Bifurcation

A bifurcation is a qualitative change in a dynamical system’s behavior as a parameter varies. It occurs when the system’s equilibria or their stability change.

Bifurcation diagram: A plot showing a dynamical system’s long-term states as a parameter varies. It displays the branches of states that appear, disappear, or change stability at bifurcations.
Dynamical system: A system whose state changes over time according to a specified rule. Bifurcation describes qualitative changes in the behavior of such systems.
Saddle-node bifurcation: A bifurcation in which a stable and an unstable equilibrium collide and disappear or emerge. It is the simplest example of equilibria being created or destroyed as a parameter changes.
Logistic map: A discrete-time model of population growth defined by a nonlinear recurrence relation. Varying its growth parameter produces period doublings and chaotic behavior.
Linear stability analysis: A method that estimates local stability by examining a system’s linearization near a state. It can detect local stability changes, but a bifurcation describes the resulting qualitative reorganization.
Equilibrium point: A state of a dynamical system that remains unchanged under its evolution. Bifurcations often occur when equilibria emerge, collide, or change stability.
Parameter: A quantity in a model that can be fixed or varied to define different system behaviors. Changing a parameter can move a system across a bifurcation point.
Pitchfork bifurcation: A bifurcation in which one equilibrium branch splits into three, typically under a symmetry condition. Its branch structure makes symmetry-breaking transitions explicit.
Lorenz system: A three-dimensional system of differential equations originally derived from a simplified model of atmospheric convection. Its parameter changes reveal transitions among steady states, periodic orbits, and chaos.
Structural stability: The property that a system’s qualitative dynamics persist under small perturbations of the system itself. Bifurcations mark parameter values where structural stability typically fails.

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