Bifurcation diagram

Symmetry breaking in bifurcation sets

In a dynamical system such as $${\displaystyle \ddot{x}+f(x;\mu )+\epsilon g(x)=0,}$$ which is structurally stable when ${\displaystyle \mu \ne 0}$, if a bifurcation diagram is plotted, treating ${\displaystyle \mu }$ as the bifurcation parameter, but for different values of ${\displaystyle \epsilon }$, the case ${\displaystyle \epsilon =0}$ is the symmetric pitchfork bifurcation. When ${\displaystyle \epsilon \ne 0}$, we say we have a pitchfork with broken symmetry. This is illustrated in the animation on the right.

Applications

Consider a system of differential equations that describes some physical quantity, that for concreteness could represent one of three examples: 1. the position and velocity of an undamped and frictionless pendulum, 2. a neuron's membrane potential over time, and 3. the average concentration of a virus in a patient's bloodstream. The differential equations for these examples include *parameters* that may affect the output of the equations. Changing the pendulum's length will affect its oscillation frequency, changing the magnitude of injected current into a neuron may transition the membrane potential from resting to spiking, and the long-term viral load in the bloodstream may decrease with carefully timed treatments.

In general, researchers may seek to quantify how the long-term (asymptotic) behavior of a system of differential equations changes if a parameter is changed. In the dynamical systems branch of mathematics, a bifurcation diagram quantifies these changes by showing how fixed points, periodic orbits, or chaotic attractors of a system change as a function of bifurcation parameter. Bifurcation diagrams are used to visualize these changes.