Phase trapping and slipping in a forced hydrodynamically self-excited jet

2013 ◽  
Vol 735 ◽  
Author(s):  
Larry K. B. Li ◽  
Matthew P. Juniper
Keyword(s):  
Natural Frequency ◽  
Nonlinear Dynamical Systems ◽  
Phase Locking ◽  
Full Range ◽  
Laser System ◽  
Frequency Locking ◽  
Nonlinear Dynamical ◽  
Phase Dynamics ◽  
Low Dimensional ◽  
Similar Phase

AbstractIn a recent study on a coupled laser system, Thévenin et al. (Phys. Rev. Lett., vol. 107, 2011, 104101) reported the first experimental evidence of phase trapping, a partially synchronous state characterized by frequency locking without phase locking. To determine whether this state can arise in a hydrodynamic system, we reanalyse the data from our recent experiment on a periodically forced self-excited low-density jet (J. Fluid Mech., vol. 726, 2013, pp. 624–655). We find that this jet exhibits the full range of phase dynamics predicted by model oscillators with weak nonlinearity. These dynamics include (i) phase trapping between phase drifting and phase locking when the jet is forced far from its natural frequency and (ii) phase slipping during phase drifting when it is forced close to its natural frequency. This raises the possibility that similar phase dynamics can be found in other similarly self-excited flows. It also strengthens the validity of using low-dimensional nonlinear dynamical systems based on a universal amplitude equation to model such flows, many of which are of industrial importance.

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.

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).

scholarly journals Mathematical relations between measures of brain connectivity estimated from electrophysiological recordings for Gaussian distributed data

2019 ◽  
Author(s):  
Guido Nolte ◽  
Edgar Galindo-Leon ◽  
Zhenghan Li ◽  
Xun Liu ◽  
Andreas K. Engel
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Brain Connectivity ◽  
Phase Locking ◽  
Distributed Data ◽  
Phase Lag ◽  
Volume Conduction ◽  
Nonlinear Dynamical ◽  
Functional Relations ◽  
Electrophysiological Recordings ◽  
Envelope Correlation

AbstractA large variety of methods exist to estimate brain coupling in the frequency domain from electrophysiological data measured e.g. by EEG and MEG. Those data are to reasonable approximation, though certainly not perfectly, Gaussian distributed. This work is based on the well-known fact that for Gaussian distributed data, the cross-spectrum completely determines all statistical properties. In particular, for an infinite number of data, all normalized coupling measures at a given frequency are a function of complex coherency. However, it is largely unknown what the functional relations are. We here present those functional relations for six different measures: the weighted phase lag index, the phase lag index, the absolute value and imaginary part of the phase locking value (PLV), power envelope correlation, and power envelope correlation with correction for artifacts of volume conduction. With the exception of PLV, the final results are simple closed form formulas. We tested in simulations of linear and nonlinear dynamical systems and for empirical resting state EEG on sensor level to what extent a model, namely the respective function of coherency, can explain the observed couplings. For empirical data w e found that for measures of phase-phase coupling deviations from the model are in general minWor, while power envelope correlations systematically deviate from the model for all frequencies. For power envelope correlation with correction for artifacts of volume conduction the model cannot explain the observed couplings at all. We also analyzed power envelope correlation as a function of time and frequency in an event related experiment using a stroop reaction task and found significant event related deviations mostly in the alpha range.

scholarly journals Analytical CPG model driven by single-limb velocity input generates accurate temporal locomotor dynamics

2018 ◽  
Author(s):  
Sergiy Yakovenko ◽  
Anton Sobinov ◽  
Valeriya Gritsenko
Keyword(s):  
Full Range ◽  
Nonlinear Dynamical System ◽  
Underlying Mechanism ◽  
Neural Pathways ◽  
Nonlinear Dynamical ◽  
Model Driven ◽  
Body Velocity ◽  
Leaky Integration ◽  
Low Dimensional ◽  
Walking Speeds

The ability of vertebrates to generate rhythm within their spinal neural networks is essential for walking, running, and other rhythmic behaviors. The central pattern generator (CPG) network responsible for these behaviors is well-characterized with experimental and theoretical studies, and it can be formulated as a nonlinear dynamical system. The underlying mechanism responsible for locomotor behavior can be expressed as the process of leaky integration with resetting states generating appropriate phases for changing body velocity. The low-dimensional input to the CPG model generates the bilateral pattern of swing and stance modulation for each limb and is consistent with the desired limb speed as the input command. To test the minimal configuration of required parameters for this model, we reduced the system of equations representing CPG for a single limb and provided the analytical solution with two complementary methods. The analytical and empirical cycle durations were similar (R2=0.99) for the full range of walking speeds. The structure of solution is consistent with the use of limb speed as the input domain for the CPG network. Moreover, the reciprocal interaction between two leaky integration processes representing a CPG for two limbs was sufficient to capture fundamental experimental dynamics associated with the control of heading direction. This analysis provides further support for the embedded velocity or limb speed representation within spinal neural pathways involved in rhythm generation.

scholarly journals Reverse-engineering Recurrent Neural Network solutions to a hierarchical inference task for mice

2020 ◽  
Author(s):  
Rylan Schaeffer ◽  
Mikail Khona ◽  
Leenoy Meshulam ◽  
Ila Rani Fiete ◽  
Keyword(s):  
Latent Variables ◽  
Nonlinear Dynamical Systems ◽  
Dimensional Subspace ◽  
Mammalian Brain ◽  
Nonlinear Dynamical ◽  
Model Compression ◽  
Knowledge Distillation ◽  
Interpretable Model ◽  
Low Dimensional ◽  
Inference Task

AbstractWe study how recurrent neural networks (RNNs) solve a hierarchical inference task involving two latent variables and disparate timescales separated by 1-2 orders of magnitude. The task is of interest to the International Brain Laboratory, a global collaboration of experimental and theoretical neuroscientists studying how the mammalian brain generates behavior. We make four discoveries. First, RNNs learn behavior that is quantitatively similar to ideal Bayesian baselines. Second, RNNs perform inference by learning a two-dimensional subspace defining beliefs about the latent variables. Third, the geometry of RNN dynamics reflects an induced coupling between the two separate inference processes necessary to solve the task. Fourth, we perform model compression through a novel form of knowledge distillation on hidden representations – Representations and Dynamics Distillation (RADD)– to reduce the RNN dynamics to a low-dimensional, highly interpretable model. This technique promises a useful tool for interpretability of high dimensional nonlinear dynamical systems. Altogether, this work yields predictions to guide exploration and analysis of mouse neural data and circuity.

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 Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems

2014 ◽  
Vol 470 (2166) ◽  
pp. 20130754 ◽  
Author(s):  
D. Venturi ◽  
G. E. Karniadakis
Keyword(s):  
Dynamical Systems ◽  
Stochastic Analysis ◽  
Evolution Equations ◽  
Nonlinear Dynamical Systems ◽  
Coarse Graining ◽  
Perturbation Series ◽  
Stochastic Nonlinear Systems ◽  
Nonlinear Dynamical ◽  
Major Interest ◽  
Low Dimensional

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima–Zwanzig–Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection–reaction problems.

NONLINEAR TIME-SERIES ANALYSIS OF THE GREEK EXCHANGE-RATE MARKET

2000 ◽  
Vol 10 (07) ◽  
pp. 1729-1758 ◽  
Author(s):  
A. S. ANDREOU ◽  
G. PAVLIDES ◽  
A. KARYTINOS
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Deterministic Chaos ◽  
Data Series ◽  
Nonlinear Dynamical ◽  
Series Analysis ◽  
Low Dimensional ◽  
Bds Test

Using concepts from the theory of chaos and nonlinear dynamical systems, a time-series analysis is performed on four major currencies against the Greek Drachma. The R/S analysis provided evidence for fractality due to noisy chaos in only two of the data series, while the BDS test showed that all four systems exhibit nonlinearity. Correlation dimension and related tests, as well as Lyapunov exponents, gave consistent results, which did not rule out the possibility of deterministic chaos for the two possibly fractal series, rejecting though the occurrence of a simple low-dimensional attractor, while the other two series seemed to have followed a behavior close to that of a random signal. SVD analysis, used to filter away noise, strongly supported the above findings and provided reliable evidence for the existence of an underlying system with a limited number of degrees-of-freedom only for those series found to exhibit fractality, while it revealed a noise domination over the remaining two. These results were further confirmed through a forecasting attempt using artificial neural networks.

scholarly journals Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators

2021 ◽  
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

Abstract Although 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 $10^3\times$ in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs.

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