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Normal form coefficients

The numerical methods used in MATCONT to compute normal form coefficients of codimension 1 bifurcations of limit cycles are discussed in [28].

The periodic normal form at the Limit Point of Cycles (LPC) bifurcation is

| (51)

where , is a real coordinate on the center manifold that is transverse to the limitcycle, , and dots denote nonautonomous -periodic -terms. For each detected LPC point the normal form coefficient is computed. If then the LPC bifurcation is nondegenerate, i.e., the cycle manifold is quadratically tangential to the hyperplane orthogonal to the parameter direction. In practice, by round-off errors is never zero, so it is better to check if the testfunction changes sign. (for the LPC degenerates to a cusp of cycles (CP) but this will not be detected on a branch of limit cycles since it is a codimension 2 phenomenon.)

The periodic normal form at the Period Doubling (PD) bifurcation is

| (52)

where , is a real coordinate on the center manifold that is transverse to the limit cycle, , and dots denote nonautonomous -periodic -terms. The coefficient determines the stability of the period doubled cycle in the center manifold and is computed during the processing of each PD point.

If then the PD bifurcation is supercritical, i.e., within the center manifold on one side of the bifurcation only stable cycles exist and on the other side unstable cycles coexist with stable period-doubled cycles. If the PD bifurcation is subcritical, i.e., within the center manifold on one side of the bifurcation stable cycles coexist with unstable period-doubled cycles and on the other side only unstable cycles exist. (for the PD degenerates to Generalized Period Doubling bifurcation (GPD) but this will not be detected on a branch of limit cycles since this is a codimension 2 phenomenon).

The periodic normal form at the Neimark-Sacker (NS) bifurcation is

| (53)

where , is a complex coordinate on the center manifold that is complementary to , , and dots denote nonautonomous -periodic -terms. The critical coefficient in the periodic normal form for the NS bifurcation is computed during the processing of a NS point.

If then the bifurcation is supercritical, i.e. in the center manifold the limit cycle is stable on one side of the bifurcation point and unstable on the other side and the unstable cycles coexist with stable invariant tori. If then the bifurcation is subcritical, i.e. in the center manifold the limit cycle is stable on one side of the bifurcation point and unstable on the other side and the stable cycles coexist with unstable invariant tori. (for the NS bifurcation degenerates to a Chenciner bifurcation (CH) but this will not be detected on a branch of limit cycles since it is a codimension 2 phenomenon).