On the global structure of period doubling flows
For a wide class of period doubling flows on \(R^ 3\) the global structure of the invariant manifolds and the topology of the bifurcating periodic orbits are analyzed. The global manifold structure implies constraints for the subsequent bifurcational behavior of the flows. The period doubled orbits are classified using the theory of iterated torus knots. This classification reveals an infinite number of topologically distinct period doubling flows and raises the important question as to which sequences of torus knots appear in the asymptotic limit of a period doubling cascade. If the asymptotic sequence of torus knots produced is not unique, then the self-similarity of the Poincaré map does not imply a self-similar structure for the flows. In particular, at the accumulation point for the cascade, one would not expect to observe a universal power spectrum for the flow. This possibility is of some experimental interest.
scientific article; zbMATH DE number 3956053
scientific article