TABLE OF CONTENTS A two-dimensional autonomous dynamical system is one defined by a pair of coupled, first-order differential equations
We assume that P and Q have a sufficient number of partial derivatives such that a unique solution exists for any given problem of interest. For many applications, P and Q are polynomials, and, as a consequence, all derivatives exist.
The phase space is the ( x,y) plane for this case and the trajectories in this phase space, y = y( x), are determined by the solutions to the following first-order differential equation
This result can be derived by noting that if y = y( x), then
The general solution to Eq. (4.4.2) is called a first-integral of the system given by Eq. (4.4.1).
The fixed-points are the constant solutions
to Eq. (4.4.1). Since d x/ dt = 0 and d y/ dt = 0, it follows that they are the simultaneous solutions to the equation
For the study of dynamical systems in the natural and engineering sciences, only the real solutions have "physical meaning." Thus, we only consider such solutions. Also, it is to be expected that Eq. (4.4.5) may have multi-fixed-points. We will present examples in the next section to illustrate this fact.
Nullclines are curves in the ( x, y) phase space along which the solution trajectories, y = y( x), have...
