Editing Dynamical system (section)

==Bifurcation theory==
{{Main|Bifurcation theory}}
When the evolution map Φ<sup>''t''</sup> (or the [[vector field]] it is derived from) depends on a parameter μ, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the [[phase space]] until a special value ''μ''<sub>0</sub> is reached.  At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.

Bifurcation theory considers a structure in phase space (typically a [[Fixed point (mathematics)|fixed point]], a periodic orbit, or an invariant [[torus]]) and studies its behavior as a function of the parameter&nbsp;''μ''. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures.  By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

The bifurcations of a hyperbolic fixed point ''x''<sub>0</sub> of a system family ''F<sub>μ</sub>'' can be characterized by the [[eigenvalues]] of the first derivative of the system ''DF''<sub>''μ''</sub>(''x''<sub>0</sub>) computed at the bifurcation point. For a map, the bifurcation will occur when there are eigenvalues of ''DF<sub>μ</sub>'' on the unit circle. For a flow, it will occur when there are eigenvalues on the imaginary axis. For more information, see the main article on [[Bifurcation theory]].

Some bifurcations can lead to very complicated structures in phase space. For example, the [[Ruelle–Takens scenario]] describes how a periodic orbit bifurcates into a torus and the torus into a [[strange attractor]]. In another example, [[Bifurcation diagram|Feigenbaum period-doubling]] describes how a stable periodic orbit goes through a series of [[period-doubling bifurcation]]s.