Topological structure of integral manifolds and period-doubling bifurcation
The authors introduce a particular local coordinate system near a nondegenerate periodic orbit of an autonomous system, in such a way that in the new coordinates the system is described by a nonautonomous system with the following properties: (i) the dimension is one less than the dimension of the original system; (ii) the origin is an equilibrium point, corresponding to the given periodic orbit of the original system; (iii) the new system is invariant under a transformation of the form \((x,t)\mapsto (Qx,t+1)\), where Q is a constant matrix with \(Q^ 2=I\); the multiplicity of -1 as an eigenvalue of Q is equal to the number of negative real Floquet multipliers of the given periodic solution. The classical center manifold for the new system can then easily be related to the center manifold of the periodic orbit; in case -1 is a characteristic multiplier with odd multiplicity, then this center manifold has the structure of a Möbius strip. In the last section it is shown how one can easily describe period-doubling bifurcations via this formalism.
scientific article; zbMATH DE number 3994615
scientific article