Heteroclinic bifurcation
Encyclopedia
In mathematics
, particularly dynamical systems, a heteroclinic bifurcation is a global bifurcation involving a heteroclinic cycle
. Heteroclinic bifurcations come in two types, resonance bifurcations, and transverse bifurcations. Both types of bifurcation will result in the change of stability of the heteroclinic cycle.
At a resonance bifurcation, the stability of the cycle changes when an algebraic condition on the eigenvalues of the equilibria in the cycle is satisfied. This is usually accompanied by the birth or death of a periodic orbit.
A transverse bifurcation of a heteroclinic cycle is caused when the real part of a transverse eigenvalue of one of the equilibria in the cycle passes through zero. This will also cause a change in stability of the heteroclinic cycle.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly dynamical systems, a heteroclinic bifurcation is a global bifurcation involving a heteroclinic cycle
Heteroclinic cycle
In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system. It is a topological circle of equilibrium points and connecting heteroclinic orbits...
. Heteroclinic bifurcations come in two types, resonance bifurcations, and transverse bifurcations. Both types of bifurcation will result in the change of stability of the heteroclinic cycle.
At a resonance bifurcation, the stability of the cycle changes when an algebraic condition on the eigenvalues of the equilibria in the cycle is satisfied. This is usually accompanied by the birth or death of a periodic orbit.
A transverse bifurcation of a heteroclinic cycle is caused when the real part of a transverse eigenvalue of one of the equilibria in the cycle passes through zero. This will also cause a change in stability of the heteroclinic cycle.