Analysis Of Ordinary Differential Equations Research Articles

A practical way to apply a technique that accelerates time history analysis of structures under digitised excitations

Time history analysis using direct time integration is a versatile and widely accepted tool for analysing the dynamic behaviour of structures. In 2008, a technique was proposed to accelerate thetime history analysis of structural systems subjected to digitised excitations. Recently, this techniquehas been named as the SEB THAAT* (Step-Enlargement-Based Time-History-Analysis-Acceleration-Technique), and the determination of appropriate values for its parameter is introduced as the mainchallenge. To overcome this challenge, a procedure is proposed in this paper. The basis of the procedure is the comments on accuracy control in structural dynamics and numerical analysis ofordinary differential equations, legalised in the New Zealand Seismic Code, NZS 1170.5:2004. As themain achievement, by using the proposed procedure, we can apply the SEB THAAT and carry out thetime history analysis clearly and with less parameter setting compared to the ordinary time historyanalysis. The proposed procedure is always applicable and, except when the behaviour is very complex, oscillatory and non-linear, the reductions in analysis run-time are considerable while the changes in accuracy are negligible. The performance can be sensitive to the problem, the integration method, the target response, and the severity of the non-linear behaviour. Compared to the previous tests on the SEB THAAT, the efficiency of applying the SEB THAAT using the proposed procedure is better, the sensitivity of the performance to the problem is lower, and a measure of accuracy is available. Compared to other techniques for accelerating structural dynamic analyses, the use of the SEB THAAT according to the proposed procedure has several positive points, including the simplicity of implementation.

Analytic one-dimensional maps and two-dimensional ordinary differential equations can robustly simulate Turing machines

In this paper, we analyze the problem of finding the minimum dimension n such that an analytic map/ordinary differential equation over R n can simulate a Turing machine in a way that is robust to perturbations. We show that one-dimensional analytic maps are sufficient to robustly simulate Turing machines; but the minimum dimension for the analytic ordinary differential equations to robustly simulate Turing machines is two, under some reasonable assumptions. We also show that any Turing machine can be simulated by a two-dimensional C ∞ ordinary differential equation on the compact sphere S 2 .

Stability analysis of sampled-data ODE–Schrödinger systems and its applications to output-feedback stabilization

In this paper, we are concerned with a sampled-data cascades system, which consists of finite-dimensional dynamics described by ordinary differential equation (ODE) and infinite-dimensional dynamics represented by a Schrödinger equation, based on a zero-order-holder (ZOH) measurement signal. The Schrödinger equation includes reaction and convection terms, and allows more general boundary conditions. Then, a small-gain approach is used to analyze the exponential stability of this hybrid system. This stability analysis is shown to be very useful in the design of sampled-data exponentially convergent observers and exponentially stable output-feedback controllers for some classes of ODE–Schrödinger cascaded system.

Mathematical Investigations of a Kinetic Model for Glycerol Hydrogenolysis Via Heterogeneous Catalysis

In this paper we report on some mathematical investigations of the chemical process for the hydrogenolysis of glycerol over a heterogeneous metal catalyst. The main interest of this process is related to the fact that glycerol is produced as a by-product in the production of biodiesel in huge amounts that are expected to exceed the projected demands. This makes the sustainability of biodiesel production depend on the conversion of the glycerol into useful products hence it is a desirable goal to have effective conversion methods. A reaction model from literature is used to derive a system of ordinary differential equations (ODE) which is then analysed using methods from qualitative analysis of ODEs. Numerical solutions of the system are simulated to try and find out the solution’s behaviour in the chemistry point of view. It was found that all solutions of the model converge to some stable limit point in a 2-dimensional plane in the positive cone of the phase space, and the limit point depends on the values of rate constants as well as on the hydrogen to glycerol initial ratios. Even though the results are based on a specific kinetic model, it is hoped that they may help in providing tools for better understanding and description of the reaction.

Learning Dynamic Systems Using Gaussian Process Regression with Analytic Ordinary Differential Equations as Prior Information

Recently the data-driven learning of dynamic systems has become a promising approach because no physical knowledge is needed. Pure machine learning approaches such as Gaussian process regression (GPR) learns a dynamic model from data, with all physical knowledge about the system discarded. This goes from one extreme, namely methods based on optimizing parametric physical models derived from physical laws, to the other. GPR has high flexibility and is able to model any dynamics as long as they are locally smooth, but can not generalize well to unexplored areas with little or no training data. The analytic physical model derived under assumptions is an abstract approximation of the true system, but has global generalization ability. Hence the optimal learning strategy is to combine GPR with the analytic physical model. This paper proposes a method to learn dynamic systems using GPR with analytic ordinary differential equations (ODEs) as prior information. The one-time-step integration of analytic ODEs is used as the mean function of the Gaussian process prior. The total parameters to be trained include physical parameters of analytic ODEs and parameters of GPR. A novel method is proposed to simultaneously learn all parameters, which is realized by the fully Bayesian GPR and more promising to learn an optimal model. The standard Gaussian process regression, the ODE method and the existing method in the literature are chosen as baselines to verify the benefit of the proposed method. The predictive performance is evaluated by both one-time-step prediction and long-term prediction. By simulation of the cart-pole system, it is demonstrated that the proposed method has better predictive performances.

Invariant ideals of local analytic and formal vector fields

The objective of this paper is to analyze analytic invariant sets of analytic ordinary differential equations (ODEs). For this purpose we introduce semi-invariants and invariant ideals as well as the notion of vector fields in Poincaré-Dulac normal form (PDNF). We prove that all invariant ideals of a vector field in PDNF are already invariant for its semi-simple linear part. Additionally, this paper provides a natural characterization of invariant ideals via semi-invariants.

On the asymptotic analysis of bounded solutions to nonlinear differential equations of second order

In this paper, we consider two different models of nonlinear ordinary differential equations (ODEs) of second order. We construct two new Lyapunov functions to investigate boundedness of solutions of those nonlinear ODEs of second order. By using the Lyapunov direct or second method and inequality techniques, we prove two new theorems on the boundedness solutions of those ODEs of second order as t to infty . When we compare the conditions of the theorems of this paper with those of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), we can see that our theorems have less restrictive conditions than those in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002) because of the two new suitable Lyapunov functions. Next, in spite of the use of the Lyapunov second method here and in (Meng in J. Syst. Sci. Math. Sci. 15(1):50–57, 1995; Sun and Meng in Ann. Differ. Equ. 18(1):58–64 2002), the proofs of the results of this paper are proceeded in a very different way from that used in the literature for the qualitative analysis of ODEs of second order. Two examples are given to show the applicability of our results. At the end, we can conclude that the results of this paper generalize and improve the results of Meng in (J. Syst. Sci. Math. Sci. 15(1):50–57, 1995), Sun and Meng in (Ann. Differ. Equ. 18(1):58–64 2002), and some other that can be found in the literature, and they have less restrictive conditions than those in these references.

Delayed loss of stability due to the slow passage through Hopf bifurcations in reaction-diffusion equations.

This article presents the delayed loss of stability due to slow passage through Hopf bifurcations in reaction-diffusion equations with slowly-varying parameters, generalizing a well-known result about delayed Hopf bifurcations in analytic ordinary differential equations to spatially-extended systems. We focus on the Hodgkin-Huxley partial differential equation (PDE), the cubic Complex Ginzburg-Landau PDE as an equation in its own right, the Brusselator PDE, and a spatially-extended model of a pituitary clonal cell line. Solutions which are attracted to quasi-stationary states (QSS) sufficiently before the Hopf bifurcations remain near the QSS for long times after the states have become repelling, resulting in a significant delay in the loss of stability and the onset of oscillations. Moreover, the oscillations have large amplitude at onset, and may be spatially homogeneous or inhomogeneous. Space-time boundaries are identified that act as buffer curves beyond which solutions cannot remain near the repelling QSS, and hence before which the delayed onset of oscillations must occur, irrespective of initial conditions. In addition, a method is developed to derive the asymptotic formulas for the buffer curves, and the asymptotics agree well with the numerically observed onset in the Complex Ginzburg-Landau (CGL) equation. We also find that the first-onset sites act as a novel pulse generation mechanism for spatio-temporal oscillations.

A method for the Computational Frequency Sweep Analysis of Nonlinear ODEs using GPU Acceleration

ABSTRACTComputational sweep analysis of nonlinear ODEs (ordinary differential equations) is of importance in engineering system analysis and design. Sweep analyses usually demand intense computational power according to the number of points and the number of system parameters. This paper presents an efficient parallel algorithm for the sweep analysis of nonlinear ODEs based on graphical processing unit acceleration. The developed method preserves the jump phenomenon characteristics intrinsic to nonlinear ODEs and reduces the effects of irregular computational load. Experiments were realized using Duffing equation by sweeping frequency, amplitude, and equation coefficients. Directly, data parallel implementation and proposed implementations are compared to show the efficiency of the proposed method. Experimental results show that the new method provides significant reductions in the computational durations when compared to sequential implementation.

Exponential Stability Analysis of Sampled-Data ODE–PDE Systems and Application to Observer Design

A small-gain approach is presented for analyzing exponential stability of a class of (dynamical) hybrid systems. The systems considered in the paper are composed of finite-dimensional dynamics, represented by a linear ordinary differential equation (ODE), and infinite-dimensional dynamics described by a parabolic partial differential equation (PDE). Exponential stability is established under conditions involving the maximum allowable sampling period (MASP). This new stability result is shown to be useful in the design of sampled-output exponentially convergent observers for linear systems that are described by an ODE–PDE cascade. The new stability result also proves to be useful in designing practical approximate observers involving no PDEs.

Weierstrass traveling wave solutions for dissipative Benjamin, Bona, and Mahony (BBM) equation

In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified Benjamin, Bona, and Mahony (BBM) equation by viscosity. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant makes the equation integrable in terms of Weierstrass ℘ functions. We will use a general formalism based on Ince's transformation to write the general solution of dissipative BBM in terms of ℘ functions, from which all the other known solutions can be obtained via simplifying assumptions. Using ODE (ordinary differential equations) analysis we show that the traveling wave speed is a bifurcation parameter that makes transition between different classes of waves.

First integrals of local analytic differential systems

We investigate formal and analytic first integrals of local analytic ordinary differential equations near a stationary point. A natural approach is via the Poincaré–Dulac normal forms: If there exists a formal first integral for a system in normal form then it is also a first integral for the semisimple part of the linearization, which may be seen as “conserved” by the normal form. We discuss the maximal setting in which all such first integrals are conserved, and show that all first integrals are conserved for certain classes of reversible systems. Moreover we investigate the case of linearization with zero eigenvalues, and we consider a three-dimensional generalization of the quadratic Dulac–Frommer center problem.

Asymptotic Analysis of Proportionally Fair Scheduling in Rayleigh Fading

This paper is concerned with the analysis of proportionally fair scheduling (PFS), and we provide an analytical approximation for the PFS throughput over Rayleigh fading channels. Though quite accurate, the ordinary differential equation (ODE) analysis, typically used to analyze the PFS throughput, is highly time-consuming when there are lots of users. On the other hand, due to the intricate interplay among these ODE equations, the ODE analysis generally fails to provide a closed-form approximation for estimating the PFS throughput unless with simplified models such as the linear rate model to characterize channel capacity. Our aim is to provide a novel framework to evaluate PFS in Rayleigh fading without the above-mentioned limitations. To put our work on a firm base, we use results of stochastic approximation in the analysis and take the Gaussian approximation for capacity modeling for fading channels. Simulations validate this approach and show that our analytic result provides highly accurate estimate of the PFS throughput. Compared to existing studies, our work advances the state of the art in three ways. First, it goes beyond the linear rate model and applies to the commonly used Shannon rate model. Second, it provides accurate estimate of the PFS throughput without the need for the time-consuming ODE analysis. Third, it provides a unified closed-form expression for estimating the PFS throughput for both the linear rate model and the Shannon rate model. It is interesting to note that our analysis provides the same result as existing studies when assuming the linear rate model. More importantly, our formula is intuitive yet easy to evaluate numerically.

Quasi-analytic solutions of analytic ordinary differential equations and o-minimal structures

It is well known that the non-spiraling leaves of real analytic foliations of codimension 1 all belong to the same o-minimal structure. Naturally, the question arises if the same statement is true for non-oscillating trajectories of real analytic vector fields. We show, under certain assumptions, that such a trajectory generates an o-minimal and model complete structure together with the analytic functions. The proof uses the asymptotic theory of irregular singular ordinary differential equations in order to establish a quasi-analyticity result from which the main theorem follows. As applications, we present an infinite family of o-minimal structures such that any two of them do not admit a common extension, and we construct a non-oscillating trajectory of a real analytic vector field in dimension 5 that is not definable in any o-minimal extension of the reals

Exponential averaging for Hamiltonian evolution equations

We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.

Mathematical simulation and analysis of cellular metabolism and regulation.

A better understanding of the biological phenomena observed in cells requires the creation and analysis of mathematical models of cellular metabolism and physiology. The formulation and study of such models must also be simplified as far as possible to cope with the increasing complexity demanded and exponential accumulation of the metabolic reconstructions computed from sequenced genomes. A mathematical simulation workbench, DBsolve, has been developed to simplify the derivation and analysis of mathematical models. It combines: (i) derivation of large-scale mathematical models from metabolic reconstructions and other data sources; (ii) solving and parameter continuation of non-linear algebraic equations (NAEs), including metabolic control analysis; (iii) solving the non-linear stiff systems of ordinary differential equations (ODEs); (iv) bifurcation analysis of ODEs; (v) parameter fitting to experimental data or functional criteria based on constrained optimization. The workbench has been successfully used for dynamic metabolic modeling of some typical biochemical networks (Dolgacheva et al., Biochemistry (Moscow), 6, 1063-1068, 1996; Goldstein and Goryanin, Mol. Biol. (Moscow), 30, 976-983, 1996), including microbial glycolytic pathways, signal transduction pathways and receptor-ligand interactions. DBsolve 5. 00 is freely available from http://websites.ntl.com/ approximately igor.goryanin. gzz78923@ggr.co.uk

Investigation of the Foias–Saut Normalization in the Finite-Dimensional Case

We study the normalization obtained by C. Foias and J. C. Saut for the Navier–Stokes equations in the classical case of analytic ordinary differential equations in the neighborhood of a stationary point. We show that in this general (finite-dimensional) case, this normalization coincides with the distinguished normalization in the sense of A. D. Brjuno. Moreover, their approach leads to a spectral formula for the (inverse of the) distinguished normalizing map. In the particular case of an asymptotically stable, by linearization, stationary point, the Foias–Saut device gives the inverse of the already known link between the Lyapunov exponential expansion and normalization.

An algorithm of multisummation of formal power series, solutions of linear ODE equations

Abstract In the neighbourhood of irregular singularities the formal power series solutions of complex analytic ordinary differential equations are generally divergent. The resummation theory developed by J. Ecalle and J.P. Ramis enables us to sum these series by multisummation effective algorithms. In particular we illustrate the computation of solutions of linear differential equations using the principle of multiple summation by iterated Laplace integrals proposed by W. Balser.

On the adaptive lattice notch filter for the detection of sinusoids

A lattice notch filter for the detection and enchancement of sinusoids in noise is presented, and the convergence properties of the algorithm are investigated using ODE (ordinary differential equation) analysis. It is shown that the frequency estimate of the algorithm for a single sinusoid is unbiased, regardless of the pole-zero contraction factor and the noise variance. An expression for the MSE (mean square error) of the frequency estimate and the output SNR of the algorithm is derived. Simulation results are presented to verify the theoretical analysis. Based on cascading the proposed second-order adaptive notch filter, three different approaches for the detection of multiple sinusoids are presented. The first approach uses a linear cascade structure, and the second and third approaches use a triangular cascade structure. The performance of the algorithms for multiple sinusoids is investigated, and computer simulation results are presented and discussed. >

Newton polygons and gevrey indices for linear partial differential operators

This paper is a continuation of Miyake [7] by the first named author. We shall study the unique solvability of an integro-differential equation in the category of formal or convergent power series with Gevrey estimate for the coefficients, and our results give some analogue in partial differential equations to Ramis [10, 11] in ordinary differential equations.In the study of analytic ordinary differential equations, the notion of irregularity was first introduced by Malgrange [3] as a difference of indices of a differential operator in the categories of formal power series and convergent power series. After that, Ramis extended his theory to the category of formal or convergent power series with Gevrey estimate for the coefficients. In these studies, Ramis revealed a significant meaning of a Newton polygon associated with a differential operator.