Definition:Bifurcation/Hopf
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Definition
A Hopf bifurcation is a bifurcation in which a family of flows $\map {x_\lambda} t$, indexed by a real bifurcation parameter $\lambda$, has an attractor consisting of:
- a fixed point replaced by a circle
- a repelling fixed point for a small change in the index.
Example Diagram
This is a diagram illustrating the effect of a Hopf bifurcation:
On the left is a depiction of the flow lines before the bifurcation.
On the right is a depiction of the flow lines after the bifurcation.
In both cases the flow lines are in blue and the attractor is in red.
Also see
- Results about Hopf bifurcations can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bifurcation
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hopf bifurcation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bifurcation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hopf bifurcation