Lyapunov exponents, stability and information

Lyapunov exponents, stability and information
Hugo Touchette Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, MA 02141
can be readily calculated for any system (linear or not) and can be used as an alternative method
to de ne several stability classes ( xed points, limit cycle, strange attractors). In addition, we shall
what constitutes Lyapunov's direct or second method. Basically, given a dynamical system with
x 2 RD as
that: (i) V
its state-variable, the idea of this has only one minimum located at
htouchet@
Abstract.|This report reviews the theory of Lyapunov exponents and its relevance for the study
of the stability of dynamical systems. The concept of a Lyapunov exponent is rst introduced informally by considering a simple discrete time evolution equation. The de nition of these exponents is then extended to continuous time dynamics and formalized within the density picture of dynamical systems. In addition, we show that these exponents can be systematically calculated and used to determine the stability properties of any systems. They can also be associated with some entropy concepts. In this way, an intuitive connection between stability and information is established and can be extended to an information-theoretical study of control.
distance between two in nitely close points is stretched after each iteration of f. Now, one may be
concerned about the existence of the limit involved in eq.(7) and its uniqueness. For instance, if there
"n = jfn(x0 + "0) ? fn(x0)j
(2)
where fn(x) = f(f( f(x) )) n times, or recursively, fn(x) = f(fn?1(x)). In order to relate "n to the initial separation, we introduce an exponential scaling factor Ln such that
in the next section.
Example 1. Consider the triangular map (also called the tent map):
f(x) = r ?1 ? 2 21 ? x ; x 2 0; 1]
(8)
2.1. Evolution of distances in discrete maps
The starting point of Lyapunov exponent theory is the analysis of the separation in time of in nitesimally close points subjected to an evolution equation. In order to illustrate this concept, let us consider a one dimensional discrete autonomous map of the form
d dx
f
n(x0)
=
f
0(xn?1)
d dx
f
n?1(x0)
=
nY?1 i=0
f
0(xi);
(6)
eq.(5b) may be written as
(x0)
=
nl!im1
1 n
nX ?1 ln
i=0
f 0(xi)
:
(7)
This last expression is of particular signi cance. It de nes e (x0) as a time average factor by which the
following equalities
(x0)
=
nl!im1
"l0i!m0
1 n
ln
fn(x0 + "0) ? fn(x0)
"0
(5a)
=
nl!im1
1 n
ln
df n(x0) dx0
;
(5b)
which de ne the Lyapunov exponent of a map. Furthermore, using the chain rule for the derivative
a locally stable equilibrium point. It is globally stable if is the whole phase space 11, 16].
In theory, the construction of a Lyapunov function is su cient to infer all the stability properties
is a Lyapunov exponent associated with each sample starting point x0, then would appear to be a rather useless quantity. The next example shows that the so-called triangular map possesses only one
"n = Ln"0 = en (x0)"0;

(3)
where
(x0) =
1 n
ln
fn(x0 + "0) ? fn(x0)
"0
:
(4)
2
To make the expression of more precise, we can take the limits "0 ! 0 and n ! 1 to obtain the
xn+1 = f(xn);
(1)
where n is the discrete time (n = 0; 1; 2; : : :). Under the action of this map, the separation of two adjacent points x0 and x0 + "0 becomes after n steps
Since the publication of Lyapunov's monograph \Probleme general de la stabilite du mouvement" in
1892 12], the stability of the xed points of dynamical systems has been extensively analysed using
method is to a xed point
nd a x in
smooth a region
scalar function V RD and (ii)
V(_x
)
such 0 in
. If these requirements are full lled, V is called a Lyapunov function and in this case, x must be
the last section of this work, we argue that this picture can be extended to the study of control from
the point of view of information theory.
2. Lyapunov exponent theory
see that the Lyapunov exponents are closely connected to some information quantities. From this