Nonlinear two-dimensional systems are introduced in this reading. In general there is no way to study them analytically, so their qualitative behavior is determined. The phase portraits are entirely filled with trajectories and there is a enormous varieties of phase portraits that arise. Vector fields become cluttered for nonlinear systems, so direction fields are used, which are usually made by computer. When quantitive aspects are needed, the Runge-Kutta method is used. Trajectories never intersect due to the uniqueness theorem. Linearization can be used to approximate the phase portrait near a fixed and the Jacobian matrix of a system is used to to analyze the dynamics of fixed points and phase portraits.
Challenges
Example 6.3.2 moved a little fast for me. It jumped through a lot of steps because we either did them earlier in the book, or we will do them in the exercises. I wish it would have walked through the linearization process more in depth.
Reflection
I like the phase portraits that get generated by nonlinear systems. It is clear how they arise in nature and have scientific applications. Its also exciting that we are finally into nonlinear systems.
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