bifurcations

Bifurcation refers to a qualitative change in a dynamical system as a parameter is varied. It occurs when small changes in a parameter alter the number or stability of equilibria or periodic orbits. Bifurcations are studied in continuous-time systems (flows) and discrete-time systems (maps) using bifurcation diagrams that plot fixed points or cycles against the parameter.

Bifurcations are typically categorized as local or global. Local bifurcations involve changes in the behavior near an invariant set such as a fixed point or a periodic orbit, often detected by eigenvalues crossing a critical boundary. Global bifurcations involve broader changes in the phase space structure, such as the creation or destruction of connecting orbits.

Common local bifurcations include saddle-node (fold), transcritical, pitchfork, and Hopf bifurcations. In a saddle-node bifurcation, two fixed points collide and annihilate as a parameter passes a critical value. In a transcritical bifurcation, two fixed points exchange stability. In a pitchfork bifurcation, a symmetric fixed point loses stability and one or more nonzero fixed points appear or disappear, with the outcome depending on the system’s symmetry. In a Hopf bifurcation, a pair of complex conjugate eigenvalues crosses the imaginary axis, creating or destroying a small-amplitude periodic orbit.

One-dimensional examples illustrate the idea: a saddle-node can occur in dx/dt = r − x^2, a transcritical in dx/dt = r x − x^2, and a pitchfork in dx/dt = r x − x^3. Hopf bifurcations require at least two dimensions and lead to oscillatory behavior.

Codimension describes how many parameters must be varied to realize the bifurcation. Bifurcation theory provides normal forms, stability criteria, and unfolding methods applicable across physics, biology, engineering, and economics.