In (Stark 1997), theorem 4.5 was applied to quasiperiodically forced systems, and in particular to results on strange nonchaotic attractors. As mentioned in section 3.2, there is a variety of different definitions in the literature of the term s strange, chaotic and attractor. We do not need to formulate any precise definitions to su percede any existing definitions, since the results presented in chapters 4 and 5 here on strange nonchaotic attractors hold under any remotely reasonable definition of the term. In particular, all th a t we require is th at a smooth invariant or periodic circle does not satisfy such a definition and we shall use the term SNA informally with such an understanding throughout chapters 4 and 5. On such a basis, (Stark 1997) gave the first results which are generally applicable to strange nonchaotic attractors in all (two-frequency) quasiperiodically forced systems. The framework in which this is set is th a t of maps of the form (2.5), with w irrational, where
9n G T^, the unit circle and yn ^ Y which is assumed to be a finite-dimensional
manifold. It will be convenient to write
F i e , y ) = {f{e),g {e,y)) (4.6)
for the skew product map on x Y . We shall also define by F ^{9,y) = ( r( ^ ) ,p W ( ^ ,p ) ) .
Due to the aperiodicity of / it is clear th a t F cannot have any periodic orbits. The simplest class of orbits is therefore given by invariant or periodic curves which are the graphs of a continuous function. If Y is one dimensional, the definition is straightforward, but in higher dimensions we have to allow for the possibility of the curve winding several times around x T before closing up. Thus,
C H A P T E R 4. I N V A R I A N T G R A P H S 65
F
F
C
' I
c
q=l, Q=\
q=2, Q=l
q=\, Q=2
Figure 4.1: Different types of invariant and periodic circles
for some Q G N. Let vr : R x y —> T ^ x y be the natural covering 7r{x,y) = {x
(mod l) ,y ) and recall that the graph of ^ is defined by graph = {(x, ^ (x )) : x G R}. I f 7T{graph 0 ) is invariant under F, we say that 0 is an invariant curve for
the skew product map (4-6). I f n{graph 0 ) is invariant under fo r some q e N , we say that 0 is a periodic curve of period q (see figure 4-V-
By working in the Q-fold cover of and replacing F by F^ if necessary, we shall henceforth take Q = q = 1 w ithout loss of generality. Clearly, if 0 is smooth
then its graph will be a smooth manifold and as indicated above cannot possibly be regarded as a strange nonchaotic attracto r under any sensible definition of the term . On the other hand, if 0 is not smooth then its graph can potentially have a
fractal dimension strictly greater than one, and hence be a candidate for a strange non-chaotic attractor. It has long been an open question whether this could indeed happen, which has only recently been answered in the negative by (Stark 1997). Recall th a t the normal or conditional Lyapunov exponents of a system of the form (2.5) are the exponents corresponding to the F-direction. The largest of these can be defined by
X { 9 ,y )= lim -logWDyp'^^^W (4.7)
n^oo n
where : F —>■ F is the function defined by 9 0^\y) = g^^\9^y) and Dyg^^^ is its
derivative at y. Given an F-invariant measure //, the limit (4.7) is guaranteed to converge for /i-almost every (0, ^) G x y by the sub-additive ergodic theorem (theorem 2.5 above). If p is ergodic (which is usually assumed in discussions of Lyapunov exponents) then A is constant //-almost everywhere. In such a case if A < 0, then all the (standard) Lyapunov exponents are negative, apart from the one in the ^-direction, which is always trivially 0.
If ^ —>■ y is a continuous invariant curve and u is the Lebesgue measure on then graph 0 (which is necessarily closed if it is continuous) supports a unique invariant measure p = (7d, 0)*(//), which is necessarily ergodic. This is defined by p{B) = p{{Id, <E>)“ ^(F)) = p{9 G : {9,0(^)) G B } for all measurable sets F C X y, and can also be characterised by f 'tpdp = f tp o ( I d , ^ ) d p = J ijj{9,^{9))dp for all ip G C®(T^ x y,R). In such a case we can informally refer to A as the maximal Lyapunov exponent of the curve 0 . The condition A < 0 (or A < 0) is usually assumed as part of any definition of a strange nonchaotic attracto r (for example as in (Grebogi, O tt, Pelikan & Yorke 1984)). This seems reasonable, since although A > 0 may not be considered as a sufficient condition for the definition of chaos, few would regard a system with A < 0 as chaotic.
C H A P T E R 4. I N V A R I A N T G R A P H S 67
The following theorem thus shows th at such an attracto r cannot be a continuous invariant curve (apart possibly from the transitional case A = 0):
T h e o re m 4.7 (Stark 1997) Suppose that F : x Y -> x T is a smooth quasiperiodically forced map of the form (2.5). Let ^ ^ Y he a continuous invariant curve, p = {Id ,^)*(p ) and A the maximal Lyapunov exponent of p. I f A < 0; then 0 is as smooth as F is.
When p = {Id,^)*{i/) we may define the maximal normal exponent with respect to p by
A = lim — log
n->cxD n D^{9)9q (n)
which converges for Lebesgue almost every 0. When Y is one dimensional, theorem 4.7 follows as a trivial corollary of theorem 4.5. In such a case ||D^p0|| = \L>y9 9\
and hence
log
n —1
(4.8) z= 0
where (j){6, y) = lo g\DyQ9\. If for simplicity we assume th a t DyQ9 ^ 0 on the graph
of 0 , then since / is uniquely ergodic (see theorem 2.4) we can apply theorem 4.5 to show th at given any e > 0, there exists a constant c such th a t
(n)
(4.9)
for all ^ G TL By taking e > 0 sufficiently small, we can obviously ensure th a t A + e < 0, and hence the graph of 0 is uniformly attracting. It then follows from the section theorem (Hirsch & Pugh 1970; Hirsch, Pugh & Shub 1977), or see (Robinson 1995) th a t 0 is as smooth as F . Note th at since / has no contraction.
equation (4.9) implies th at the graph of 0 is a normally hyperbolic manifold, and indeed the section theorem is the fundamental result required to prove the smoothness and structural stability of such manifolds. If DyÇe is singular, a simple technical device still allows us to obtain the crucial estim ate (4.9) (see the proof of theorem 5.4 in section 5.6.4 below).
When we try to extend theorem 4.7 to higher dimensions we immediately encounter a fundamental obstacle. The key to proving of theorem 4.7 in one dimension is th a t in this case we have ||D^^g|| = \Dyge\. In general \\AB\\ ^ ||^ ||||B ||, so we cannot write log as an ergodic sum, as in (4.8). This is the key reason why Lyapunov exponents cannot be defined using the Birkhoff ergodic theorem (theorem 2.3), and one needs recourse to more powerful results such as the sub additive ergodic theorem, or the multiplicative ergodic theorem (see for example (Arnold 1998)). The simplest for our purposes is the sub-additive ergodic theorem (Kingman 1968) (see theorem 2.5).
Unfortunately, it is known th a t the generalisation of theorem 4.5 to the sub additive case (or indeed the multiplicative case) is false (Derriennic & Krendel 1981). In other words one does not get uniform convergence in (2.6) even for uniquely ergodic systems.
However, in order to apply the section theorem, we do not actually require uniform convergence, only a uniform estim ate from above, as in 4.9. It turns out (Stark 1997) th a t this can be obtained using elementary estimates together with theorem 4.5. In effect this gives a proof of the following semi-uniform version of the sub-additive ergodic theorem, though this is not explicitly stated in (Stark 1997):
T h e o re m 4.8 Suppose that T : X ^ X is a uniquely ergodic measurable map on a compact metrizable space X , and {(j)n} is a sub-additive sequence of continuous
C H A P T E R 4. I N V A R I A N T G R A P H S 69
functions : X ^ R. Then given e > 0, there exists an N E N such that for all n > N we have
—(f)n{x) < 0 + e
n
for all X E X , where <j) is the limit (2.6), which is necessarily constant almost
everywhere with respect to the unique T-invariant measure.
A direct elementary proof, using theorem 4.5, can be found in (Stark, Feudel, Glendinning & Pikovsky 2000), where it is used to give an integral characterisation of the rotation number of quasiperiodically forced circle maps, as well as give a more elementary proof of the existence and uniform convergence of such rotation numbers than th a t given by (Herman 1983) (which makes use of theorem 4.6). To prove theorem 4.7, we simply apply theorem 4.8 to the sub-additive sequence 0n(^) = log with the same provisos as above if DyÇg is singular on the graph of 0 . This allows us to obtain a uniform estim ate as in (4.9), and we can then apply the section theorem as before.
The disadvantage of theorem 4.7 is th at it requires the a priori existence of a con tinuous invariant curve. It can thus give no information about strange nonchaotic attractors in general. One attem pt to surmount this was suggested in (Stark 1997). This was to require some uniformity in each fibre {^} x Y . Thus, suppose th at there exists an integrable function ^ —)• R such th at
for all 0 G T^. Then it is easy to see th a t —> R defined by
f„((9) = logsup
yeY
is a sub-additive sequence of integrable functions. Hence by the sub-additive er godic theorem (theorem 2.5)
Â(0) = lim — logsup
n-+ o o n yçY
exists for Lebesgue almost every 6. Furthermore, since the Lebesgue measure is
ergodic, A(0) is constant for almost all 6. Now if T is, for instance, homeomorphic
to a closed disc in R"^ and DyQQ is non-singular for at least one point in each fibre, then is continuous. If DyQg is singular on a whole fibre we use the same technical device as in theorem 4.7 above (for example see the proof of theorem 5.4 in section 5.6.4 below). Then theorem 4.8 and the C section theorem imply
T h e o re m 4.9 (Stark 1997) Suppose that F : x Y -> x T is a smooth quasiperiodically forced map of the form (2.5) with Y homeomorphic to a closed disc in MF. Suppose that A < 0. Then the unique invariant set m x T is a smooth invariant curve.
4.3.4
Strange nonchaotic attractors
This suggests th a t any strange nonchaotic attra cto r must have non-expanding orbits arbitrarily close. It thus seems reasonable th a t in fact it should contain such orbits. This is consistent with the only known examples where a strange nonchaotic attra cto r can rigorously be proved to exist, namely those analysed in (Grebogi, O tt, Pelikan & Yorke 1984) and (Keller 1996) (see chapter 3). In
C H A P T E R 4. I N V A R I A N T G R A P H S 71
these, the attractor always contains the circle y = 0, which has a positive normal
Lyapunov exponent.
If we want to prove some such result about general attractors, or invariant sets, of a quasiperiodically forced map, it seems unlikely th a t theorem 4.8 will be of much use. This is because typically any such invariant set will not be uniquely ergodic, but will rather support many different invariant measures. This is exactly analo gous to an ordinary chaotic attra cto r for an unforced system, which apart from its natural measure (assuming there is one) will support many others, corresponding to periodic orbits, horseshoes etc. Our aim is therefore to generalise theorem 4.8 in such a way as to allow us to deduce a uniform estim ate analogous to (4.9) from non-uniform hypotheses on such a collection of invariant measures.
As discussed in chapter 3, the notion of ‘strange’ in a strange nonchaotic attracto r is one for which universal agreement has not been reached. The theorems in this chapter, however, prove the existence of smooth invariant curves given the presence of an invariant graph. W hatever definition of ‘strange’ is taken when defining an SNA, it is clear th at a sm ooth invariant curve will not fall into th a t category. Hence a straightforward corollary of theorem 4.7 is
C o ro lla ry 4.1 (Stark 1997) A strange nonchaotic attractor in a quasiperiodically forced system is not the graph of any continuous function.
This had been a conjecture for many years. We should note the relationship of this result to the work of (Keller 1996) described in section 3.3, in which an SNA is constructed, but as a graph of a discontinuous function.