GENERALIZED TEMPLATE FOR PERIOD DOUBLINGS

Abstract

It is shown how one can derive the formula for the local crossing number with the help of the symbolic dynamics with two letters knowing that, on the extended (ξ, η)-template, any period-doubling sequences in continuous dynamical systems is characterized by the periodic block [Formula: see text] The formulas of the global crossing number and of the linking number are derived in terms of the local crossing number which can be extracted from the power spectrum of periodic orbits. The topological characterization of periodic orbits in continuous dynamical systems has come to be at hand for experimenters as well as for theorists. The formulas are applied to the situations of the numerical experiments for particular systems, and the results are investigated in connection with the concept of the irreducible and reducible templates. It is also shown how the formulas can be applicable to the forced Lorenz system, the Brusselator equation, the parametric pendulum, T(2, 5) resonant torus knot and P(7, 3, −2) pretzel knot.