arXiv subjects

Yoshitaka Saiki

Publications and source records attributed to Yoshitaka Saiki.

27 records · Page 2Linked to original sources

arXiv2019

Machine-learning construction of a model for a macroscopic fluid variable using the delay-coordinate of a scalar observable

We construct a data-driven dynamical system model for a macroscopic variable the Reynolds number of a high-dimensionally chaotic fluid flow by training its scalar time-series data. We use a machine-learning approach, the reservoir computing for the construction of the model, and do not use the knowledge of a physical process of fluid dynamics in its procedure. It is confirmed that an inferred time-series obtained from the model approximates the actual one in each of various time-intervals, and that some characteristics of the chaotic invariant set mimic the actual ones. We investigate the appropriate choice of the delay-coordinate, especially the delay-time and the dimension, which enables us to construct a model having a relatively high-dimensional attractor easily.

physics.flu-dyn↗

arXiv2018

Quasiperiodic orbits in Siegel disks/balls and the Babylonian problem

We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasiperiodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this "linearization" (or conjugacy) from knowledge of a single quasiperiodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian Problem: determining the characteristics of a quasiperiodic set from a trajectory. Our computation of rotation rates and Fourier coefficients depends on the very high speed of our computational method "the weighted Birkhoff average".

math.DS↗

arXiv2018

Machine-learning inference of fluid variables from data using reservoir computing

We infer both microscopic and macroscopic behaviors of a three-dimensional chaotic fluid flow using reservoir computing. In our procedure of the inference, we assume no prior knowledge of a physical process of a fluid flow except that its behavior is complex but deterministic. We present two ways of inference of the complex behavior; the first called partial-inference requires continued knowledge of partial time-series data during the inference as well as past time-series data, while the second called full-inference requires only past time-series data as training data. For the first case, we are able to infer long-time motion of microscopic fluid variables. For the second case, we show that the reservoir dynamics constructed from only past data of energy functions can infer the future behavior of energy functions and reproduce the energy spectrum. It is also shown that we can infer a time-series data from only one measurement by using the delay coordinates. These implies that the obtained two reservoir systems constructed without the knowledge of microscopic data are equivalent to the dynamical systems describing macroscopic behavior of energy functions.

physics.comp-ph↗

arXiv2018

Low-dimensional paradigms for high-dimensional hetero-chaos

The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some regions than others. Some regions of a chaotic attractor can be expanding in more dimensions than other regions. Imagine a situation where two such regions and each contains trajectories that stay in the region for all time while typical trajectories wander throughout the attractor. Such an attractor is "hetero-chaotic" (i.e. it has heterogeneous chaos) if furthermore arbitrarily close to each point of the attractor there are points on periodic orbits that have different unstable dimensions. This is hard to picture but we believe that most physical systems possessing a high-dimensional attractor are of this type. We have created simplified models with that behavior to give insight to real high-dimensional phenomena.

nlin.CD↗

arXiv2018

The continuous route to multi-chaos

For low-dimensional chaotic attractors there is usually a single number of unstable dimensions for all of its periodic orbits and we can say such attractors exhibit "mono-chaos". In high-dimensional chaotic attractors, trajectories are prone to travel through quite different regions of phase space, some far more unstable than others. This heterogeneity makes predictability even more difficult than in low-dimensional homogeneous chaotic attractors. A chaotic attractor is "multi-chaotic" if every point of the attractor is arbitrarily close to periodic points with different numbers of unstable dimensions. We believe that most physical systems possessing a high-dimensional attractor are of this type. We make three conjectures about multi-chaos which we explore using three two-dimensional paradigmatic examples of multi-chaotic attractors. They can be thought of as small-scale examples that give insight for real high-dimensional phenomena. We find a single route from mono-chaos to multi-chaos if an attractor changes continuously as a parameter is varied. This multi-chaos bifurcation (MCB) is a periodic orbit bifurcation; one branch of periodic orbits is created with a number of unstable dimensions that is different from the mono-chaos.

nlin.CD↗

arXiv2017

Quantitative Quasiperiodicity

The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, $Σ_{n=0}^{N-1} f(x_n)/N$ of a function $f$ along a length $N$ ergodic trajectory $(x_n)$ of a function $T$ converge to the space average $\int f dμ$, where $μ$ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We introduce a modified average of $f(x_n)$ by giving very small weights to the "end" terms when $n$ is near $0$ or $N-1$. When $(x_n)$ is a trajectory on a quasiperiodic torus and $f$ and $T$ are $C^\infty$, we show that our weighted Birkhoff averages converge 'super" fast to $\int f dμ$ with respect to the number of iterates $N$, i.e. with error decaying faster than $N^{-m}$ for every integer $m$. Our goal is to show that our weighted Birkhoff average is a powerful computational tool, and this paper 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 accuracy.

math.DS↗

arXiv2017

Solving the Babylonian Problem of quasiperiodic rotation rates

A trajectory $u_n := F^n(u_0), n = 0,1,2, \dots $ is quasiperiodic if the trajectory lies on and is dense in some $d$-dimensional torus, and there is a choice of coordinates on the torus $\mathbb{T}$ for which $F$ has the form $F(θ) = θ+ ρ\bmod1$ for all $θ\in\mathbb{T}$ and for some $ρ\in\mathbb{T}$. There is an ancient literature on computing three rotation rates $ρ$ for the Moon. %There is a literature on determining the coordinates of the vector $ρ$, called the rotation rates of $F$. (For $d>1$ we always interpret $\bmod1$ as being applied to each coordinate.) However, even in the case $d=1$ there has been no general method for computing $ρ$ given only the trajectory $u_n$, though there is a literature dealing with special cases. Here we present our Embedding Continuation Method for computing some components of $ρ$ from a trajectory. It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. Rotation rates are often called "rotation numbers" and both refer to a rate of rotation of a circle. However, the coordinates of $ρ$ depend on the choice of coordinates of $\mathbb{T}$. We explore the various sets of possible rotation rates that $ρ$ can yield. We illustrate our ideas with examples in dimensions $d=1$ and $2$.

math.DS↗

arXiv2016

Measuring quasiperiodicity

The Birkhoff Ergodic Theorem asserts under mild conditions that Birkhoff averages (i.e. time averages computed along a trajectory) converge to the space average. For sufficiently smooth systems, our small modification of numerical Birkhoff averages significantly speeds the convergence rate for quasiperiodic trajectories -- by a factor of $10^{25}$ for 30-digit precision arithmetic, making it a useful computational tool for autonomous dynamical systems. Many dynamical systems and especially Hamiltonian systems are a complex mix of chaotic and quasiperiodic behaviors, and chaotic trajectories near quasiperiodic points can have long near-quasiperiodic transients. Our method can help determine which initial points are in a quasiperiodic set and which are chaotic. We use our {\bf weighted Birkhoff average} to study quasiperiodic systems, to distinguishing between chaos and quasiperiodicity, and for computing rotation numbers for self-intersecting curves in the plane. Furthermore we introduce the Embedding Continuation Method which is a significantly simpler, general method for computing rotation numbers.

math-ph↗

arXiv2015

Quantitative Quasiperiodicity

The Birkhoff Ergodic Theorem concludes that time averages, that is, Birkhoff averages, $Σ_{n=1}^N f(x_n)/N$ of a function $f$ along an ergodic trajectory $(x_n)$ of a function $T$ converges to the space average $\int f dμ$, where $μ$ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We introduce a modified average of $f(x_n)$ by giving very small weights to the "end" terms when $n$ is near $0$ or $N$. When $(x_n)$ is a trajectory on a quasiperiodic torus and $f$ and $T$ are $C^\infty$, we show that our weighted Birkhoff averages converge "super" fast to $\int f dμ$, {\em 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 paper 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.

math.DS↗