Bifurcation Family Explorer
The phase line of \(\dot{x}=f(x;r)\) is one vertical slice through the \((r,x^*)\) diagram. Drag the \(r\) slider to see both representations change together.
What Is a Bifurcation?
A bifurcation is a qualitative change in the phase portrait of a family of dynamical systems \(\dot{x}=f(x;r)\) as the parameter \(r\) passes through a critical value \(r_c\). At \(r_c\) the number or stability of equilibria changes — equilibria are born, collide, split, or exchange stability.
In one dimension, every generic bifurcation is one of three types. With symmetry (reflection \(x\mapsto -x\)), two more appear. This tool walks through all four canonical normal forms.
The Four Normal Forms
Two equilibria for \(r<0\), collide at \(r=0\), none for \(r>0\). The generic way equilibria are created or destroyed.
Two equilibria swap stability at \(r=0\). The generic way equilibria can pass through each other.
One stable branch at \(r<0\) splits into two stable branches at \(r=0\). Symmetry-breaking, soft transition.
Symmetry-breaking with hysteresis: trajectory jumps up at \(r=0\), back down only at \(r=-1/4\) (saddle-nodes of equilibria).
Reading the Coupled View
- The left panel shows the graph of \(f(x;r)\) for the current \(r\), with the phase line on the \(x\)-axis. Intersections with \(y=0\) are equilibria; arrows show the sign of \(\dot{x}\). Green dots are stable (\(f'<0\)), red dots unstable (\(f'>0\)).
- The right panel plots all branches \(x^*(r)\). Solid green = stable; dashed red = unstable. Purple dots mark bifurcation points. A vertical amber line tracks the current \(r\).
- The intersections of the vertical \(r\)-line with the branches are exactly the equilibria shown on the phase line. This is the whole point.
Hysteresis in the Subcritical Case
For \(\dot{x}=rx+x^3-x^5\), between \(r=-1/4\) and \(r=0\) there are five equilibria: the origin, two small unstable, two large stable. Start at \(x=0\) with \(r<0\) and increase \(r\): the origin loses stability at \(r=0\) and the trajectory jumps up to the large stable branch. Now decrease \(r\): the large branches persist until \(r=-1/4\), where they collide with the unstable branches in a saddle-node and vanish — the trajectory jumps back down. The up-jump and down-jump happen at different \(r\): that is hysteresis, and the tool's slider reveals it visually when you animate back and forth.
How to Use
- Pick a normal form from the preset bar.
- Drag the \(r\) slider (or type in the number box) to change \(r\) by hand.
- Click Play to animate \(r\) from left to right; Reverse plays right to left (useful for seeing hysteresis in the subcritical case).
- The readout bar reports the current equilibria and their stability.
Saddle-node
Two equilibria collide and annihilate. Typical \"window\" in parameter space where equilibria only exist on one side of \(r_c\). Normal form: \(\dot{x}=r+x^2\).
Transcritical
Two equilibria exchange stability by passing through each other. One branch (often \(x=0\)) exists for all \(r\); the other crosses it at \(r_c\). Normal form: \(\dot{x}=rx-x^2\).
Supercritical pitchfork
Single stable branch splits into two stable branches (unstable middle). \"Soft\" symmetry breaking — small amplitude growth near \(r_c\). Normal form: \(\dot{x}=rx-x^3\).
Subcritical pitchfork
Unstable branches bend back and collide with stable branches at \(r