scholarly journals Statistical analysis of nonlinear dynamical systems using differential geometric sampling methods

2011 ◽  
Vol 1 (6) ◽  
pp. 821-835 ◽  
Author(s):  
Ben Calderhead ◽  
Mark Girolami
Keyword(s):  
Statistical Analysis ◽  
Nonlinear Dynamical Systems ◽  
Probability Distributions ◽  
Nonlinear Differential Equations ◽  
Cell Signalling ◽  
Probabilistic Approach ◽  
Nonlinear Ordinary Differential Equation ◽  
Model Parameters ◽  
Nonlinear Dynamical ◽  
Biological Phenomena

Mechanistic models based on systems of nonlinear differential equations can help provide a quantitative understanding of complex physical or biological phenomena. The use of such models to describe nonlinear interactions in molecular biology has a long history; however, it is only recently that advances in computing have allowed these models to be set within a statistical framework, further increasing their usefulness and binding modelling and experimental approaches more tightly together. A probabilistic approach to modelling allows us to quantify uncertainty in both the model parameters and the model predictions, as well as in the model hypotheses themselves. In this paper, the Bayesian approach to statistical inference is adopted and we examine the significant challenges that arise when performing inference over nonlinear ordinary differential equation models describing cell signalling pathways and enzymatic circadian control; in particular, we address the difficulties arising owing to strong nonlinear correlation structures, high dimensionality and non-identifiability of parameters. We demonstrate how recently introduced differential geometric Markov chain Monte Carlo methodology alleviates many of these issues by making proposals based on local sensitivity information, which ultimately allows us to perform effective statistical analysis. Along the way, we highlight the deep link between the sensitivity analysis of such dynamic system models and the underlying Riemannian geometry of the induced posterior probability distributions.

Updating Nonlinear Dynamical Models Using Response Measurements Only

2003 ◽  
Author(s):  
Ka-Veng Yuen ◽  
James L. Beck
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Probabilistic Approach ◽  
Simulated Data ◽  
Model Parameters ◽  
Response Signal ◽  
Dynamical Models ◽  
Nonlinear Dynamical ◽  
Uncertain Input ◽  
Measured Response

A spectral density approach for the identification of linear systems is extended to nonlinear dynamical systems using only incomplete noisy response measurements. A stochastic model is used for the uncertain input and a Bayesian probabilistic approach is used to update the uncertainties in the model parameters. The proposed spectral-based approach utilizes important statistical properties of the Fast Fourier Transform and their robustness with respect to the probability distribution of the response signal in order to calculate the updated probability density function for the parameters of a nonlinear model conditional on the measured response. This probabilistic approach is well suited for the identification of nonlinear systems and does not require huge amounts of dynamic data. The formulation is presented directly for multiple-degree-of freedom systems. Examples using simulated data for a Duffing oscillator and a four-DOF inelastic structure are presented to illustrate the proposed approach.

Determining Upconversion Parameters and Invertibility of Nonlinear Dynamical Systems

2015 ◽  
Vol 1753 ◽  
Author(s):  
Makhin Thitsa ◽  
Thanh Q. Ta
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Energy Levels ◽  
Classical System ◽  
Rate Equations ◽  
Unknown Parameters ◽  
Emission Data ◽  
Nonlinear Dynamical ◽  
Material Development ◽  
System Inversion

ABSTRACTDetermining upconversion parameters is of high interest in laser material development. For many materials these parameters cannot be directly measured by experimental methods. These upconversion coefficients appear as unknown parameters in the laser rate equations, which are a system of coupled nonlinear differential equations that are used to model the dynamics of population densities in different energy levels. In this paper we propose the well-established system theoretic tools pertaining to the system inversion to be applied in this case. The unknown parameters can be considered as the inputs and the fluorescence signals can be considered as the outputs of the dynamical system. Therefore the determination of the unknown upconversion rates in the system equations from the output data is a classical system inversion problem. In this paper we demonstrate how to compute the unknown coefficients in the rate equations from the experimental emission data utilizing this method.

Mapping the dynamics of a bursting neuron

1993 ◽  
Vol 341 (1298) ◽  
pp. 345-359 ◽  
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Model Neuron ◽  
Neural Model ◽  
Model Parameters ◽  
Nonlinear Dynamical ◽  
Bursting Neuron ◽  
Channel Blocking ◽  
Real Neuron ◽  
Intrinsic Firing

The anterior burster (AB) neuron of the lobster stom atogastric ganglion displays varied rhythmic behavior when treated with neuromodulators and channel-blocking toxins. We introduce a channelbased model for this neuron and show how bifurcation analysis can be used to investigate the response of this model to changes of its parameters. Two dimensional maps of the parameter space of the model were constructed using com putational tools based on the theory of nonlinear dynamical systems. Changes in the intrinsic firing and oscillatory properties of the model AB neuron were correlated with the boundaries of Hopf and saddle-node bifurcations on these maps. Complex rhythmic patterns were observed, with a bounded region of the parameter plane producing bursting behavior of the model neuron. Experiments were performed by treating an isolated AB cell with 4-am inopyridine which selectively reduces gλ, the conductance of the transient potassium channel. The model accurately predicts the qualitative changes in the neuronal voltage oscillations that are observed over a range of reduction of gλ in the neuron. These results dem onstrate the efficacy of dynamical systems theory as a means of determ ining the varied oscillatory behaviors inherent in a channel-based neural model. Further, the maps of bifurcations provide a useful tool for determining how these behaviors depend upon model parameters and comparing the model to a real neuron.

scholarly journals Bayesian differential programming for robust systems identification under uncertainty

2020 ◽  
Vol 476 (2243) ◽  
pp. 20200290
Author(s):  
Yibo Yang ◽  
Mohamed Aziz Bhouri ◽  
Paris Perdikaris
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Monte Carlo Sampling ◽  
Model Parameters ◽  
Nonlinear Dynamical ◽  
Predator Prey ◽  
Learning Framework ◽  
Numerical Studies ◽  
Recent Developments ◽  
Systems Identification ◽  
Robust Systems

This paper presents a machine learning framework for Bayesian systems identification from noisy, sparse and irregular observations of nonlinear dynamical systems. The proposed method takes advantage of recent developments in differentiable programming to propagate gradient information through ordinary differential equation solvers and perform Bayesian inference with respect to unknown model parameters using Hamiltonian Monte Carlo sampling. This allows an efficient inference of the posterior distributions over plausible models with quantified uncertainty, while the use of sparsity-promoting priors enables the discovery of interpretable and parsimonious representations for the underlying latent dynamics. A series of numerical studies is presented to demonstrate the effectiveness of the proposed methods, including nonlinear oscillators, predator–prey systems and examples from systems biology. Taken together, our findings put forth a flexible and robust workflow for data-driven model discovery under uncertainty. All codes and data accompanying this article are available at https://bit.ly/34FOJMj .

scholarly journals Stability of The Equilibrium of Nonlinear Dynamical Systems

2018 ◽  
Vol 71 (1) ◽  
pp. 71-80
Author(s):  
Irada A. Dzhalladova ◽  
Miroslava Růžičková
Keyword(s):  
Lyapunov Functions ◽  
Lyapunov Method ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Fractional Part ◽  
Stability Domain ◽  
Nonlinear Dynamical ◽  
Quadratic Lyapunov Functions ◽  
The Stability ◽  
Zero Equilibrium

Abstract The algorithm for estimating the stability domain of zero equilibrium to the system of nonlinear differential equations with a quadratic part and a fractional part is proposed in the article. The second Lyapunov method with quadratic Lyapunov functions is used as a method for studying such systems.

scholarly journals Physics-informed Spline Learning for Nonlinear Dynamics Discovery

2021 ◽  
Author(s):  
Fangzheng Sun ◽  
Yang Liu ◽  
Hao Sun
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Climate Science ◽  
Analytical Form ◽  
Optimization Strategy ◽  
Governing Equations ◽  
Nonlinear Dynamical ◽  
High Level

Dynamical systems are typically governed by a set of linear/nonlinear differential equations. Distilling the analytical form of these equations from very limited data remains intractable in many disciplines such as physics, biology, climate science, engineering and social science. To address this fundamental challenge, we propose a novel Physics-informed Spline Learning (PiSL) framework to discover parsimonious governing equations for nonlinear dynamics, based on sparsely sampled noisy data. The key concept is to (1) leverage splines to interpolate locally the dynamics, perform analytical differentiation and build the library of candidate terms, (2) employ sparse representation of the governing equations, and (3) use the physics residual in turn to inform the spline learning. The synergy between splines and discovered underlying physics leads to the robust capacity of dealing with high-level data scarcity and noise. A hybrid sparsity-promoting alternating direction optimization strategy is developed for systematically pruning the sparse coefficients that form the structure and explicit expression of the governing equations. The efficacy and superiority of the proposed method have been demonstrated by multiple well-known nonlinear dynamical systems, in comparison with two state-of-the-art methods.

A New Approach to Stable Optimal Control of Complex Nonlinear Dynamical Systems

2013 ◽  
Vol 81 (3) ◽  
Author(s):  
Firdaus E. Udwadia
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Nonlinear Dynamical System ◽  
Rate Of Change ◽  
Definite Function ◽  
Constrained Motion ◽  
Nonlinear Controller ◽  
Control Cost ◽  
Nonlinear Dynamical ◽  
Negative Definite Function

This paper gives a simple approach to designing a controller that minimizes a user-specified control cost for a mechanical system while ensuring that the control is stable. For a user-given Lyapunov function, the method ensures that its time rate of change is negative and equals a user specified negative definite function. Thus a closed-form, optimal, nonlinear controller is obtained that minimizes a desired control cost at each instant of time and is guaranteed to be Lyapunov stable. The complete nonlinear dynamical system is handled with no approximations/linearizations, and no a priori structure is imposed on the nature of the controller. The methodology is developed here for systems modeled by second-order, nonautonomous, nonlinear, differential equations. The approach relies on some recent fundamental results in analytical dynamics and uses ideas from the theory of constrained motion.

scholarly journals Interconnections between various analytic approaches applicable to third-order nonlinear differential equations

2015 ◽  
Vol 471 (2176) ◽  
pp. 20140720 ◽  
Author(s):  
R. Mohanasubha ◽  
V. K. Chandrasekar ◽  
M. Senthilvelan ◽  
M. Lakshmanan
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Nonlinear Differential Equations ◽  
Lie Point Symmetries ◽  
Third Order ◽  
Nonlinear Odes ◽  
Darboux Polynomials ◽  
Nonlinear Dynamical ◽  
Starting Point ◽  
The Family ◽  
Integrating Factors

We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ODEs. We establish an important interconnection between the extended Prelle–Singer procedure and λ-symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjoint-symmetries, λ-symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.

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