Dynamics MOC

Bifurcation

A bifurcation occurs when a smooth variation of the parameters of a Dynamical system results in a sudden qualitative change in behaviour.

Local bifurcations

Local bifurcations occur at non-hyperbolic fixed points and change the number of fixed points.¹ The codimension refers to the number of parameters.

• 1D
  • Saddle-node bifurcation ˙𝑥 =𝜇 −𝑥2
  • Transcritical bifurcation ˙𝑥 =𝜇𝑥 −𝑥2
  • Pitchfork bifurcation ˙𝑥 =𝜇𝑥 −𝑥3
• 2D
  • Hopf bifurcation ˙𝑟 =𝛼𝑟 −𝑟3 and ˙𝜔 =1

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