The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 257 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 257

Consider an equilibrium solution of a dynamical system, that is, a

solution that is time invariant. If a stable system is perturbed when it is

in a position of equilibrium, it tends to return to the equilibrium posi-

tion or, in any case, not to diverge indefinitely from its equilibrium posi-

tion. For example, a damped pendulum—if perturbed from a position of

equilibrium—will tend to go back to an equilibrium position. If the pen-

dulum is not damped it will continue to oscillate forever.

Consider a system of n equations of first order. (As noted above,

systems of higher orders can always be reduced to first-order systems by

enlarging the set of variables.) Suppose that we can write the system

explicitly in the first derivatives as follows:

dy 1

---------= f 1 (xy,,, 1 ...yn)

dx

dy

--------^2 -

 = f^2 (xy^1 ...

dx

,,, yn)



.



.

.

dyn

---------= f(xy 1 ...

dx

n ,,, yn)



If the equations are all linear, a complete theory of stability has been

developed. Essentially, linear dynamical systems are stable except possi-

bly at singular points where solutions might diverge. In particular, a

characteristic of linear systems is that they incur only small changes in

the solution as a result of small changes in the initial conditions.

However, during the 1970s, it was discovered that nonlinear sys-

tems have a different behavior. Suppose that a nonlinear system has at

least three degrees of freedom (that is, it has three independent nonlin-

ear equations). The dynamics of such a system can then become chaotic

in the sense that arbitrarily small changes in initial conditions might

diverge. This sensitivity to initial conditions is one of the signatures of

chaos. Note that while discrete systems such as discrete maps can

exhibit chaos in one dimension, continuous systems require at least

three degrees of freedom (that is, three equations).

Sensitive dependence from initial conditions was first observed in

1960 by the meteorologist Edward Lorenz of the Massachusetts Institute

of Technology. Lorenz remarked that computer simulations of weather

forecasts starting, apparently, from the same meteorological data could