bifurcation

Bifurcation is a qualitative change in the behavior of a dynamical system as a parameter is varied. It is studied in both continuous-time systems, described by differential equations, and discrete-time systems, described by iterated maps. Bifurcations mark parameter values where the system undergoes a fundamental change in its long‑term dynamics, such as the creation or destruction of equilibria or periodic orbits, or a change in their stability.

Local bifurcations occur near equilibria and involve changes in their number or stability. Common examples include saddle-node (fold) bifurcations, where two fixed points collide and annihilate each other; transcritical bifurcations, where stability is exchanged between fixed points; and pitchfork bifurcations, where a symmetric fixed point loses stability and gives rise to new symmetric equilibria. In continuous-time systems, the stability change is tied to eigenvalues crossing zero.

A Hopf bifurcation is another central local type, occurring when a pair of complex eigenvalues crosses the imaginary axis, causing an equilibrium to lose stability and a small-amplitude periodic orbit to emerge or disappear. Period-doubling (flip) bifurcations in discrete maps produce cycles whose period doubles, often leading to complex, potentially chaotic dynamics under parameter variation.

Global bifurcations involve changes in the overall structure of invariant sets and cannot be detected by linearization alone. Examples include homoclinic and heteroclinic bifurcations, crises that abruptly alter attractor size, and border-collision bifurcations in piecewise-smooth systems.

Bifurcation analysis is aided by bifurcation diagrams, which plot long-term behavior versus a parameter, and by normal form theory, which reduces dynamics near critical points to simple canonical models. Applications span physics, biology, engineering, and economics, wherever systems exhibit regimes that change qualitatively with parameters.