Point Neurons

Bistability in the Hodgkin-Huxley model of the action potential in squid axon.
Supercritical Hopf bifurction in the Morris-Lecar model of barnacle muscle.
Saddle-Node bifurcation on a Limit Cycle in the Morris-Lecar model.
Regular spiking in a minimal neural model by Pospischil, et al.
Intrinsic Bursting in the Hindmarsh-Rose neuron.

Neural Masses

Wilson-Cowan model of reciprocally-coupled excitatory and inhibitory neural populations.
Chaos in the Breakspear-Terry-Friston model.
Perceptual opponency in the Heitmann-Ermentrout E-I-E model.

Spatial Models

The 2D Wave Equation with reflecting boundary conditions.
Initiation of spiral rotors in a sheet of FitzHugh-Nagumo neurons.
Arrhythmogenesis in a heterogeneous sheet of FitzHugh-Nagumo neurons with a tissue boundary.

Normal Forms

Normal form of the Saddle-Node bifurcation on a Limit Cycle. The basis of Type-1 neural excitability.
Normal form of the Supercritical Hopf bifurcation. The basis of Type-2 neural excitability.

Classics

Determistic chaos in the Lorenz Attractor.
The damped-and-driven pendulum.
The Van Der Pol relaxation oscillator.
The Lotka-Volterra predator-prey equations.
Arithmetic Brownian Motion. Drift and diffusion.
Geometric Brownian Motion. Multiplicative noise. Mathematical finance.

Strogatz Textbook Examples

Example 5.1.1. Closed orbits in the simple harmonic oscillator.
Example 5.1.2. A pair of uncoupled linear equations.
Example 5.2.5. A degenerate node has only one eigenvector.
Example 5.3.1. Love affairs.
Example 6.3.1. Saddles and stable nodes.
Example 6.4. Rabbits versus Sheep.
Example 6.5.2. Particle moving in a double well potential.
Example 6.6.1. System with a nonlinear centre at the origin.
Example 6.6.2. Symmetrical system with a homoclinic orbit.
Example 6.6.3. A symmetrical system that is reversible but not conservative.
Example 6.7. The linearly damped pendulum.