Manifold Learning Approach for Chaos in the Dripping Faucet

Dripping water from a faucet is a typical example exhibiting rich nonlinear phenomena. For such a system, the time stamps at which water drops separate from the faucet can be directly observed in real experiments, and the time series of intervals \tau_n between drop separations becomes a subject of analysis. Even if the mass m_n of a drop at the onset of the n-th separation, which cannot be observed directly, exhibits perfectly deterministic dynamics, it sometimes fails to obtain important information from time series of \tau_n. This is because the return plot \tau_n-1 vs. \tau_n may become a multi-valued function, i.e., not a deterministic dynamical system. In this paper, we propose a method to construct a nonlinear coordinate which provides a "surrogate" of the internal state m_n from the time series of \tau_n. Here, a key of the proposed approach is to use ISOMAP, which is a well-known method of manifold learning. We first apply it to the time series of $\tau_n$ generated from the numerical simulation of a phenomenological mass-spring model for the dripping faucet system. It is shown that a clear one-dimensional map is obtained by the proposed approach, whose characteristic quantities such as the Lyapunov exponent, the topological entropy, and the time correlation function coincide with the original dripping faucet system. Furthermore, we also analyze data obtained from real dripping faucet experiments which also provides promising results.Comment: 9 pages, 10 figure

Detecting Generalized Synchronization Between Chaotic Signals: A Kernel-based Approach

A unified framework for analyzing generalized synchronization in coupled chaotic systems from data is proposed. The key of the proposed approach is the use of the kernel methods recently developed in the field of machine learning. Several successful applications are presented, which show the capability of the kernel-based approach for detecting generalized synchronization. It is also shown that the dynamical change of the coupling coefficient between two chaotic systems can be captured by the proposed approach.Comment: 20 pages, 15 figures. massively revised as a full paper; issues on the choice of parameters by cross validation, tests by surrogated data, etc. are added as well as additional examples and figure

Chaotic Phase Synchronization in Bursting-neuron Models Driven by a Weak Periodic Force

We investigate the entrainment of a neuron model exhibiting a chaotic spiking-bursting behavior in response to a weak periodic force. This model exhibits two types of oscillations with different characteristic time scales, namely, long and short time scales. Several types of phase synchronization are observed, such as 1 : 1 phase locking between a single spike and one period of the force and 1 : l phase locking between the period of slow oscillation underlying bursts and l periods of the force. Moreover, spiking-bursting oscillations with chaotic firing patterns can be synchronized with the periodic force. Such a type of phase synchronization is detected from the position of a set of points on a unit circle, which is determined by the phase of the periodic force at each spiking time. We show that this detection method is effective for a system with multiple time scales. Owing to the existence of both the short and the long time scales, two characteristic phenomena are found around the transition point to chaotic phase synchronization. One phenomenon shows that the average time interval between successive phase slips exhibits a power-law scaling against the driving force strength and that the scaling exponent has an unsmooth dependence on the changes in the driving force strength. The other phenomenon shows that Kuramoto's order parameter before the transition exhibits stepwise behavior as a function of the driving force strength, contrary to the smooth transition in a model with a single time scale

リドルド ベイスン ト オン オフ カンケツセイ ノ スケーリング コウゾウ

京都大学0048新制・課程博士博士(情報学)甲第9217号情博第45号新制||情||11(附属図書館)UT51-2001-R766京都大学大学院情報学研究科複雑系科学専攻(主査)教授 藤坂 博一, 教授 木上 淳, 教授 船越 満明学位規則第4条第1項該当Doctor of InformaticsKyoto UniversityDA

Pulse Dynamics in a Model of Coupled Excitable Fibers : A Variety of Patterns and Spatio-temporal Chaos

この論文は国立情報学研究所の電子図書館事業により電子化されました。研究会報

Non-Markov-Type Analysis and Diffusion Map Analysis for Molecular Dynamics Trajectory of Chignolin at a High Temperature

We apply the non-Markov-type analysis of state-to-state transitions to nearly microsecond molecular dynamics (MD) simulation data at a folding temperature of a small artificial protein, chignolin, and we found that the time scales obtained are consistent with our previous result using the weighted ensemble simulations, which is a general path-sampling method to extract the kinetic properties of molecules. Previously, we also applied diffusion map (DM) analysis, which is one of a manifold of learning techniques, to the same trajectory of chignolin in order to cluster the conformational states and found that DM and relaxation mode analysis give similar results for the eigenvectors. In this paper, we divide the same trajectory into shorter pieces and further apply DM to such short-length trajectories to investigate how the obtained eigenvectors are useful to characterize the conformational change of chignolin