Kei-ichi Hirose

Integrable Duffing’s maps and solutions of the Duffing equation

Abstract In the numerical integration of nonlinear differential equations, discretization of the nonlinear terms poses extra ambiguity in reducing the differential equation to a discrete difference equation. As for the cubic nonlinear Schrodinger equation, it was well known that there exists the corresponding discrete soliton system. Here, representing the discrete systems by the mappings, we explore structure of the integrable mappings. We introduce the first kind and the second kind of Duffing’s map, and investigate temporal evolution of the orbits. Although the smooth periodic orbits are in accord with the solutions of the Duffing equation, the integrable Duffing’s maps provide much wider variety of orbits.

Sequence of global period doubling bifurcation in the Hénon maps

Abstract Though the onset of instability is determined by a local and linear property of the system, a region of bifurcated area in the phase space expands gradually when the control parameter increases over the critical threshold. Relative prevalence of the bifurcated orbit is measured in terms of the relative ratio of the area covered by the bifurcated daughter orbit to that of the mother orbit. Power law scaling relation of the growth process of the period doubling bifurcation have been elucidated for the sequence up to the third generation of the period doubling bifurcation of the periodic orbit in the Henon maps.

Global periodic structure of integrable Duffing’s maps

Abstract Integrable Duffing’s mappings have been constructed on the basis of a scheme presented by Suris. The Poincare–Birkhoff resonance condition determines the periodic behavior of the orbit close to the fixed points of the integrable mappings. In the region far apart from the fixed points, the nonlinear effect modifies the property of the periodic orbits. Here, the global behavior of periodic orbits of the integrable Duffing’s maps is investigated by applying the Fourier analysis on the individual orbits.

Global periodic structure of integrable hyperbolic map

Abstract Integrable hyperbolic mappings are constructed within a scheme presented by Suris. The Cosh map is a singular map, of which fixed point is unstable. The global behavior of periodic orbits of the Sinh map is investigated referring to the Poincare–Birkhoff resonance condition. Close to the fixed point, the periodicity is indeed determined from the Poincare–Birkhoff resonance condition. Increasing the distance from the fixed point, the orbit is affected by the nonlinear effect and the average periodicity varies globally. The Fourier transformation of the individual orbits determines overall spectrum of global variation of the periodicity.

Symmetry structure of the hyperbolic bifurcation without reflection of periodic orbits in the standard map

Abstract For the area preserving maps, the linearized tangent map determines the stability of the fixed point. When the trace of the tangent map is less than −2, the fixed point is inversion hyperbolic, thus the subsequent points of mapping alternate across the destabilized fixed point. That is to say, the fixed point undergoes periodic doubling bifurcation. While for the trace of the tangent map is larger than +2, the fixed point undergoes the hyperbolic bifurcation without reflection. Here, the processes of the hyperbolic bifurcation without reflection in the standard map have been examined in terms of the higher order symmetry in the momentum inversion. It is shown that the higher order symmetry lines approach asymptotically to the separatrix of the hyperbolic fixed point, and the existing symmetry lines cannot determine the structure of the periodic islands born after the hyperbolic bifurcation without reflection.

Global Period-Doubling Bifurcation in the Standard Map

While there has been great effort to establish universal behavior of the sequence of period-doubling bifurcation in Hamiltonian systems with few degrees of freedom, the nature of the period-doubling bifurcation is far more complicated in two-dimensional maps. Though the onset of instability is determined by a local, linear property of the system, the area of a bifurcated region in the phase space increases gradually when the control parameter increases beyond the critical threshold. Scaling laws for the growth process of the period-doubling bifurcation are elucidated for the period-2 step-1 accelerator mode and for the fundamental fixed orbit in the standard map.