Introduction to Nonlinear Dynamics, and Flows on the Line

Corresponds to Strogatz, Nonlinear Dynamics and Chaos, Chapters 1–2. Opens with the book’s thesis — most of the interesting behavior in the world (and in this book) comes specifically from nonlinearity, which linear systems theory (introduction-and-autonomous-linear-systems) can’t produce — then starts the geometric (rather than closed-form) approach to differential equations that the rest of the book builds on: for $\dot x = f(x)$, just plot $f(x)$ against $x$ and read off the flow directly, without solving anything. This geometric view immediately gives fixed points, their stability (from the sign of $f'$ at the fixed point — “linear stability analysis”), and a clean topological reason 1D flows can never oscillate (the line has no room to loop back).

Outline (TODO — flesh out each)

• Chaos, fractals, and dynamics — capsule history; why nonlinearity matters; a dynamical view of the world
• A geometric way of thinking about $\dot x = f(x)$
• Fixed points and stability
• Population growth models
• Linear stability analysis
• Existence and uniqueness
• Impossibility of oscillations in one dimension
• Potentials
• Solving equations on the computer