for determining the existence of homoclinic orbits in single-degree-of.freedom quasiperiodically forced oscillators• These techniques are special cases of more general techniques for multi-degreeof-freedom systems, the details of which can be found in Wiggins [17]. We study systems of the form

Yc=JDH(x)+cg(x,t;gt),

x ~ n 2,

(2.1)

an open set (note:/~ represents the system parameters),

01)

J= -1

and 0 < c ~: 1. Thus (2.1) represents a class of perturbed single-degree-of-freedom Hamiltonian systems (note: the perturbation may or may not be dissipative). We further assume that for each fixed x e U, /~ ¢ V, g(x, t;/~) is quasiperiodic in time. This means that for some integer i satisfying 2 <: I < oo, g ( x , t;/~) can be written as

0,,..., 0,;,), where for each fixed x e U, /~ e V, g is 2~r periodic in each of the 0~ coordinates, i - 1,..., ! and 0~- ,0d, i = 1,..., I. The to~ are called the fundamental frequencies. Thus for each fixed x e U, l~ ~ V, g( x, 01,..., Of;/L) can be viewed as a function on the/-torus, T t (see Moser [1966] for more details on quasiperiodic functions). We can therefore rewrite (2.1) as an autonomous system as follows:

= JOH(

Oi = (~i,

and the phase space of (2.2) is IR2 X T t. The unperturbed system is given by

=son(x), 01 = (01,

is C ' + l ( r >_ 2) on some open X

(2.3)

0I --" (01,

H: U ~ [R1

UXRI

) + ,g(x, o,,...,o,;t,), (2.2)

where

g"

0

set U c ~

2,

and we make the following structural assumption regarding (2.3).

V~--~R 2

Assumption. The x-component of (2.3) has a hyis C'(r> 2) with U c

R 2 an

open set and V c Rv

perbolic fixed point, x o, which is connected to

171

K. Ide and $. Wiggins~ Bifurcation to homoclinic tori in the During oscillator

We define a global cross-section to the phase space R 2 x T t by fixing any one of the phases of a forcing frequency, say 0~, as follows:

O X .... X O ~,,

..

Ac~cu~s

Fig. 1. Homoclinic geometry of the unperturbed phase space.

itself by a homodinic trajectory, Xo(t), i.e., limt _. ±® Xo(t)= x o. Thus in the full phase space (2.3) has a normally hyperbolic/-toms, To, given by To ffi

{ x ~ R 2, 0 ~

The Poincar6 map generated by the flow of (2.2) is then given by Pc: ,~O,o~ ,~o,0,

(x,(O), O,o,..., o,_,o,

( (2 )2 ol X( tO-T '

rqxffiXo},

whose (l + 1)-dimensional stable and unstable manifolds, denoted WS(To) and W u (To), respectively, coincide aion;~ ~he ~~+ 1)-dimensional homoclinic manifold ~ven by

w ' (to) n w " (to) = ( x + R z, O~

z'.o= {(x,O) Cae x r' 10,= 0,o}.

Ttlx=xo(t), t ~ U } ,

see fig. 1 for an illustration of the geometry of the homoclinic manifold of the unperturbed system. We remark that the term "normally hyperbolic" is a technical term which implies that the expansion and contraction rates of the flow generated by (2.3) normal to To dominate those tangent to To. For (2.2) this is clear since Xo is a hyperbolic fixed point so that trajectories approach To (in positive or negative time) exponentially fast but the flow on To evolves only linearly in time. For more technical definitions of normal hyperbolicity for general systems see Fenichel [5] or Hirsch et al. [81. UJo ~ i l ho t,r~noorrtoH t~'._'Lh how thi.~ s t ~ ¢ t u r e breaks up under the perturbation since this will provide a mechanism for chaos in this class of systems. In describing the geometrical structure, the mechanism for chaos, and the generalization of Melnikov's method it is more convenient to reduce the study of (2.1) to the study of an associated Poincar6 map.

+0,_1o,2~ ~°'+l

,o,..., O,o)

2 °,1

~0i + 0 1 0 ' " "

tO~

+ 0,+1o,... 2 'at ~+OlO ] , '

tO i

l

and we denote the Poincar6 map generated by the unperturbed vector field (2.3) by P0- For more information concerning Poincar6 maps of this type see Wiggins [17]. Thus the unperturbed Poincar6 map has a normally hyperbolic (! - 1)-dimensional torus given by -

Ze'° n

To.

Moreover, the stable and unstable manifolds of %, denoted WS (%) and W'* (%), respectively, are /-dimensional and coincide along an/-dimensional homoclinic manifold. The following results enable us to determine what parts of this structure go over for P,.

Proposition 2.1. For c sufficiently small, P, possesses a C r (1-D-dimensional normally hyperbolic invariant toms, r,, whose local, /-dimensional, C r stable and unstable manifolds, denoted Wl~ (~-,) and Wl~ (~-,), respectively, are C r e - close to Wl~: (%) and Wl~: ( To), respectively. Proof. This is an immediate consequence of the persistence theory for normally hyperbolic invafiant manifolds, see Fenichel [5], Hffsch et al. [8], or Wiggins [17].

K. Ide and S. Wiggins / Bifurcation to homoclinic tori in the DuJfing oscillator

172

We next want to determine whether or not W~ (~,) and W~ (~,) intersect. In Wiggins [17] it is shown that, up to a known normalization factor, the first order term in the Taylor expansion in c for the distance between W s (~,) and W u (s-,) is given by

wU(po)

__

M(Olo,..., Oto; IX) = f

Dn(xo(t))

Do,1 ~._.WSlP)o ~

I

O0

• g(xo(t ), ~olt + Ozo,..., ~ott+ Oto;Ix) dr,

(2.4)

where 00 --- (Olo,..., 0t0) can be viewed as parameters along the unperturbed homoclinic manifold and " . " denotes the usual ~e,::v~~dot product, see Wiggins [17] for a detailed discussion of the geometry associated with (2.4) and the parametrization of the homoelinic manifold by the independent variables of (2.4). We have the following theorem.

Theorem

2.2. Suppose there (01o,..., Or0; ~) such that

exists

a

point

1) 2) DooM(01o,..., 0-to; ~) is of rank one. Then, for c sufficiently small, W ~(~-,) and W" (T,) intersect transversely near this point. Moreover, considering ~ = ~ fixed, if DaoM(01o..... Oto;~) has rank one for all 00 ~ T t, then this zero of M can be continued to an ( l - D-dimensional torus.

Proof. See Wiggins [17] We refer to this (1-1)-dimensional torus of zero's of M as a transverse homoclinic torus and we next explore their dynamical consequences.

3. The nature and mechanism of chaos We will now discuss how the intersection of the stable and unstable manifolds of an invariant torus of a C'(r >_2) diffeomorphism can give rise to chaos. However first we will discuss the more familiar situation. Namely, how the intersection of the stable and unstable manifolds of a hyperbolic fix.d point of a Cr(r > 2) diffeomorphism gives

Fig. 2. The intersection of WS(po) and W" (Po).

rise to chaos. This gives the usual horseshoe construction. Hopefully this approach will allow the reader to develop some intuition for the more complicated situation.

Orbits homoclinic to a hyperbolic fixed point Suppose we have a diffeomorphism of R 2, f, possessing a hyperbolic fixed point Po whose stable and unstable manifolds intersect transversely at some point p as shown in fig. 2. (Recall- see Arnol'd [2]-that transversality means that the vector space sum of the tangent spaces of the stable and unstable manifolds of P0 at the point p is equal to R2.) We remark that unlike the case of orbits homoclin.;c to hyperbolic fixed points of ordinary differential equations it is possible for the stable and unstable manifolds of a hyperbolic fixed point of a map to intersect in a discrete set of points without violating uniqueness of solutions. This is becaase orbits of maps are infinite sequences of discrete points whereas orbits of ordinary differential equations are smooth curves. Now p lies simultaneously in the invariant manifolds WS(po) and WU (po), hence the orbit of p must lie in both WS(po) and W u (Po)- Thus iterating fig. 2 gives us me nomocumc tangte, part of which is shown in fig. 3. p is called a transverse homoclinic point. So one transverse homoclinic point implies the existence of a countable infinity of transverse homoclinic points due to the invariance of W~(po) and WU(po) i-or a more detailed and careful discussion of fig. 3 we refer the reader to Abraham and Shaw [1].

K. lde and S. Wiggins~ Bifurcation to homoclinic tori in the Duffing oscillator

173

) ___

DO

Fig. 4. The domain D.

Fig. 3. The homodinic tangle.

N o w we want to show how a Smale-horseshoe

map and its attendant chaotic dynamics arises from this geometric structure. Consider the domain D shown in fig. 4, whose left "vertical" side lies in W" (P o) and whose right "verticle" side touches WS(po). By invariance D must maintain this contact with WS(po) and W"(po) under all iterations by f. This is an important point to remember.

Next we consider f ( D ) which appears a~ ha fig. 5(a). Now we deduce the behavior of f ( D ) by noting the portions of f ( D ) which must remain on WU(p0) and WS(po), respectively. However an obvious question is why can't f ( D ) appear as in fig. 5(b-d) since these situations still respect invariance of the manifolds? The answer is that these are indeed possible and we have only chosen fig. 5(a) for definiteness.

\

(

a

b

d

0 Fig. 5. The dynamics of D in the homoclinic tangle.

K. lde and S. Wiggins / Bifurcation to homoclinic tori in the Duffing oscillator

174

/ _ f 3 (D)

~--TransverseHomoctinicTorus. ~"

... - 7~'----Z.---2 2 2 - - - - .

f 7(D)

Fig. 7. A transverse h o m o c l i n i c torus (cut-away view).

Fig. 6. The horseshoe map.

However we make the following comments regarding the remaining figures. Fig. 5(b, d). These situations cannot occur if f preserves orientation. In Wiggins [17] it is shown that Poincar6 maps arising from ordinary differential equations must preserve orientation. Fig. 5(c). It is certainly possible for the image of a "lobe" formed by pieces of WS(p0) and WU (P0) to "jump" over many other lobes under iteration. We have chosen the situation in fig. 5(a) where the lobe goes to the nearest possible lobe under iteration by f while preserving orientation. Thus the remaining iterates of D appear as in fig. 6. Fig. 6 shows, for example, that f 7(D) intersects D and has the shape oi" a horseshoe. Using techniques found in Wiggins [17] it can be shown that D contains an invariant Cantor set A on which f 7 is topologically conjugate to a full shift on two symbols. This implies that f 7 has: 1) A countable infinity of periodic points of all possible periods. 2) An uncountable infinity of nonperiodic points. 3) A dense orbit, i.e., a point in A whose orbit under f 7 approaches every point in A arbitrarily closely. Moreover, all orbits in A are unstable of saddle type. A is an ex,~mple of a chaotic invariant set. Recall from Devaney [1986] that a map is said to

be chaotic on an invariant set if: 1) The map has sensitive dependence to initial conditions on the invariant set. 2) The periodic points are dense in the invariant set. 3) The invariant set contains a dense orbit. For our map, these three properties follow immediately from the fact that f 71a is topologically conjugate to a Bernoulli shift on two symbols, for details see Wiggins [17].

Orbits homoclinic to normally hyperbolic invariant tori Now consider the situation of a diffeomorphism of R 3, f, having a normally hyperbolic invariant 1-torus, %, (i.e., a circle) whose stable and unstable manifolds intersect transversely in a 1-torus, ~-, such that ~- is homotopic to % see fig. 7. We call T a transverse homoclinic toms. By invariance of WS (¢o) and WU (T0) the orbit of T must always lie in both W s (To) and W u (To). Hence one transverse homoclinic torus implies the existence of a countable infinity of transverse homoclinic tori. Thus itt.rating fig. 7 gives fig. 8. UsinL arguments similar to those given in the previous case for a hyperbolic fixed point we can find a region D which is mapped over itself by some iterate of f in a horseshoelike shape as shown in fig. 9. However in this case we will get a circle's worth of horseshoes. The normal hyperbol-

K. Ide and S. Wiggins~ Bifurcation to homoclinic tori in the DuJ~ng oscillator

the hypothesis of normal hyperbolicity does not rule out the possibility of complicated dynamics on the tori ;n A and it would be interesting to see how the dynamics along the toil couple to the chaotic dynamics normal to the toil. Now the two examples discussed here are simple cases of a more general result which we state in the following theorem.

Fig. 8. The homoclinic toms tangle.

! a {al -

v

To

175

ttm

Fig. 9. A toms worth of horseshoes.

icity insures that the dynamics normal to the invariant torus dominate the dynamics on the toms so that the region D does not "kink up" in the direction of the invariant torus as it is being mapped back onto itself by some iterate. Using techniques from Wiggins [17] it can be shown, for example, that f s I o contains ~m invariant Cantor set of 1-tori, A, i.e., A is the Cartesian product of a Cantor set of points with a 1-toms. Moreover, the dynamics on A is chaotic in the sense that, viewing the toil in A as points, fsIA is topologically conjugate to a Bernoulli shift on two symbols. This implies the following for the dynamics of fSla: 1) A contains a countable infinity of periodic 1-tori of all possible periods. 2) A contains an uncountable infinity of nonperiodic 1-ton. 3) A contains a 1-toms whose orbit under f5 is dense in A. Heuristically, the dynamics in directions normal to the toil in A are chaotic while the dynamics in directions tangent to the tori in A may or may not be chaotic. In the quasiperiodically forced oscillators described in eq. (2.2) the dynamics along the tori in A is trivial; it's just irrational flow if the ,% i = 1 , . . . , 1 are incommensurate or rational flow if the , i = 1 , . . . , 1 are commensurate. However,

Theorem 3.1. Let f: R" X R " x T i ~ R" x R " x T t be a C ' ( r > 2 ) diffeomorphism having an /-dimensional normally hyperbolic invariant toms, To, possessing an (n +/)-dimensional stable manifola, W S(To), and an (m +/)-dimensional unstable manifold, W"(T0). Suppose WS(T0) and WU(%) intersect transversely along an /-dimensional "transverse homoclinic toms". Then, for some k > 1, f k has an invafizmt Cantor set of tori, A. Moreover, there exists a homeomorphism, q~, taking tori in A to bi-infinite sequences of N symbols such that the following diagram commutes A

fk

,A

ol X

1o 0

,X

Proof. The proof can be found in Wiggins [17]. A result similar to theorem 3.1 can be found in Silnikov [15] and Meyer and Sell [10].

4. The quasiperiodically forced Dulling oscillator We can consider the following equation: "~1 "-- X 2 '

+ ' [ f l c ° s O l + "'" + f , c ° s O t - ~'x2]"

Ol =WI" 1,4.1)

Oz=cot,

(Xl,X2,01 . . . . . Or) ~ N 1 X N l X T t,

where V, f,, w,, >_0, i = 1. . . . . I.

K. Ide and S. Wiggins~ Bifurcation to homoclinic tori in the Duffing oscillator

176

Our goal is to study the bifurcation to homoclinic tori in (4.1); however, (4.1) contains 21 + 1 parameters. For this reason we will concentrate on the i = 2 case (i.e., 2 frequencies) which only has five parameters and later on make some observations concerning the case for general 1. It is well known (see Guckenheimer and Holmes [7]) that for ¢---0 the x t - x 2 component of (4.1) has a symmetric pair of homoclinic orbits given by

by the four-dimensional surface =

3,~

~l

s ch - E +

3~ V-'

~2

s ch E

(4.6)

in the next section we will study in detail the structure of this surface.

5. The homodinie toms bifurcation set

=

( ± 4:

t t) , =1:sech ~ - tanh ~ - .

,

(4.2) Using (4.2) along with (2.4), we have

• t(o,0, e2o; v, A,

',',,

A,

T ~V~7 5: 2~rflcoI sech -~I - ~ sin 0t0

=

q~2

5:2'~f2o~2 sech ~

(4.3)

sin 02o.

V:

The zeros of (4.3) are given by

V=

3q'l"

ql"¢.O 1

3~r

~r~o2

:t: ~-.f1~ol sech - ~

±

sin 01o

sin 6 o.

(4.4)

As mentioned in section 4, the homoelinic torus bifurcation set is given by a four-dimensional surface defined by eq. (4.6) in (7, fl, f2, col, co2) space. Obviously, there is some ambiguity in the best way to represent this surface in order for it to be visually accessible and convey the most information. We choose to present bifurcation curves in the col-=o2 plane for fixed values of y,/'1, and f2We do this because eq. (4.6) has a structure which divides (Y, f l, f2) space into a collection of regions defining topologically equivalent bifurcation curves in the ~01-~ 2 plane. The regions are separated by planes which, most importantly, are independent of ,o~ and to2. We relegate the necessary calculations in order to justify these statements to an appendix and now present the results. Fig. 10 shows (y, fl, f2) space (note: y, fl and 1~ are > 0) divided into five regions by a collection of planes. Fig. 11 shows a section of fig. 10 in ( f l, Y) space for f2 = constant. First, we explain

It's easy to see that (4.4) has solutions provided the parameters satisfy 3~r Y -< ~"f,~n sech ~~r¢°l + - ~

f2~o2sech

~a°2 ¢~-

(4.5) ,

•

and that each solution can be continued to a |-torus (note: it is possible that at isolated values of (81o, 020) (such as (810, 02, ) = (~r/2, 3~/2)) (4.3) will not have rank 1, however at these values either D~,M or D~M will be nol~zero so .that the solution will continue through these points also). Now in the five-dimensional parameter space (i.e., (¥, .ft, f2. ~'~, ~o2)) the bifurcation set is given

I

..,/o

L2 pz

"'~'i~P*'

s,

~i I

L; Rz

"" . '

Poz 3

R o

Fig. 10. Bifurcation set in (y, fl, f2 ) space.

fa

K. lde and S. Wiggins~ Bifurcation to homoclinic tori in the Duffing oscillator ¢

pot Ro pt '2 RI R aI

?

p;,

I

L2 Rz

I J" P~" L,~

RZ

' ; fl=

F," f2

Fig. 11. Bifurcation set in (7, fl ) space, f2 = constant.

the notation. The five regions (i.e., open sets in (~', fl, f2) space) are denoted R °, R 1, RI2, R 2, and R 3 (note: the subscript i on R 2, i - 1,2, indicates the constant f~ > fj, j = 1, 2, this will be explained in the appendix). The regions are separated by the planes pOt, p~2, p123, p~2, and P ~ . The latter four of these planes meet at the line L2~2 whose )'1 and f2 coordinates satisfy fl = f2- The plane p4 is given by 7 - 0. We also want to consider the boundaries of these regions in the 7-ft plane and the 7-f2 plane. This will enable us to understand how two-frequency forcing goes to one-frequency forcing in the limit of one of the amplitudes going to zero. In the ~-ft plane, we denote the boundary of R 0 by S ° (S for "side"), pOX and p~2 meet in the line L~, p23 and p4 meet in the line L~, and S~ is the region between L~ and L~. Similar notation holds in the 7-f2 plane. It should be clear that the equations (and, hence, the bifurcation set) are unchanged under the simultaneous interchange of fl and f2 and `01 and `02-Therefore, fig. 10 is invariant under the interchange of

f, with/ . Now the five regions represent values of ~, ft, and f2 which give rise to qualitatively distinct bifurcation curves m the `0~-`02 plane. These are represented in fig. 12 which we will shortly describe. In this figure we note that 00m is the unique

177

maximum of the function X(~o) = ~osech ~ , o / ¢ 2 . Also, fig. 12 contains information on the cases with either)'1, f2, `0z, or ,02 zero. In these cases the mechanism for chaos is the usual homoclinic orbit to a fixed point of the Poincar~ map rather than an invariant 1-torus since in these cases the Dufling o s ~ a t o r is excited only by a single frequency. We now give a brief description of the bifurcation curves in the `0z-`02 plane for (7, ft, f2) values in some of the regions shown in fig. 12. R°: In this re#on the damping y is so large that, despite the value of fl and f2, there are no values of o t and ~o2 for which homoclinic tort exist. pOX: For choices of (-y, fx, f2) on this plane homoclinic tort occur at exactly one point in the `0t-w2 plane given by (o i, 602) "-- (`0m, 00m)" RZ: As we move across pOZ into R z the point (`0z,`02)=(`0m,~Om) opens up into a closed curve. Inside this closed curve transverse homoclinic tort exist. Note that in this re#on we must have both ~ot and ~o2 nonzero in order for homoclinic tort to exist. Pzz2: Passing from R 1 onto the plane p~2 the closed curve in the ~oz-~o2 plane tow'hes the ,0~ axis at ~'z = ~°m and "breaks open" along the line co1 = ~om at co2 = 00. Inside this re#on transverse homoclinic tort exist. However, now it is possible for transverse homoclinic orbits to occur at `02 = 0, but only at ~0~= `ore- A sitmlar scenario occurs on p~2 with ,02. Rz2: Moving from p~2 into Rz2, the basic structure of the region in the ~ol-`02 plane where transverse homoclinic tori exist persists, however the re#on Mong the ~o~ axis at which homoclinic orbits exist for ,02 = 0 has increased to an interval around ,0~ = ,%. p~3 " On tiffs - ' . . . . i l'-" "i"" l~; tlz~ i - - 6 ~ "2 ~ " " inside which transverse homoclinic tori exist now touches the ~o2 axis at ~o2 = ~om. This indicates that homoclinic orbits are possible at ~ = 0 but only at ,02 = ~om. R3: In 13, the re#on in the ~ - ~ 2 plane inside which transverse homoclinic toil exists now

K. /de and S.

178

Wiggins/Bifurcation to homocfinic tori in the DuJ~ng oscillator

includes an interval on the ~o~ axis around ~oz = wm. This indicates that homoclinic orbits can exist at ,o~ = 0 for ,o~ in this interval. p4: At y = 0, transverse homoclinic tori exist for all values of ~01 and w,. Fig. 12 indicates several interesting possibilities for a single-degree-of-freedom system undergoing two-frequency excitation. 1) For some values of (y, fl, f2) (e.g., in R 1) and for a given value of ~o1, there may be no values of o , such that homoclinic orbits occur. However, for some values of ,01, there may be a IN (y,f~ .I'~]

SET IN Wl~'to~ ,

'to:h

IN (Ylfl 'f2 ~ ,,,

3)

LOCATION IN ( ¥ , f , ,f2]

SET IN

~"l" to~

Bl~CK110~

LOCAllON DIRJRCATION IN SET IN (y,f, .f~) wp.wZ

SET IN

w! " to~

,

wZ

toe

wmi '

R°

2)

window of w2 values in which homoclinic orbits occur. For some values of (y, f~, f2) (e.g., in R~) and for any value of w2 there is a window of ~ values in which homoclinic orbits occur. Outside of this window no homoclinic orbits occur. For some values of (y, f~, f~) (e.g., in R 3) and for some values of w2, there is a window of wz values in which homoclinic orbits occur. Outside of this window no homoclinic orbits occur. For other values of w~, homoelinic orbits occur for all w I values.

S°

tom

WI

tom

~m

Wl

to'

to:~

pO, .,..|.

to2

R' wm

w I

to, j

p,, 2

2

2

12

L,

P~

L2

I~lm i tom

=

to I

w m

w I

~um

(alm ~om

wI

%

, I

R2

sl'

I

tom

tol

rum

to

s:

wmq

i

~ ~m

tol

I11m

tom

t.ot

%.L tom

p,~

p223 turn

R~

Wl

wl

,

toll1" I

wm

w I

~2

w2

p4

L:

fu~m " tun,

tol

tom

to

Fig. 12. The complete bifurcation set.

to m

t&

K Ide and S. Wiggins~ Bifurcation to homoclinic tori in the Duffing oscillator

This points to the value of our techniques for giving a criteria for chaos (although this could not be done unless the geometrical mechanism responsible for the chaos, i.e., homoelinic tori, was understood). Evidently these results show how chaos may be created or destroyed in periodically excited single-degree-of-freedom systems by the introduction of additional excitation frequencies.

179

carries over. Thus we see that the effect of adding many forcing frequencies is to reduce the damping and, hence, to increase the likelihood of homoclinic orbits and their attendant chaotic dynamics.

Appendix We now discuss how figs. 10, 11 and 12 were obtained. Rewriting eq. (4.6), the bifurcation set is given by

6. The case of an arbitrary number of frequencies - a y + f l X ( ~ t ) +f2X(~2) - 0 , We now consider the case of an arbitrary, but finite, number of frequencies and show how this falls into the framework of the results in section 5. For i frequencies,

0,o,

A,

M( 01o,..., Oto, y; fl, ~1,'" ", ft, ~°t) "

7 + 2~r ~ f'~°' sech ~ - sin 0~°

(6.1)

ilt

and the bifurcation surface in the (21 + 1)dimensional parameter space is given by 31I

i

where X(,o)

= ,o s e c h b , o ,

a =

f,,

is given by

3

'ffoi

A key feature will be that (A.1) is linear in ft, f2, X(c°l), and X(c02). The strategy for our analysis of (A.1) will consist of two parts: 1) Denote X(¢l) - X l, X(c02) - X2 and regard (A.1) as a surface in (y, fl, f2, Xl, X2) space. Then draw bifurcation curves in the X~-X 2 plane for different values of ~,, fl, and f22) Using the properties of X(co) and its inverse, interpret the bifurcation curves in the X I - X 2

(6.2)

"1'= V~- E.t fi~°, sech

x (~,)

L~;t us choose any two of the 1 frequencies, say ~x ,C0iz.Then if we define Y=Y

3~r ! 'ff~i ~ ,--rE f~o~,sech ¢~- .

Xm

(6.3)

We see that (6.2) can be rewritten as V = --~-fi~,, sech

(A.1)

+

3'~

sech

"ffOi2 (6.4) ~om

So (6.4) has the same structure as (4.6) and ~herefore the bifurcation set described ia section 5

Fig. 13. Graph of X{.~).

K. /de aa'd S. Wiggins~Bifurcation

180

W

homoclinic tori in the Duffing oscillator

Rewriting (A.1), the bifurcation curve in the Xx-X~ plane is Oven by

plane as bifurcation curves in the to~-to2 plane. Step 1 is easy since the bilurcation curve will just be a line. Fkst we need to note some important properties of the function ,1"(to). X(to) has a maximum at to = tom, is strictly increasing in 0 < to < tom and stMctly decreasing in tom < to < oo. We denote X(tom) ~ Xm. NOW if bifurcation c~rves in the X t - X 2 phne have counterparts in the tox-to2 plane we must have X t < X m and X2 ~ Xm; see fig. 13 for the graph of X~,~0). j

LOCATION IN ( )'~ft . t l )

x2=l(-f A

Xt+a?),

0
So for a given value of (?, fx,/2), points in the Xt-X2 plane above (see eq. (4.5)) the curve given by (A.2) correspond to transverse homoclinic toil. There are twelve different bifurcation fines in the X t - X 2 plane which define the five regions, six

BIFt.~CATION SET~I

BllRJRCATION SET IN

W I - W|

w I -w: t

LOCATION

IN (~',(t .fzt

BIFURCATION SET IN wl-

W I - W2

ut2

i |

wz Jr_

> :~'[t, + h) /: >0,It >0

I

'

~m

'~

X'_

:

"~= :'-~-(,t: + h) h >o,I, >o Xm X ,

~ (/: + hi > "~> R*

' ~m

wI

tom

Wl

w2

"2

Xm "" " * ~ ,

~ , * p ( I - hi tt > 0, t'a >0

x.i--

Xt

Xm

pi 2

-t - ff-~ h

to~,

X2

X2

¢

X~

'T-, 7 h to,,,

f=>yt >o

l:>h>0

xm X,

tom

m

tol

q

xm2,

x21 --;- tt >'~> It " - h > o

~,..

to,

.,m

,5!

to2

2

R2 .X_m -X,

tom

I~1

to m

to

h>I,

! i

>o

,gin X,

X2 2

I-, 2 ×m X . i X2 Z3

.~= x../,

Xm

.....

-t

p~

3

ot

It > /'a > 0

~m x

Rs

Xm . . . . . .

:~-iaf{l,, I~}

X2 Xm---

-i, ,~ 0

,

tom

W2

to

.

t, i i

ft > 0 , f= > 0

tol

~I"-T

x___i_

It > O , J ' = > 0

p4

i

Xm

e°mj" X0

, wm

X2 X~r

]

= 7/1 f2>/L>0

tozl ~ !\ $! I , ,_x~.

x2I

0<'T<

(,~m

Xm~

to!

Fig. 14. The bifurcation sets.

,,h'' Xm

X,

to m

to

K. Ide and $. Wiggins~Bifurcation to homoclinic tori in the Duffing oscillator ix)

planes, and one line shown in fig. 14. These are Oven below: pOt__

181

/ (v, ft, f2) Iv__ ..~_(ft +° + f2), fl > O, f2 > O/ ,

~+~=(<, ,, ,~>i,=~,, ,, >,~>0)

to'ix)

', i

Xm

e'= {(v,A, A)I~ = o, A >o, A>o},

~o-(<,,:,,:~,I,>~:~:,÷:~,, :,>o,:~>o) >Xm } a s u p { f l ' f 2 } ' fl > 0 , f 2 > 0 ,

R2-- (Y,ft,f,)

.:+~"Y> "~'f2 ft >f2 >0 ,

R2= (~',ft,fz) "~-fz>Y>-'-d"ft,O
R3--((V. fl.f2)IO<'0, A >o},

+/

L'f2= (v,fl,/2 so=

V'- a

2

/

'A=f2>0 '

I(¥,fx,f2)¥>--d--ft, I ~m f1>O, fz=O,1

S°=(('y,f,,f2)l¥>--d--f2Xm,ft =

(

i ~m

t

I

0,/2

>0},

1

L~= ('y,f,,/z)y=--a--ft,ft>O, f2=O , X_

,~ = i~, :,,:,)i~= --::2,:1=o, :2>0 I, ('y,fl,fz) O<'y<--d"fx, (I ~m ft>O, f2=O 1, Sf= (('y, Ift,L) O<'~<'d-f2, ~m ft=O, f2>O 1,

S?=

L~= {(y, ft,/2)1~,'- 0 , / , > 0 , / 2 = 0}, =:~-- {(~,A,A)I~ = o, f+ =o, A > o }

X

Fig. 15. Graph of ~(X).

In fig. 14 we label the different regions, planes, and line in (~, fl, f2) space and alongside each we show the corresponding bifurcation line in the X t - X 2 plane while next to each of these we show the corresponding region in the ,0t-to 2 plane. We now briefly describe how we go from the bifurcation lines in the X1-X 2 plane to the corresponding bifurcation curve(s) in the tol-to2 plane. First let us consider the inverse of X(to) which we denote to(X). This is a double-valued "'function" whose graph we display in fig. 15. We denote the decreasing branch of ~ ( X ) by ~o-(X) and the increasing branch of ~0(X) by to+(X). Now let us draw the bifurcation hne in the X t - X 2 plane corresponding to region R t. We are interested in the points above this line with the constraint that Xt < X m, X 2 < X,.~, i.e., the triangle shown in fig. 16. Now we want to convert the triangle shown in fig. 16 (inside which transverse homoclinic tori occur) into a bifurcation diagram in th+~ tol-to2 plane. From fig. 15 for each value of ( Xl, X2) inside the triangle there are four values o[ (tot, to2). This fact, along with the shape of ~0(X) shown in fig. 15, allows us to graphically construct the bifurcation curve in the to~-to2 plane inside which transverse homoclinic toil occur. In fig. 17 we show the ,iorresponding region in the ~01-~02 plane.

K. lde and S. I$ iggins / Bifurcation to homoclinic tori in the Duffing oscillator

182

References

xz

N

%

.....

% . . . . . . . . . . . . . . .

-I

l I I l I

^

X 2

....

,.X2°

I l I I I

I........

Ii

,

:

i

|

"" "T' ......

-

\

, "~

]' .......

I

t

I

I

;

k

o

Xi

I

__'X~x %% %

""

XI

Xm

X,

Fig. 16 Bifurcation set in Xt-X,, corresponding to Rt.

A

(.0 m

4, ^

to (X 21

.... :-,X--;- / ,

w+lX~)

_ _ _J_ I

~ I }

i \ ~ /

I

_1 _ _ % ~ /

t

I

I

I

I

t~+t~l) ton,

l t

.

w'(~l)

~'lx 71

-tO I

Fig. 17. Bifurcation set in o l - ~ corresponding to R1.

This graphical construction technique can be repeated for all the bifurcation lines in the Xt-X2 plane corresponding to the different regions, planes, and line in (~',fl, f2) space in order to yield the remainder of the bifurcation curves in the ~ t - " : plane shown in fig. 14.

[1] R.H. Abraham and C.D. Shaw, Dynamics-The Geometry of Behavior, Part Three: Global Behavior (Aerial, Santa Cruz, 1984). [2] V.I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, New York, Heidelberg, Berlin, 1983). [3] R. Devaney, An Introduction to Chaotic Dynamical Systems (Benjamin/Cummings, Menlo Park, CA, 1986). [4] ff.D. Farmer, E. Ott and ].A. Yorke, The dimension of chaotic attractors, Physica D 7 (1983) 153. [5] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Ind. Univ. Math ft. 21 (1971) 193-225. [6] C. Grebogi, E. Ott and J.A. Yorke, Crises, sudden changes in chaotic attractors, and transient chaos, Physica D 7 (1983) 181-200. [7] 3. Guckenheimer and PJ. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, Heidelberg, Berlin, 1983). [8] M.W. Hirsch, C.C. Pugh and M. Shut), Invariant Manifolds, Springer Lecture Notes in Mathematics, vol. 583 (Springer, New York, Heidelberg, Berlin, 1977). [9] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. IHES 51 (1980) 137-173. [to] K.R. Meyer and G.R. Sell, Melnikov transforms, Bernoulli bundles, and almost periodic perturbations, Univ. of Minnesota preprint (19R7) [11] F.C. Moon and W.T. Holmes, Double Poincar6 sections of a quasiperiodically forced, chaotic attractor, Phys. Lett. A 111 (1985) 157-160. [12] J. Moser, On the theory of quasiperiodic motions, SIAM Key. 8 0966) 145-172. [13] FJ. Romeiras and E. Ott, Strange nonchaotic attractors o[ the damped pendulum with quasiperiodic forcing, Phys. Rev. A 35 (1987) 4404-4413. [14] J. Scheufle, Chaotic solutions of systems with almost periodic forcing, ZAMP 37 (1986) 12-26. [15] L.P. Silnikov, Structure of the neighborhood of a homoclinic tube of an invariant torus, Sov. Math. Dokl. 9 (1968) 624-628. [16] S. Wiggins, Chaos in the quasiperiodically forced Dufling oscillator, Phys. Lett. A 124 (1987), 138-142. [17] S. Wiggins, Global Bifurcations and Chaos-Analytical Methods (Springer, New York, Heidelberg, Berlin, 1988).