Bifurcations

Mathematical Modeling · Axiom Academy

Understanding qualitative changes in dynamical systems as parameters vary A bifurcation occurs when a small smooth change in a parameter causes a sudden qualitative change in the system's behavior. The word "bifurcation" comes from Latin, meaning "to divide into two branches." Consider a discrete dynamical system defined by the map: where r is a parameter. As r changes, the number and stability of fixed points can change dramatically. A bifurcation diagram is a visual representation showing how the long-term behavior of a system changes as a parameter varies. It plots the asymptotic values (fixed points, periodic orbits, or chaotic attractors) against the parameter. The parameter value (e.g., r in the logistic map) Long-term values of the state variable x At each parameter value, we iterate the system many times and plot the attractor values. This reveals the structure of bifurcations across the entire parameter space. In a saddle-node bifurcation (also called a fold bifurcation ), two fixed points - one stable and one unstable - collide and annihilate each other as the parameter crosses the bifurcation point. For r > 0: Two fixed points exist at x* = r (stable) and x* = -r (unstable) For r = 0: The two fixed points merge For r < 0: No real fixed points exist In a transcritical bifurcation , two fixed points exchange their stability as the parameter passes through the bifurcation point. Unlike the saddle-node, both fixed points continue to exist - they just swap roles.

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