Hiroaki Daido

Analytical Conditions for the Appearance of Homoclinic and Heteroclinic Points of a 2-dimensional Mapping The Case of the Hénon Mapping

A method is proposed to obtain analytical conditions for the appearance of homoclinic and heteroclinic points of a class of 2-dimensional mappings with a small parameter. It consists of solving a functional equation, which is satisfied by invariant curves, perturbatively and combining the results with simple geometrical consideration. The mapping proposed by Henan is treated here, for which it is also found that the appearance of heteroclinic points results in the disappearance of a strange attractor. The results are in good agreement with those of numerical simulations.

Theory of the period-doubling phenomenon of one-dimensional mappings based on the parameter dependence

Abstract A theory is presented of the period-doubling phenomenon of one-dimensional mappings of the form xn+1 = F(xn, r), which is different from that of Feigenbaum mainly in that it is based on the r dependence of various quantities rather than on their x dependence. Consequently, it enables us to evaluate, for example, the Lyapunov numbers of periodic orbits as a function of r as well as the Feigenbaum ratio. It is shown that the results of our theory are in good agreement with those of numerical simulations.

Universal relation of a band-splitting sequence to a preceding period-doubling one

Abstract Based on a theory for mappings of the form x n +1 = F ( x n , r ) developed previously by the author, we explain the numerically observed fact that band-splitting transition points r k convergence to r c at the same rate as period-doubling bifurcation points r k , where r c ≡ lim r k , and moreover show that θ ≡ ( r k −r c ) (r c −r k ) is universal. In addition, it is shown that the theory gives a sufficiently good value of θ. The behavior of the Lyapunov number at r k is also discussed in its relation to r k .

Thermal Fluctuation of a Self-Oscillating Reaction System under a Periodic External Force. II: Quasi-Periodic Region

A reductive perturbation scheme is proposed for describing the quasi-periodic behavior exhibited by a self-oscillating reaction system under a weak periodic force. Using this scheme and the system size expansion the behavior of thermal fluctuation is investigated. As a result it is found that variance of thermal fluctuation has a component making a secular increase with time. This phenomenon, similar to the ensemble dephasing or phase diffusion of a free limit cycle, is interpreted as ensemble dephasing in a stroboscopic phase space, i.e., stroboscopic ensemble dephasing.