Time Series from a Nonlinear Dynamical Systems Perspective

2017 ◽  
pp. 199-263
Author(s):  
Daniel Durstewitz
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Systems Perspective

scholarly journals Phase Synchronization Is the Amplified Result by the Hilbert Transform

2015 ◽  
Vol 2015 ◽  
pp. 1-3 ◽  
Author(s):  
Ming-Chi Lu ◽  
Hsing-Chung Ho ◽  
Chen-An Chan ◽  
Chia-Ju Liu ◽  
Jiann-Shing Lih ◽  
...  
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Hilbert Transform ◽  
Phase Synchronization ◽  
Nonlinear Dynamical Systems ◽  
The State ◽  
Nonlinear Dynamical ◽  
Large Correlation ◽  
Current Usage ◽  
The Hilbert Transform

We investigate the interplay between phase synchronization and amplitude synchronization in nonlinear dynamical systems. It is numerically found that phase synchronization intends to be established earlier than amplitude synchronization. Nevertheless, amplitude synchronization (or the state with large correlation between the amplitudes) is crucial for the maintenance of a high correlation between two time series. A breakdown of high correlation in amplitudes will lead to a desynchronization of two time series. It is shown that these unique features are caused essentially by the Hilbert transform. This leads to a deep concern and criticism on the current usage of phase synchronization.

Sparse Recovery and Dictionary Learning to Identify the Nonlinear Dynamical Systems: One Step Toward Finding Bifurcation Points in Real Systems

2019 ◽  
Vol 29 (03) ◽  
pp. 1950030 ◽  
Author(s):  
Fahimeh Nazarimehr ◽  
Aboozar Ghaffari ◽  
Sajad Jafari ◽  
Seyed Mohammad Reza Hashemi Golpayegani
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dictionary Learning ◽  
Nonlinear Dynamical Systems ◽  
Sparse Recovery ◽  
Governing Equations ◽  
Nonlinear Dynamical ◽  
Bifurcation Points ◽  
Long Time

Modeling real dynamical systems is an important challenge in many areas of science. Extracting governing equations of systems from their time-series is a possible solution for such a challenge. In this paper, we use the sparse recovery and dictionary learning to extract governing equations of a system with parametric basis functions. In this algorithm, the assumption of sparsity in the functions of dynamical equations is used. The proposed algorithm is applied to different types of discrete and continuous nonlinear dynamical systems to show the generalization ability of this method. On the other hand, transition from one dynamical regime to another is an important concept in studying real world complex systems like biological and climate systems. Lyapunov exponent is an early warning index. It can predict bifurcation points in dynamical systems. Computation of Lyapunov exponent is a major challenge in its application in real systems, since it needs long time data to be accurate. In this paper, we use the predicted governing equation to generate long time-series, which is needed for Lyapunov exponent calculation. So the proposed method can help us to predict bifurcation points by accurate calculation of Lyapunov exponents.

scholarly journals Extracting Nonlinear Dynamics from Psychological and Behavioral Time Series Through HAVOK Analysis

2020 ◽  
Author(s):  
Robert Glenn Moulder ◽  
Elena Martynova ◽  
Steven M. Boker
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Time Series Data ◽  
Nonlinear Dynamical Systems ◽  
Series Data ◽  
Alternative View ◽  
Unified Framework ◽  
Nonlinear Dynamical ◽  
Depth Analysis ◽  
Analysis Of Time Series

Analytical methods derived from nonlinear dynamical systems, complexity, and chaos theories offer researchers a framework for in-depth analysis of time series data. However, relatively few studies involving time series data obtained from psychological and behavioral research employ such methods. This paucity of application is due to a lack of general analysis frameworks for modeling time series data with strong nonlinear components. In this article, we describe the potential of Hankel alternative view of Koopman (HAVOK) analysis for solving this issue. HAVOK analysis is a unified framework for nonlinear dynamical systems analysis of time series data. By utilizing HAVOK analysis, researchers may model nonlinear time series data in a linear framework while simultaneously reconstructing attractor manifolds and obtaining a secondary time series representing the amount of nonlinear forcing occurring in a system at any given time. We begin by showing the mathematical underpinnings of HAVOK analysis and then show example applications of HAVOK analysis for modeling time series data derived from real psychological and behavioral studies.

AUTOMATIC DETECTION OF SLIGHT CHANGES IN NONLINEAR DYNAMICAL SYSTEMS USING MULTIRESOLUTION ENTROPY TOOLS

2001 ◽  
Vol 11 (04) ◽  
pp. 967-981 ◽  
Author(s):  
M. E. TORRES ◽  
M. M. AÑINO ◽  
L. G. GAMERO ◽  
M. A. GEMIGNANI
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Wavelet Analysis ◽  
Numerical Simulations ◽  
Nonlinear Dynamical Systems ◽  
Automatic Detection ◽  
Nonlinear Dynamical ◽  
Dynamical Complexity ◽  
Changes Detection ◽  
Classical Entropy

The continuous multiresolution entropy, which combines advantages stemming from both classical entropy and wavelet analysis, has shown to be sensitive to dynamical complexity changes. The addition of classical statistical changes detection tools gives rise to a new tool that allows their automatic detection. In this paper, a new tool for the automatic detection of slight parameter changes in nonlinear dynamical systems from the analysis of the corresponding time series is proposed. The relevance of the approach, together with its robustness in the presence of moderate noise, is discussed in numerical simulations and it is applied to biological signals.

Segmentation of time series from nonlinear dynamical systems

2011 ◽  
Vol 44 (1) ◽  
pp. 13209-13214 ◽  
Author(s):  
Tillmann Falck ◽  
Henrik Ohlsson ◽  
Lennart Ljung ◽  
Johan A.K. Suykens ◽  
Bart De Moor
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical

IMPROVED PARAMETER ESTIMATION FROM NOISY TIME SERIES FOR NONLINEAR DYNAMICAL SYSTEMS

2007 ◽  
Vol 17 (05) ◽  
pp. 1741-1752 ◽  
Author(s):  
TOMOMICHI NAKAMURA ◽  
YOSHITO HIRATA ◽  
KEVIN JUDD ◽  
DEVIN KILMINSTER ◽  
MICHAEL SMALL
Keyword(s):  
Time Series ◽  
Parameter Estimation ◽  
Dynamical Systems ◽  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Parameter Estimates ◽  
Nonlinear Dynamical ◽  
Observational Noise

In this paper we consider the problem of estimating the parameters of a nonlinear dynamical system given a finite time series of observations that are contaminated by observational noise. The least squares method is a standard method for parameter estimation, but for nonlinear dynamical systems it is well known that the least squares method can result in biased estimates, especially when the noise is significant relative to the nonlinearity. In this paper, it is demonstrated that by combining nonlinear noise reduction and least squares parameter fitting it is possible to obtain more accurate parameter estimates.

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