version of 18 May 1992
A quantitative notion of sensitive dependence on initial conditions relies on the concept of LYAPUNOV (Lqpunocentsv) characteristic exponents. This is a measure of the tendency of neighboring points to converge or diverge as the dynamics progresses. Consider the flow [[phi]]t of a dynamical system (with the understanding that t is an integer variable, t = 0, 1, 2, ..., when the dynamical system is discrete in time), and initially suppose that the system is one-dimensional, S Ì R . We would like to be able to quantify the tendency of nearby points to diverge. The subtle point here is that if we are interested in the behavior of nearby points x and y at relatively large times - long enough for divergence but not so long that [[phi]]t(x) no longer has much to do with [[phi]]t(y), then we must balance two different limits, y -> x and t -> [[infinity]]. If x and y are neighboring points and t is large, then we might expect to approximate the difference with the first term in the Taylor series:
[[phi]]t(y) = [[phi]]t(x) + (y-x) f(d[[phi]]t(x),dx) + ...,
[[phi]]t(y) - [[phi]]t(x) = (y-x) f(d[[phi]]t(x),dx) + ....
If we keep only the first-order term in y-x, then we observe that bbc|(f(d[[phi]]t(x),dx)) represents the factor by which |y-x| is stretched in time t, independently of the point y. Therefore it allows us to set aside the limit y -> x. Our experience with linearization in the case where x is an equilibrium leads us to expect exponential growth, i.e., that
If this were the case, then . This reasoning leads to the following definition:
Definition. Let (S, [[phi]]t) be a dynamical system with S Ì R , and
x [[propersubset]] S. The LYAPUNOV, or characteristic, exponent is defined as
whenever this limit exists.
Some sources prefer to define [[lambda]](x) as the limit superior of this expression, to ensure that it is defined. A positive LYAPUNOV exponent indicates divergence of nearby points, whereas a negative LYAPUNOV exponent indicates convergence of nearby points. In the computer age some people prefer to define the LYAPUNOV exponent with log2, since it then indicates the rate of loss of bits of information. This merely differs by a scale factor of approximately 0.693147181 from (6.1).
Logically, the LYAPUNOV exponent depends not on an individual x but on the trajectory through x. Experience shows that it frequently has the same value for whole regions of S or even S in its entirety. We shall discuss some of the reasons for this in the next section.
Example. Consider the tent map T(x) on [0,1], [[phi]]n(x) = T[[ring]]n(x). If
x != k/2n, then no iterate of x ever becomes equal to f(1,2), so |T'(T[[ring]]n(x)| = 2 for all n.
f(d[[phi]]n(x),dx) = ipr(j=0,n,T'(T[[ring]]j(x))) = 2n
by the chain rule, we find that [[lambda]] = ln 2. If x = k/2m, then f(d[[phi]]m(x) ,dx) is not defined.
This example illustrates an important point about the LYAPUNOV exponent of an iterated function F on R: it is a kind of time-average of ln|F'| over the trajectory. If xn := F[[ring]]n(x0), then the chain rule implies that
b(f(1,N)) ln bbc|(f(dF[[ring]]N(x0),dx)) = b(f(1,N)) ln ipr(k=0,N-1, |F'(xk)|) = b(f(1,N)) isu(k=0,N-1, ln|F'(xk)|) (6.2)
Exercise VI.1. Calculate LYAPUNOV exponents for the dyadic map and the quadratic map (the latter computationally).
The LYAPUNOV exponent can be determined computationally from (6.2) if it is easy to evaluate the derivative F'. This is not always easy, however, if F is not known simply as a formula. A common approximate method is just to consider two nearby points x and y and a large fixed n, and calculate
b(f(1,n)) ln b(f(dist([[phi]]n(x),[[phi]]n(y)),dist(x,y))).
It is, of course, uncertain whether you are near the right answer. A more sophisticated approach is to choose different y's at different times, scaled so as to effectively produce a finite-difference approximation to the derivative in (6.2).
Another useful fact for computation is that the notion of a LYAPUNOV exponent is robust against changes of variables. Suppose that S is a diffeomorphism from S to another state space T, and that 0 < m <= |S'(x)| <= M for some m and M independent of x. The dynamical system [[phi]]t(x) for x [[propersubset]] S is equivalent to [[psi]]t(w) := S([[phi]]t(S-1(w)) for w = S(x) [[propersubset]] T. We can call this relationship between dynamical systems a diffeomorphic conjugacy. By the chain rule,
f(d[[psi]]t(w),dw) = S'([[phi]]t(S-1(w)) f(d[[phi]]t(S-1(w)),dx) f(dS-1(w),dw)
= f(d[[phi]]t(x),dx) f(S'(S-1([[psi]]t(w))),S'(x)),
bbc|(lnf(d[[psi]]t(w),dw) - lnf(d[[phi]]t(x),dx) ) <= bbc|(lnb(f(M,m))).
Since the right side is independent of t, from (6.1) we conclude: Diffeomorphically conjugate dynamical systems have the same LYAPUNOV exponent.
While the dynamical system ([[sigma]],[[Sigma]]2) does not immediately allow differentiation in the set [[Sigma]]2 and therefore does not admit the definition of the LYAPUNOV exponent according to 6.1, it would not be too difficult to modify that definition for this case, and we would conclude that the LYAPUNOV exponent was ln 2. We know that ([[sigma]],[[Sigma]]2) is topologically but not diffeomorphically conjugate to (Fu,[[Lambda]]) for large u, and a calculation would show that the LYAPUNOV exponents of these dynamical systems are different from ln 2.
In more than one dimension, there may be different LYAPUNOV exponents in different directions, and this fact is still true for all of them, by much the same argument. The definition of a LYAPUNOV exponent in this case works as follows. If F (or [[phi]]t) is any function from S to S, let DF(x) denote its linearization, i.e., the matrix with entries. [[partialdiff]]Fj(z)/[[partialdiff]]zi. This plays the rôle of d[[phi]]t/dx.
Definition. Let (S, [[phi]]t) be a dynamical system, and
z [[propersubset]] S. If for some vector v != 0, the limit
[[lambda]]v(z) = a( ,lim,t->[[infinity]]) b(f(1,t)) ln (||D[[[phi]]t(z)] v||), (6.3)
exists, then it is said to be a LYAPUNOV exponent for z. The largest such value obtained for all v is called the principal LYAPUNOV exponent for z.
Observe that the LYAPUNOV exponents depend only on the direction of v and not its length, since a scale factor [[alpha]] in v will lead to an additive contribution to (6.3) (ln [[alpha]])/t -> 0. It certainly does depend importantly on the direction. A system will be said to have sensitive dependence on initial conditions at x if its principal LYAPUNOV exponent is positive.
The analogue of Equation (6.2) reads:
b(f(1,N)) ln ||DF[[ring]]N(x0) v|| = b(f(1,N)) ln ||DF(xN-1)DF(xN-2)...DF(x0) v||,
but the product is now a product of matrices, and so there is no easy way to write it as a sum (except in the easy case when all the matrices commute).
Example. Let [[phi]]t be the flow of a continuous linear dynamical system generated by a matrix A with eigenvalues [[lambda]]j. In other words, the equation of motion is
f(dz,dt) = A z
For simplicity suppose that the eigenvalues are real-valued and the eigenvectors form a basis of S = Rn. Then the LYAPUNOV exponents are [[lambda]]j, and the characteristic directions v are the eigenvectors of A. If the dynamical system is discrete, with mapping F = [[phi]]1 = B linear with real eigenvalues uj and a basis of eigenvectors, then the LYAPUNOV exponents are [[lambda]]j = ln|uj|. In the next section we shall learn of some conditions under which LYAPUNOV exponents are guaranteed to exist and be essentially constant on all of S or on some distinguished part of S.
Exercise VI.2. Show that just as in one dimension, higher-dimensional diffeomorphically conjugate dynamical systems have the same LYAPUNOV exponents.
Recalling that given certain nonresonance conditions a differentiable dynamical system near an equilibrium can be diffeomorphically transformed into a linear one, this often makes it possible to evaluate LYAPUNOV exponents for nonlinear systems.
Finally, notice another simplification that often arises. If a dynamical system is preserves the ordinary Euclidean volume on phase space, then the Jacobian determinant is always 1, and it follows that the product of all the LYAPUNOV exponents must likewise be 1, and therefore , provided that they exist. In two dimensions, for example, the two exponents will be negatives of each other.
DEVANEY  suggests a working definition of chaos for a dynamical system on a state space S:
a) S has a dense set of periodic trajectories;
b) S is topologically transitive; and
c) S has sensitive dependence on initial conditions.
Let us call a dynamical system with these properties D-chaotic. DEVANEY introduces a less precise notion of sensitive dependence than the LYAPUNOV exponent, but for our purposes we will interpret c) to mean that the principle LYAPUNOV exponent is positive. Other related peoperties abound in the literature:
Definition. A dynamical system satisfies Axiom A if S is nonwandering, hyperbolic, and has a dense set of periodic points.
SMALE has shown that if the dynamical system satisfies Axiom A, then S is the union of finitely many invariant sets, each of which is dynamically transitive.
Exercise VI.3. Show that the shift on [[Sigma]]2 and the quadratic map are chaotic. As remarked above, the shift map can be considered to have a generalized LYAPUNOV exponent equal to ln 2.
Recall that the SMALE-BIRKHOFF homoclinic theorem states that dynamical systems with transverse homoclinic points have subsets that are equivalent to BERNOULLI-type systems. The motion is therefore chaotic on these subsets.
Exercise VI.4. Calculate the LYAPUNOV exponents for the cat map. Show that the cat map has all the characteristics of chaos. Do the same for the baker's transformation.
Hint. In order to show that the cat map is topologically transitive, recall that it has a dense set of homoclinic points. Hence any two open sets will contain two homoclinic points q and r and some [[epsilon]]-neighborhoods of q and r (i.e., open balls B(q,[[epsilon]]) and B(r,[[epsilon]]) of radius [[epsilon]] centered at q,r). Consider the forward iterates of the piece of Iu := Wu(0) [[intersection]] B(q,[[epsilon]]) and the backward iterates of Is := Wu(0) [[intersection]] B(r,[[epsilon]]). For any d>0 there are arbitrarily large m,n, such that A[[ring]]m(q) and A[[ring]]-n(r) will be at a distance < d from the origin. The same will obviously be true for A[[ring]]m(Iu) and A[[ring]]n(Is). But these two sets are perpendicular and of lengths that are arbitrarily long, so they must intersect. Now step forward by time n.
Attractors by definition pass one of the conditions of chaos, and are frequently chaotic, especially if they have fractal structures, in which case they are known as strange attractors. There are, however, examples of strange attractors with principal LYAPUNOV exponent 0 [GREBOGI et al., 1984].
The two-variable systems considered in [[section]]IV, the cat map and the baker's transformation, are essentially linear, except that they have been adapted to a torus. More representative might be the HéNON mapping,
which was invented as a simplified model of a POINCARÉ map for the LORENZ system of differential equations,
These equations are in turn a caricature of the NAVIER-STOKES equations, used in studies of fluid flow. In these mappings, a and b, or, respectively, [[sigma]], r, and b, are physical parameters. The HéNON mapping, e.g., for a = 1.4, b = 0.3, and the Lorenz system are famous for having strange attractors.
Exercise VI.5. Plot the HéNON attractor on a computer screen. Estimate the LYAPUNOV exponents computationally.
The HéNON and LORENZ systems are not difficult to study numerically, but many of the theoretical issues about their dynamics are not rigorously understood, and there is active research on both models.