In Chapter ?? we saw that equilibria and their stabilities completely determined the phase line dynamics for single autonomous first order differential equations. The stability of the equilibria determine the direction in which one equilibrium is connected to the next.
In Chapter ?? we discussed the extent to which equilibria and their stability determined the phase planes of systems of two autonomous systems of ordinary differential equations. When an equilibrium is hyperbolic, nonlinear planar systems behave much like their linearizations — at least on a small neighborhood of the equilibrium. However, away from equilibria, planar systems can have dynamically more interesting states: limit cycles. We have seen that qualitatively we can understand the global dynamics of most planar systems (the Morse-Smale ones) if we can find their equilibria, their periodic solutions, and their connecting orbits. Analytically, this is an impossible problem to solve in closed form — but numerically this kind of calculation is often tractable.
In this chapter we discuss briefly the dynamics of systems of three or more autonomous first order differential equations. The situation is very complicated — even on the qualitative level. On the positive side, nonlinear systems do behave like their linear counterparts on a neighborhood of hyperbolic equilibria, and the dynamics of the linearized systems can be understood as a consequence of the Jordan normal form theorem. These issues are discussed in Sections ?? and ??. On the negative side, in Sections ?? and ?? we will see that the dynamics of nonlinear systems away from equilibria are just much more complicated than their planar counterparts. In particular, we will see that quasiperiodic motion may be expected (first in linear nonhyperbolic four dimensional systems and then in nonlinear three dimensional systems). Finally, we show that even complicated ‘chaotic’ motion may be expected in three dimensions (the Lorenz equations).
The discussion in Sections ?? and ?? is predicated on being able to solve numerically systems of differential equations with more than two equations. To do this, we must use the MATLAB differential equations solver ode45 directly. We introduce this solver in Section ?? by solving certain one-dimensional differential equations. In this section we also discuss how to store functions in MATLAB m-files. In Section ?? we use ode45 to solve several sample differential equations in three and four dimensions, and we display the results using MATLAB graphics.
The discussion in this chapter continues an important theme: what information can we learn about the dynamics and solutions of nonlinear systems of ordinary differential equations from numerical simulation. Indeed, what can mathematics say that will help in interpreting numerically obtained solutions? We will see that even on the qualitative level, the situation is very complicated — complicated enough to guarantee that closed form solutions are, in general, not an option. It should be noted, however, that closed form solutions do exist for many particular types of equations (not least of which are the linear constant coefficient systems). In later chapters we will discuss some of the techniques of integration that allow us to solve certain special kinds of differential equations in closed form.