What is supercritical Hopf bifurcation?
Andrew Mclaughlin
Updated on April 25, 2026
A Hopf Bifurcation occurs when a periodic solution or limit cycle, surrounding an equilibrium point, arises or goes away as a parameter µ varies. When a stable limit cycle surrounds an unstable equilibrium point, the bifurcation is called a supercritical Hopf bifurcation.
What is Hamiltonian Hopf bifurcation?
1. Introduction. A celebrated bifurcation problem in Hamiltonian dynamics involves the loss of linear stability of a fixed point through the collision of two pairs of imaginary eigenvalues of the linearised flow, and their subsequent departure off the imaginary axis.
Who discovered the Hopf bifurcation?
Yuri A. Kuznetsov
Yuri A. Kuznetsov (2006), Scholarpedia, 1(10):1858. Figure 1: Supercritical Andronov-Hopf bifurcation in the plane.
What do you mean by Hopf bifurcation?
The term Hopf bifurcation (also sometimes called Poincaré-Andronov-Hopf bifurcation) refers to the local birth or death of a periodic solution (self-excited oscillation) from an equilibrium as a parameter crosses a critical value.
What do you mean by the Hopf bifurcation for a two dimensional dynamical system?
Hopf bifurcation occurs when a pair of complex conjugate eigenvalues of the Jacobian matrix of the dynamical system crosses the imaginary axis with a non-zero speed (all the other eigenvalues being in the left half plane) as the control parameter is varied slowly.
What does Hopf stand for?
HOPF
| Acronym | Definition |
|---|---|
| HOPF | Home Office Police Force (UK) |
What is neimark Sacker bifurcation?
Neimark-Sacker bifurcation is the birth of a closed invariant curve from a fixed point in dynamical systems with discrete time (iterated maps), when the fixed point changes stability via a pair of complex eigenvalues with unit modulus.
What does Hopf mean?
What are hophopf bifurcations?
Hopf bifurcations occur in the Lotka–Volterra model of predator–prey interaction (known as paradox of enrichment ), the Hodgkin–Huxley model for nerve membrane potential, the Selkov model of glycolysis, the Belousov–Zhabotinsky reaction, the Lorenz attractor, the Brusselator and Classical electromagnetism.
What is Hopf bifurcation analysis in railway vehicles?
In railway vehicle systems, Hopf bifurcation analysis is notably important. Conventionally a railway vehicle’s stable motion at low speeds crosses over to unstable at high speeds.
What is a complex conjugate bifurcation?
More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues —of the linearization around the fixed point—crosses the complex plane imaginary axis.
