Bifurcation theory

Bifurcation theory studies qualitative changes in a system’s equilibria, periodic orbits, or other dynamics as parameters vary.

Saddle-node bifurcation: A bifurcation in which two equilibria meet and disappear as a parameter crosses a critical value. It is the basic mechanism for equilibrium creation or loss.
Dynamical system: A mathematical model specifying how a state changes over time. Bifurcation theory tracks changes in the behavior of these systems.
Lorenz system: A three-dimensional system of differential equations introduced as a simplified model of atmospheric convection. Its parameter-dependent changes connect bifurcation structure with chaotic attractors.
Linear stability analysis: A method that estimates local stability by examining the linearization of a system near a solution. It detects stability changes but alone may not describe the nonlinear branches they create.
Hopf bifurcation: A local bifurcation in which an equilibrium changes stability as a pair of eigenvalues crosses the imaginary axis, often creating a periodic orbit. It explains how steady behavior can give way to oscillation.
Equilibrium point: A state of a dynamical system that remains unchanged under its evolution. Equilibria are common solution branches whose number or stability can change.
Logistic map: A one-dimensional discrete-time model of population growth with density-dependent reproduction. Its parameter sweep exhibits period doubling and the onset of chaos.
Catastrophe theory: The mathematical study of discontinuous changes in equilibria under smooth variations of control parameters. It emphasizes sudden transitions and singularity geometry within a broader bifurcation landscape.
Pitchfork bifurcation: A bifurcation in which a symmetric equilibrium changes stability and branches into multiple equilibria. It shows how symmetry can organize branching solution families.
Stability theory: The study of whether small disturbances around a solution remain small or grow over time. Stability changes often mark the onset of a bifurcation.

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