scholarly journals Data-driven spectral analysis for coordinative structures in periodic systems with unknown and redundant dynamics

2019 ◽  
Author(s):  
Keisuke Fujii ◽  
Naoya Takeishi ◽  
Benio Kibushi ◽  
Motoki Kouzaki ◽  
Yoshinobu Kawahara
Keyword(s):  
Spectral Analysis ◽  
Nonlinear Dynamical Systems ◽  
Data Driven ◽  
Periodic Systems ◽  
Limit Cycle Oscillation ◽  
Control Mechanisms ◽  
Nonlinear Dynamical ◽  
Reduction Methods ◽  
Dynamical Properties ◽  
Low Dimensional

AbstractLiving organisms dynamically and flexibly operate a great number of components. As one of such redundant control mechanisms, low-dimensional coordinative structures among multiple components have been investigated. However, structures extracted from the conventional statistical dimensionality reduction methods do not reflect dynamical properties in principle. Here we regard coordinative structures in biological periodic systems with unknown and redundant dynamics as a nonlinear limit-cycle oscillation, and apply a data-driven operator-theoretic spectral analysis, which obtains dynamical properties of coordinative structures such as frequency and phase from the estimated eigenvalues and eigenfunctions of a composition operator. First, from intersegmental angles during human walking, we extracted the speed-independent harmonics of gait frequency. Second, we discovered the speed-dependent time-evolving behaviors of the phase on the conventional low-dimensional structures by estimating the eigenfunctions. Our approach contributes to the understanding of biological periodic phenomena with unknown and redundant dynamics from the perspective of nonlinear dynamical systems.

scholarly journals Data-driven spectral analysis for coordinative structures in periodic human locomotion

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Keisuke Fujii ◽  
Naoya Takeishi ◽  
Benio Kibushi ◽  
Motoki Kouzaki ◽  
Yoshinobu Kawahara
Keyword(s):  
Spectral Analysis ◽  
Nonlinear Dynamical Systems ◽  
Model Simulation ◽  
Human Locomotion ◽  
Data Driven ◽  
Limit Cycle Oscillation ◽  
Double Pendulum ◽  
Control Mechanisms ◽  
Dynamical Properties ◽  
Low Dimensional

AbstractLiving organisms dynamically and flexibly operate a great number of components. As one of such redundant control mechanisms, low-dimensional coordinative structures among multiple components have been investigated. However, structures extracted from the conventional statistical dimensionality reduction methods do not reflect dynamical properties in principle. Here we regard coordinative structures in biological periodic systems with unknown and redundant dynamics as a nonlinear limit-cycle oscillation, and apply a data-driven operator-theoretic spectral analysis, which obtains dynamical properties of coordinative structures such as frequency and phase from the estimated eigenvalues and eigenfunctions of a composition operator. Using segmental angle series during human walking as an example, we first extracted the coordinative structures based on dynamics; e.g. the speed-independent coordinative structures in the harmonics of gait frequency. Second, we discovered the speed-dependent time-evolving behaviours of the phase by estimating the eigenfunctions via our approach on the conventional low-dimensional structures. We also verified our approach using the double pendulum and walking model simulation data. Our results of locomotion analysis suggest that our approach can be useful to analyse biological periodic phenomena from the perspective of nonlinear dynamical systems.

scholarly journals Non-intrusive nonlinear model reduction via machine learning approximations to low-dimensional operators

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Zhe Bai ◽  
Liqian Peng
Keyword(s):  
Machine Learning ◽  
Nonlinear Dynamical Systems ◽  
Reduced Order Models ◽  
Simulation Code ◽  
Nonlinear Dynamical ◽  
Reduced Order ◽  
Nonlinear Model Reduction ◽  
Regression Techniques ◽  
Low Dimensional ◽  
Modern Machine

AbstractAlthough projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to $$10^3{\times }$$ 10 3 × in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs. The domain of applications include both parabolic and hyperbolic PDEs, regardless of the dimension of full-order models (FOMs).

Generating Fully Bounded Chaotic Attractors

2013 ◽  
pp. 148-153
Author(s):  
Zeraoulia Elhadj
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Dynamical ◽  
Dynamical Properties ◽  
The Given

Generating chaotic attractors from nonlinear dynamical systems is quite important because of their applicability in sciences and engineering. This paper considers a class of 2-D mappings displaying fully bounded chaotic attractors for all bifurcation parameters. It describes in detail the dynamical behavior of this map, along with some other dynamical phenomena. Also presented are some phase portraits and some dynamical properties of the given simple family of 2-D discrete mappings.

scholarly journals Multidimensional Approximation of Nonlinear Dynamical Systems

2019 ◽  
Vol 14 (6) ◽  
Author(s):  
Patrick Gelß ◽  
Stefan Klus ◽  
Jens Eisert ◽  
Christof Schütte
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Measurement Data ◽  
Data Driven ◽  
High Dimensional ◽  
Data Sets ◽  
Nonlinear Dynamical ◽  
Tensor Network ◽  
Wide Range ◽  
Multidimensional Approximation

A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems.

A data-driven approach to model calibration for nonlinear dynamical systems

2019 ◽  
Vol 125 (24) ◽  
pp. 244901 ◽  
Author(s):  
C. M. Greve ◽  
K. Hara ◽  
R. S. Martin ◽  
D. Q. Eckhardt ◽  
J. W. Koo
Keyword(s):  
Dynamical Systems ◽  
Model Calibration ◽  
Nonlinear Dynamical Systems ◽  
Data Driven ◽  
Nonlinear Dynamical ◽  
Data Driven Approach

Generating Fully Bounded Chaotic Attractors

2011 ◽  
Vol 2 (3) ◽  
pp. 36-42
Author(s):  
Zeraoulia Elhadj
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Dynamical ◽  
Dynamical Properties ◽  
The Given

Generating chaotic attractors from nonlinear dynamical systems is quite important because of their applicability in sciences and engineering. This paper considers a class of 2-D mappings displaying fully bounded chaotic attractors for all bifurcation parameters. It describes in detail the dynamical behavior of this map, along with some other dynamical phenomena. Also presented are some phase portraits and some dynamical properties of the given simple family of 2-D discrete mappings.

Lock-in and quasiperiodicity in a forced hydrodynamically self-excited jet

2013 ◽  
Vol 726 ◽  
pp. 624-655 ◽  
Author(s):  
Larry K. B. Li ◽  
Matthew P. Juniper
Keyword(s):  
Natural Frequency ◽  
Nonlinear Dynamical Systems ◽  
Three Dimensional ◽  
Van Der Pol Oscillator ◽  
Natural Mode ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Frequency Pulling ◽  
Low Dimensional ◽  
Lock In

AbstractThe ability of hydrodynamically self-excited jets to lock into strong external forcing is well known. Their dynamics before lock-in and the specific bifurcations through which they lock in, however, are less well known. In this experimental study, we acoustically force a low-density jet around its natural global frequency. We examine its response leading up to lock-in and compare this to that of a forced van der Pol oscillator. We find that, when forced at increasing amplitudes, the jet undergoes a sequence of two nonlinear transitions: (i) from periodicity to ${ \mathbb{T} }^{2} $ quasiperiodicity via a torus-birth bifurcation; and then (ii) from ${ \mathbb{T} }^{2} $ quasiperiodicity to 1:1 lock-in via either a saddle-node bifurcation with frequency pulling, if the forcing and natural frequencies are close together, or a torus-death bifurcation without frequency pulling, but with a gradual suppression of the natural mode, if the two frequencies are far apart. We also find that the jet locks in most readily when forced close to its natural frequency, but that the details contain two asymmetries: the jet (i) locks in more readily and (ii) oscillates more strongly when it is forced below its natural frequency than when it is forced above it. Except for the second asymmetry, all of these transitions, bifurcations and dynamics are accurately reproduced by the forced van der Pol oscillator. This shows that this complex (infinite-dimensional) forced self-excited jet can be modelled reasonably well as a simple (three-dimensional) forced self-excited oscillator. This result adds to the growing evidence that open self-excited flows behave essentially like low-dimensional nonlinear dynamical systems. It also strengthens the universality of such flows, raising the possibility that more of them, including some industrially relevant flames, can be similarly modelled.

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