Bifurcation theory

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In mathematics, specifically in the 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 the system's long-term dynamical behaviour. Bifurcations can occur in continuous systems (described by ODEs, DDEs or PDEs), and discrete systems (described by maps).

Bifurcation theory is the study of how and when such bifurcations can occur.

Types of bifurcation

It is useful to divide bifurcations into two principal classes:

• Local bifurcations, which can be analysed entirely through changes in the local stability properties of equilibria or orbits as parameters cross through critical thresholds; and
• Global bifurcations, which often occur when larger invariant sets of the system 'collide' with each other, or with equilibria of the system. They cannot be detected purely by a stability analysis of the equilibria (fixed points).

Local bifurcations

A local bifurcation occurs when a parameter change causes the stability of an equilibrium (or fixed point) to change. In continous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero. In discrete systems (those described by maps rather than ODEs), this corresponds to a fixed point having a Floquet multiplier with modulus equal to one.

The topological changes in the phase portrait of the system can be confined to arbitrarily small neighbourhoods of the bifurcating fixed points by moving the bifurcation parameter close to the bifurcation point (hence 'local').

More technically, consider the continous dynamical system described by the ODE

[\dot x=f(x,\lambda)\quad f:\mathbb^n\times\mathbb\rightarrow\mathbb^n]

A local bifurcation occurs at [(x_0,\lambda_0)] if the matrix

[ \textrmf_]

has an eigenvalue with zero real part. If the eigenvalue is equal to zero, the bifurcation is a steady state bifurcation, if the eigenvalue is non-zero but purely imaginary, this is a Hopf bifurcation.

For discrete dynamical systems, consider the system

[x_=f(x_n,\lambda)]

Then a local bifurcation occurs at [(x_0,\lambda_0)] if the matrix

[ \textrmf_]

has an eigenvalue with modulus equal to one.

Examples of local bifurcations include:

• Fold or saddle-node bifurcation
• Transcritical bifurcation
• Pitchfork bifurcation
• Flip or period-doubling bifurcation
• Hopf bifurcation
• Neimark or secondary Hopf bifurcation

Global bifurcations

Examples of global bifurcations include:

• Homoclinic bifurcation in which a limit cycle collides with a saddle point
• Heteroclinic bifurcation in which a limit cycle collides with two or more saddle points
• Infinite-period bifurcation in which a stable node and saddle point simultaneously occur on a limit cycle

See also

• Bifurcation diagram
• Catastrophe theory

References

• [Nonlinear dynamics]

 

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