Phase Plane Analysis

Mathematical Modeling · Axiom Academy

A graphical technique for understanding the qualitative behavior of two-dimensional dynamical systems Consider a system of two first-order differential equations: The phase plane is the x-y coordinate plane where each point represents a possible state of the system. As time evolves, the state moves through the phase plane, tracing out a trajectory or orbit . 2. Direction Fields (Vector Fields) At each point (x, y) in the phase plane, the derivatives define a velocity vector: A direction field (or vector field) shows these vectors at many points, revealing the flow pattern of the system. Trajectories follow these vectors, always tangent to the field. Nullclines are curves where one of the derivatives is zero: x-nullcline: Where f(x,y) = 0. On this curve, dx/dt = 0, so motion is purely vertical. y-nullcline: Where g(x,y) = 0. On this curve, dy/dt = 0, so motion is purely horizontal. Nullclines divide the phase plane into regions. In each region, we can determine the direction of flow by testing the sign of f(x,y) and g(x,y): A phase portrait combines all the information: nullclines, equilibria, direction field, and representative trajectories to show the complete qualitative behavior of the system. 6. Classifying Equilibria Visually The local behavior near equilibria can be classified by examining nearby trajectories: Stable Node: All trajectories approach the equilibrium directly Unstable Node: All trajectories move away directly

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