Autonomous Planar Nonlinear Systems

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In Chapter ?? we discussed how the phase line for a single autonomous nonlinear differential equation is determined by the equilibria of the differential equation. Once the equilibria and their stability properties are known, we can find the long time or asymptotic behavior of every solution to a single equation, even though we do not necessarily know a closed form formula for that solution.

In this chapter we discuss the corresponding and more complicated results for planar systems of nonlinear autonomous differential equations. We discuss precisely what information about planar phase portraits is needed in order to be able to tell the (asymptotic) fate of all solutions. The information that we need includes the equilibria and their type (as discussed in Chapter ??), periodic solutions and their stability type, and connecting trajectories. Once we have the needed information, we can determine the qualitative features of all solutions to a planar system — even though we cannot write a closed form formula for these solutions.

In Section ?? we experiment numerically with nonlinear planar systems, introducing the information that is needed to draw a qualitative phase plane portrait of a differential equation. We see that we will need to know the equilibria and their type (saddle, sink, or source), time periodic solutions and their stability, and trajectories that connect these solutions (including stable and unstable orbits emanating from saddles).

In Section ?? we look more closely at the role of linearized systems of differential equations near a hyperbolic equilibrium. In particular, we introduce the Jacobian matrix and show numerically that solutions near an equilibrium behave like solutions to the linearized system. Then we use the results of Chapter ?? to understand specific features of the local behavior of nonlinear systems near a saddle, sink, or source in terms of the corresponding features of solutions to linear differential equations.

The analytic discussion of aspects of time periodic solutions is introduced in Section ??. In particular, we show how periodic solutions to certain systems of differential equations can be constructed using phase-amplitude equations in polar coordinates.

This information is then synthesized in Section ?? in terms of stylized phase portraits for Morse-Smale planar systems of autonomous differential equations. Morse-Smale systems are differential equations with properties that allow us to find qualitative phase plane portraits. Moreover, in a sense that we will not try to make precise, most planar autonomous systems are Morse-Smale.