TABLE OF CONTENTS A one-dimensional dynamical system is characterized by a single, first-order differential equation
where the initial condition, x 0, is specified. The function f only depends on the dependent variable x and not the time, t. This type of differential equation is called a first-order, autonomous equation. The related nonautonomous equation is
and, for the moment, will not be considered. We assume that f( x) has the required mathematical properties such that the existence and uniqueness theorems hold. For problems in the natural and engineering sciences, this requirement is almost always satisfied.
The fixed-points are the constant solutions for Eq. (4.2.1). This means that they are determined by the solutions to the following, in general, algebraic equation
In actual applications, the only solutions of relevance are those for which x is real, i.e., as far as the fixed-points are concerned, those having complex values do not correspond to actual states of the dynamical system. Also, it should be indicated that the number of real solutions to Eq. (4.2.3) may vary from zero to any finite integer. Clearly, purely mathematical models can be constructed such that the number of fixed-points is unbounded.
For physical systems, such as those arising in physics or mechanical engineering, the fixed-points correspond to states of equilibrium. This is the reason why only the real solutions of Eq. (4.2.3) are of interest.
Assume that we have a one-dimensional dynamical...
