add TODO stubs for planar analysis, limit cycles, and planar bifurcations

Author: jhobbs

content/applied-math/dynamical-systems/limit-cycles.md

@@ -0,0 +1,20 @@
+---
+layout: page
+title: Limit Cycles
+---
+
+## Conservative and Hamiltonian Systems
+
+<!-- TODO: Hamiltonian systems, energy conservation, potential functions, energy level sets, closed orbits -->
+
+## Lyapunov Functions
+
+<!-- TODO: Lyapunov function construction, proving global/asymptotic stability, Lyapunov's theorems -->
+
+## Limit Cycles
+
+<!-- TODO: definition, isolated closed orbits, stable/unstable/half-stable limit cycles, amplitude and frequency -->
+
+## Ruling Out and Proving Closed Orbits
+
+<!-- TODO: Bendixson's theorem (ruling out closed orbits via divergence), Poincaré-Bendixson theorem (proving existence via trapping regions) -->

content/applied-math/dynamical-systems/planar-bifurcations.md

@@ -0,0 +1,12 @@
+---
+layout: page
+title: Planar Bifurcations
+---
+
+## Zero-Eigenvalue Bifurcations
+
+<!-- TODO: saddle-node, transcritical, and pitchfork bifurcations in 2D systems, relation to 1D normal forms from [[bifurcations]] -->
+
+## Hopf Bifurcation
+
+<!-- TODO: definition, supercritical vs subcritical, birth/death of limit cycles, eigenvalues crossing imaginary axis, normal form -->

content/applied-math/dynamical-systems/planar.md

@@ -26,3 +26,20 @@ Trajectories that connect two @fixed-point are called **heteroclinic orbits.**
 :::definition "Hyperbolic Fixed Point"
 A **hyperbolic fixed point** is a @fixed-point for which the real part of both eigenvalues is non-zero.
 :::
+
+## Nullclines
+
+<!-- TODO: nullcline definition, using nullclines to find fixed points and sketch phase portraits -->
+
+## Classification of Linear Systems
+
+<!-- TODO: trace-determinant classification (saddle, stable/unstable node, stable/unstable spiral, center, degenerate node), eigenvalue analysis -->
+
+## Jacobian Linearization
+
+<!-- TODO: Jacobian matrix at fixed points, linearization of nonlinear systems, when linearization determines stability (hyperbolic case) -->
+
+## Polar Coordinate Methods
+
+<!-- TODO: converting planar systems to polar form, radial and angular dynamics, using polar coordinates for limit cycle analysis -->
+