Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.
Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior.
[1] Bifurcations occur in both continuous systems (described by ordinary, delay or partial differential equations) and discrete systems (described by maps).
The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior.
[2] It is useful to divide bifurcations into two principal classes: A local bifurcation occurs when a parameter change causes the stability of an equilibrium (or fixed point) to change.
In continuous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero.
In discrete systems (described by maps), this corresponds to a fixed point having a Floquet multiplier with modulus equal to one.
In both cases, the equilibrium is non-hyperbolic at the bifurcation point.
More technically, consider the continuous dynamical system described by the ordinary differential equation (ODE)
Examples of local bifurcations include: Global bifurcations occur when 'larger' invariant sets, such as periodic orbits, collide with equilibria.
This causes changes in the topology of the trajectories in the phase space which cannot be confined to a small neighbourhood, as is the case with local bifurcations.
In fact, the changes in topology extend out to an arbitrarily large distance (hence 'global').
Examples of global bifurcations include: Global bifurcations can also involve more complicated sets such as chaotic attractors (e.g. crises).
However, transcritical and pitchfork bifurcations are also often thought of as codimension-one, because the normal forms can be written with only one parameter.
[10] Bifurcation theory has also been applied to the study of laser dynamics[11] and a number of theoretical examples which are difficult to access experimentally such as the kicked top[12] and coupled quantum wells.
[13] The dominant reason for the link between quantum systems and bifurcations in the classical equations of motion is that at bifurcations, the signature of classical orbits becomes large, as Martin Gutzwiller points out in his classic[14] work on quantum chaos.
