Fast Computation of Birkhoff Average along a Quasi-periodic Orbit and its Applications
| Project/Area Number |
17K05360
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Research Field |
Foundations of mathematics/Applied mathematics
|
|---|
| Research Institution | Hitotsubashi University |
|---|
Principal Investigator |
SAIKI Yoshitaka 一橋大学, 大学院経営管理研究科, 教授 (20433740)
|
|---|
| Project Period (FY) |
2017-04-01 – 2021-03-31
|
| Project Status |
Completed (Fiscal Year 2020)
|
| Budget Amount *help |
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
|
|---|
| Keywords | 力学系 / 準周期軌道 / バーコフ平均 / 高速計算 / 重み付きバーコフ平均 / ヘテロカオス / 時間遅れ埋め込み / 回転数 / ジーゲル板 / ジーゲル球 / 位相共役 / フーリエ級数 / 凖周期軌道 / カオス / 数値解析 / 数値解析学 / モンテカルロ法 |
|---|
| Outline of Final Research Achievements |
The Birkhoff Ergodic Theorem concludes that time averages, that is, Birkhoff averages of a function f along an ergodic trajectory of a function T converges to the space average. Convergence of the time average to the space average is slow. We introduce a modified average by giving very small weights to the "end" terms. When (x_n) is a trajectory on a quasiperiodic torus and f and T are infinitely differentiable, we show that our weighted Birkhoff averages converge "super" fast, i.e. with error smaller than every polynomial of 1/N. Our goal is to show that our weighted Birkhoff average is a powerful computational tool, and this study illustrates its use for several examples where the quasiperiodic set is one or two dimensional. In particular, we compute rotation numbers and conjugacies (i.e. changes of variables) and their Fourier series, often with 30-digit precision.
|
| Academic Significance and Societal Importance of the Research Achievements |
回転数、リアプノフ数などをはじめとしてバーコフ平均は力学系の軌道に関するさまざまな量に関わっている。軌道長Nのバーコフ平均の収束スピードは一般に1/Nのオーダーであり実際に計算で準周期性の判断をすることは困難であった。しかし、研究代表者らは、準周期軌道上のバーコフ平均に対しては、理論的には1/(Nに関する任意の多項式)よりも速く収束する重み付きバーコフ平均を提案してその応用可能性を示した。
|
Report
(5 results)
Research Products
(27 results)
