Scalar Autonomous Equations
In this opening chapter, we present selected basic concepts about the geometry of solutions of ordinary differential equations. To keep the ideas free from technical complications, the setting is one-dimensional—the scalar autonomous differential equations. Despite their simplicity, these concepts are central to our subject and reappear in various incarnations throughout the book. Following a collection of examples, we first state a theorem on the existence and uniqueness of solutions. Then we explain what a differential equation is geometrically. To facilitate qualitative analysis, geometric concepts such as vector field, orbit, equilibrium point, and limit set are included in this discussion. The next topic is the notion of stability of an equilibrium point and the role of linear approximation in determining stability. We conclude the chapter with an example of a scalar differential equation defined on a one-dimensional space other than the real line—a circle.
KeywordsVector Field Equilibrium Point Potential Function Phase Portrait Linear Differential Equation
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