chapter11.md

The saddle-node bifurcation

Overview

This chapter's single example makes a saddle-node collision concrete: in the displayed positive-$x$ region, a stable fixed point and a saddle coalesce into one half-stable point and then disappear.

Core ideas

A saddle-node bifurcation changes the number of equilibria in a local region. Before the displayed collision, trajectories can be attracted to a node or separated by a saddle's stable manifold. At the collision, the flow is attracting from one side and repelling from the other; beyond it, the displayed positive-$x$ pair no longer remains.

Essential model

The plotted planar normal-form-like system is

$$ \dot x=-ax+y,\qquad \dot y=\frac{x^2}{1+x^2}-by. $$

Here $x$ and $y$ are state variables, $a$ is the varied control parameter, and $b=1$ is fixed. In the displayed positive-$x$ region, solving the nullclines produces two fixed points for $a<0.5$, one collision point at $a=0.5$, and none for $a>0.5$. The full system also has the equilibrium $(x,y)=(0,0)$ for every $a$; the script clips it out by starting and displaying trajectories at $x\ge0.2$.

Code examples

The example now lives in chapter11.ipynb: plot_saddle_node_bifurcation integrates trajectories for the three parameter regimes and draws their phase portraits, using saddle_node_fixed_points for the closed-form fixed points.

What to look for

Read the three panels from left to right. The black filled marker is the stable fixed point and the white marker is the saddle; their merged half-filled marker in the middle panel shows the collision. In the final panel, all shown trajectories pass through because no positive-$x$ fixed point survives; the equilibrium at the excluded origin is outside the plotted frame.

Suggested order

1. Run SADDLE_NODE_BIFURCATION and identify the fixed-point markers.
2. Compare nearby trajectories on each side of the saddle's separatrix.
3. Relate the three panels to creation or destruction of a resting state.

Prerequisites and related chapters

Chapter 10 supplies nullclines and phase portraits. Chapter 12 applies saddle-node ideas to reduced neuron dynamics, while Chapters 17--18 connect them to firing thresholds and bistability.

Running the examples

Open chapter11.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom.