Two-Dimensional Linear Systems and the Phase Plane
Stub. The classification of planar linear systems (nodes, saddles, spirals, centers, degenerate cases) via eigenvalues, with a love-affair toy example; then the phase-plane method for nonlinear planar systems — phase portraits, existence/uniqueness and the topological consequences that follow in 2D (unlike 1D), linearization at fixed points, conservative and reversible systems, the pendulum, and index theory.
Corresponds to Strogatz, Chapters 5–6. Classifies every planar linear system by its eigenvalues (real distinct, complex, repeated, zero) into the canonical pictures — node, saddle, spiral, center, star, degenerate node — the same eigenvalue/mode analysis as introduction-and-autonomous-linear-systems, specialized to where every case can be drawn. This classification is the local building block for the phase plane method applied to genuinely nonlinear 2D systems: sketch the phase portrait globally, then linearize at each fixed point to read off its local type. Unlike 1D flows, 2D phase space has room for trajectories to loop, which is why oscillation becomes possible — but also subtler topological constraints (from existence/uniqueness: trajectories can’t cross) that conservative, reversible, and Hamiltonian-like systems exploit, with index theory giving further global constraints on which fixed-point/closed-orbit combinations are even possible.
Outline (TODO — flesh out each)
- Linear systems — definitions and examples
- Classification of linear systems
- Love affairs (worked toy example)
- Phase portraits
- Existence, uniqueness, and topological consequences in 2D
- Fixed points and linearization
- Rabbits versus sheep (competition model)
- Conservative systems
- Reversible systems
- The pendulum
- Index theory