Piecewise-linear maps with heterogeneous chaos

Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos), While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two more explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.Comment: 16 pages, 9 figure

<Poster Presentation 16>Relations between statistical values along unstable periodic orbits in differential equation systems

[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPA

非双曲力学系の不安定周期軌道解析 : エノン写像の周期軌道展開

九州大学応用力学研究所研究集会報告 No.25AO-S2 「非線形波動研究の拡がり」Reports of RIAM Symposium No.25AO-S2 The breadth and depth of nonlinear wave scienceProceedings of a symposium held at Chikushi Campus, Kyushu Universiy, Kasuga, Fukuoka, Japan, October 31 - November 2, 2013カオス力学系には一般に無限個の不安定周期軌道が埋め込まれている．しかし, 不安定周期軌道はその不安定性ゆえに数値的にも検出に困難を伴うため, これまでに十分調べられているとは言えない．特に, 双曲力学系に比べて, 非双曲力学系の研究は不十分であった. 本研究では, 安定多様体と不安定多様体の接構造の存在によって双曲性が崩れた力学系を不安定周期軌道の観点で考察する．特に, カオス軌道統計量(natural measure)を不安定周期軌道の不安定性ならびに双曲性の度合いを用いて特徴づける