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Connections between saddles for the FitzHugh-Nagumo system. (English) Zbl 1090.37549

Summary: By studying the two-dimensional FitzHugh-Nagumo (F-N) dynamical system, points of Bogdanov-Takens bifurcation are detected. Two of the curves of homoclinic bifurcation emerging from these points intersect each other at a point of double breaking saddle-connection bifurcation. Numerical investigations of the bifurcation curves emerging from this point, in the parameter plane, allow us to find other types of codimension-one and -two bifurcations concerning the connections between saddles and saddle-nodes, referred to as saddle-node-saddle connection bifurcation and saddle-node-saddle with separatrix connection bifurcation, respectively. The local bifurcation diagrams corresponding to these bifurcations are presented. An analogy between the bifurcation corresponding to the point of double homoclinic bifurcation and the point of double breaking saddle connection bifurcation is presented, too.

MSC:

37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems
37N25 Dynamical systems in biology
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[1] DOI: 10.1016/S0006-3495(61)86902-6 · doi:10.1016/S0006-3495(61)86902-6
[2] Georgescu A., Dishliev, A. pp 33– (1997)
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[6] DOI: 10.1137/0518083 · Zbl 0651.58025 · doi:10.1137/0518083
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