5,336 research outputs found
Weakly coupled two slow- two fast systems, folded node and mixed mode oscillationsM
We study Mixed Mode Oscillations (MMOs) in systems of two weakly coupled
slow/fast oscillators. We focus on the existence and properties of a folded
singularity called FSN II that allows the emergence of MMOs in the presence of
a suitable global return mechanism. As FSN II corresponds to a transcritical
bifurcation for a desingularized reduced system, we prove that, under certain
non-degeneracy conditions, such a transcritical bifurcation exists. We then
apply this result to the case of two coupled systems of FitzHugh- Nagumo type.
This leads to a non trivial condition on the coupling that enables the
existence of MMOs
Hopf bifurcation with non-semisimple 1:1 resonance
A generalised Hopf bifurcation, corresponding to non-semisimple double imaginary eigenvalues (case of 1:1 resonance), is analysed using a normal form approach. This bifurcation has linear codimension-3, and a centre subspace of dimension 4. The four-dimensional normal form is reduced to a three-dimensional system, which is normal to the group orbits of a phase-shift symmetry. There may exist 0, 1 or 2 small-amplitude periodic solutions. Invariant 2-tori of quasiperiodic solutions bifurcate from these periodic solutions. The authors locate one-dimensional varieties in the parameter space 1223 on which the system has four different codimension-2 singularities: a Bogdanov-Takens bifurcation a 1322 symmetric cusp, a Hopf/Hopf mode interaction without strong resonance, and a steady-state/Hopf mode interaction with eigenvalues (0, i,-i)
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