Yoshitaka Saiki

Reconstruction of chaotic saddles by classification of unstable periodic orbits: Kuramoto-Sivashinsky equation.

The unstable periodic orbits (UPOs) embedded in a chaotic attractor after an attractor merging crisis (MC) are classified into three subsets, and employed to reconstruct chaotic saddles in the Kuramoto-Sivashinsky equation. It is shown that in the post-MC regime, the two chaotic saddles evolved from the two coexisting chaotic attractors before crisis can be reconstructed from the UPOs embedded in the pre-MC chaotic attractors. The reconstruction also involves the detection of the mediating UPO responsible for the crisis, and the UPOs created after crisis that fill the gap regions of the chaotic saddles. We show that the gap UPOs originate from saddle-node, period-doubling, and pitchfork bifurcations inside the periodic windows in the post-MC chaotic region of the bifurcation diagram. The chaotic attractor in the post-MC regime is found to be the closure of gap UPOs.

RECOGNITION OF TRANSITION PATTERNS IN A BUSINESS CYCLE MODEL USING UNSTABLE PERIODIC ORBITS

More than a thousand unstable periodic orbits with relatively low periods are identified from a continuous dynamical system which represents business cycles of two countries with different fiscal policies. Patterns of regimes and regime transitions are clearly recognized by using the detected periodic orbits. Both regimes and regime transitions are essential structures in chaotic business fluctuations generated by interactive behavior among workers, entrepreneurs and governments.

Network analysis of chaotic systems through unstable periodic orbits

A chaotic motion can be considered an irregular transition process near unstable periodic orbits embedded densely in a chaotic set. Therefore, unstable periodic orbits have been used to characterize properties of chaos. Statistical quantities of chaos such as natural measures and fractal dimensions can be determined in terms of unstable periodic orbits. Unstable periodic orbits that can provide good approximations to averaged quantities of chaos or turbulence are also known to exist. However, it is not clear what type of unstable periodic orbits can capture them. In this paper, a model for an irregular transition process of a chaotic motion among unstable periodic orbits as nodes is constructed by using a network. We show that unstable periodic orbits which have lots of links in the network tend to capture time averaged properties of chaos. A scale-free property of the degree distribution is also observed.

Quasiperiodicity: Rotation Numbers

A map on a torus is called “quasiperiodic” if there is a change of variables which converts it into a pure rotation in each coordinate of the torus. We develop a numerical method for finding this change of variables, a method that can be used effectively to determine how smooth (i.e., differentiable) the change of variables is, even in cases with large nonlinearities. Our method relies on fast and accurate estimates of limits of ergodic averages. Instead of uniform averages that assign equal weights to points along the trajectory of N points, we consider averages with a non-uniform distribution of weights, weighing the early and late points of the trajectory much less than those near the midpoint N∕2. We provide a one-dimensional quasiperiodic map as an example and show that our weighted averages converge far faster than the usual rate of O(1∕N), provided f is sufficiently differentiable. We use this method to efficiently numerically compute rotation numbers, invariant densities, conjugacies of quasiperiodic systems, and to provide evidence that the changes of variables are (real) analytic.