## ABSTRACT

Critical points, direction fields and trajectories are elements of “phase-space analysis”, a powerful analytical system for geometrically analyzing the behavior of solutions to certain common systems of differential equations. In this analytical system, solutions to suitable N×N systems of differential equations are graphically represented by their “trajectories”, paths taken by the solutions in N×N space. The simplest trajectory is that given by a constant (or “equilibrium”) solution. It is a single point, a “critical point” for the system. For the other trajectories, one can use the system of equations to construct a “direction field” (similar to a “slope field” for a single first-order equation) consisting of vectors tangent at points to the trajectories. From that, the trajectories, themselves, can be sketched and the “direction of travel” on each ascertained. And from the resulting picture, a great deal of information about the general behavior of the solutions can be determined, especially in the neighborhoods of the critical points.

This chapter introduces and develops this phase-space analysis. It is further used to study the ultimate outcomes in a mixing problem with two tanks, in the interactions of two species competing for the same resources, and in the classic damped swinging pendulum.