Stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations

Summary. We prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation by a (periodic) boundary value approach. This equation arises, e.g., in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations. We obtain convergence results for the standard collocation algorithm and for two variants. In particular, estimates of the difference between the collocation solution and the true solution are derived. For the standard collocation scheme the convergence results are “unconditional”, that is, they do not require mesh-ratio restrictions. Numerical results that support the theoretical findings are also given. Received June 9, 2000 / Revised version received December 14, 2000 / Published online October 17, 2001     

一类非线性系统的周期扰动Hopf分支

研究了小周期扰动对一类存在Hopf分支的非线性系统的影响.特别是应用平均法讨论了扰动频率与Hopf分支固有频率在共振及二阶次调和共振的情形周期解分支的存在性.表明了在某些参数区域内,系统存在调和解分支和次调和解分支,并进一步讨论了二阶次调和分支周期解的稳定性.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

Time periodic solutions of compressible Navier-Stokes equations

In this paper, we study the Navier-Stokes equations with a time periodic external force in Rn. We show that a time periodic solution exists when the space dimension n?5 under some smallness assumption. The main idea is to combine the energy method and the spectral analysis for the optimal decay estimates on the linearized solution operator. With the optimal decay estimates, we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem. In addition, we also study the time asymptotic stability of the time periodic solution.     

Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations

In this paper, we investigate a classical periodic Lotka–Volterra competing system with impulsive perturbations. The conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are given by applying Floquet theory of linear periodic impulsive equation, and we also give the conditions for the global stability of these solutions as a consequence of some abstract monotone iterative schemes introduced in this paper, which will be also used to get some sufficient conditions for persistence. By using the method of coincidence degree, the conditions for the existence of at least one strictly positive (componentwise) periodic solution are derived. The theoretical results are confirmed by a specific example and numerical simulations. It shows that the dynamic behaviors of the system we consider are quite different from the corresponding system without pulses.     

Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross–Pitaevskii Equation

We rigorously analyze the bifurcation of stationary so-called nonlinear Bloch waves (NLBs) from the spectrum in the Gross–Pitaevskii (GP) equation with a periodic potential, in arbitrary space dimensions. These are solutions which can be expressed as finite sums of quasiperiodic functions and which in a formal asymptotic expansion are obtained from solutions of the so-called algebraic coupled mode equations. Here we justify this expansion by proving the existence of NLBs and estimating the error of the formal asymptotics. The analysis is illustrated by numerical bifurcation diagrams, mostly in 2D. In addition, we illustrate some relations of NLBs to other classes of solutions of the GP equation, in particular to so-called out-of-gap solitons and truncated NLBs, and present some numerical experiments concerning the stability of these solutions.     

Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay

In this paper, a modified Holling-Tanner predator-prey model with time delay is considered. By regarding the delay as the bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. Meanwhile, we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values. In particular, we study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.     

Bifurcation analysis of a delayed epidemic model

In this paper, Hopf bifurcation for a delayed SIS epidemic model with stage structure and nonlinear incidence rate is investigated. Through theoretical analysis, we show the positive equilibrium stability and the conditions that Hopf bifurcation occurs. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. In addition, we also study the effect of the inhibition effect on the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.     

Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method

In this paper, we present a reliable algorithm to study the known model of nonlinear dispersive waves proposed by Boussinesq. The modified algorithm of Adomian decomposition method is used with an emphasis on the single soliton solution. New exact periodic solutions and polynomial solutions are obtained. The results of numerical examples are presented and only few terms are required to obtain accurate solutions.     

Bifurcation analysis of a population model and the resulting SIS epidemic model with delay

This paper deals with the model for matured population growth proposed in Cooke et al. [Interaction of matiration delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999) 332–352] and the resulting SIS epidemic model. The dynamics of these two models are still largely undetermined, and in this paper, we perform some bifurcation analysis to the models. By applying the global bifurcation theory for functional differential equations, we are able to show that the population model allows multiple periodic solutions. For the SIS model, we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution.     

Computation of periodic solutions in maximal monotone dynamical systems with guaranteed consistency

In this paper, we study a class of set-valued dynamical systems that satisfy maximal monotonicity properties. This class includes linear relay systems, linear complementarity systems, and linear mechanical systems with dry friction under some conditions. We discuss two numerical schemes based on time-stepping methods for the computation of the periodic solutions when these systems are periodically excited. We provide formal mathematical justifications for the numerical schemes in the sense of consistency, which means that the continuous-time interpolations of the numerical solutions converge to the continuous-time periodic solution when the discretization step vanishes. The two time-stepping methods are applied for the computation of the periodic solution exhibited by a power electronic converter and the corresponding methods are compared in terms of approximation accuracy and computation time.     

Linearized stability in periodic functional differential equations with state-dependent delays

In this paper, we study stability of periodic solutions of a class of nonlinear functional differential equations (FDEs) with state-dependent delays using the method of linearization. We show that a periodic solution of the nonlinear FDE is exponentially stable, if the zero solution of an associated linear periodic linear homogeneous FDE is exponentially stable.     

四阶非线性周期系统周期解的存在唯一性

本文利用Liapunov函数方法,研究了一类具有缓变系数的四阶非线性非自治周期系统的周期解的存在唯一性及其渐近稳定性.我们得到了保证这些系统存在唯一稳定周期解的充分条件,并对系数的缓变范围作了较为精确的估计.     

具有缓变系数非线性周期系统周期解的存在唯一性

本文利用Liapunov函数的方法,讨论了一类具有缓变系数的非线性非自治周期系统的周期解的存在唯一性,得到了保证系统存在唯一稳定周期解的充分条件,并对系统的缓变范围作了精确的估计.     

Periodic solutions for a weakly dissipated hybrid system

We consider the motion of a stretched string coupled with a rigid body at one end and we study the existence of periodic solution when a periodic force f acts on the body. The main difficulty of the study is related to the weak dissipation that characterizes this hybrid system, which does not ensure a uniform decay rate of the energy. Under additional regularity conditions on f, we use a perturbation argument in order to prove the existence of a periodic solution. In the last part of the paper we present some numerical simulations based on the theoretical results.     

On numerical study of periodic solutions of a delay equation in biological models

Some results are presented of the numerical study of periodic solutions of a nonlinear equation with a delayed argument in connection with themathematical models having real biological prototypes. The problem is formulated as a boundary value problem for a delay equation with the conditions of periodicity and transversality. A spline-collocation finite-difference scheme of the boundary value problem using a Hermitian interpolation cubic spline of the class C ¹ with fourth order error is proposed. For the numerical study of the system of nonlinear equations of the finitedifference scheme, the parameter continuation method is used, which allows us to identify possible nonuniqueness of the solution of the boundary value problem and, hence, the nonuniqueness of periodic solutions regardless of their stability. By examples it is shown that the periodic oscillations occur for the parameter values specific to the real molecular-genetic systems of higher species, for which the principle of delay is quite easy to implement.     

Linear stability and Hopf bifurcation analysis for exponential RED algorithm with heterogeneous delays

In this paper, the exponential RED algorithm with heterogeneous delays is considered. Local stability of the equilibrium solution of this algorithm is investigated based on analyzing the corresponding transcendental characteristic equation. Some general stability criteria involving the delays and the system parameters are derived by using generalized Nyquist criteria. In particular, using one of the delays as the bifurcation parameter, when the delays exceed a critical value, the exponential RED system undergoes a supercritical Hopf bifurcation. The explicit formulas determining the stability and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying Hassard et al’s approaches. Finally, some numerical simulations are performed to verify the theoretical results.     

Analytical solutions of the boundary layer flow of power-law fluid over a power-law stretching surface

This article discusses analytical solutions for a nonlinear problem arising in the boundary layer flow of power-law fluid over a power-law stretching surface. Using perturbation method analytical solution is presented for linear stretching surface. This solution covers large range of shear thinning and shear thickening fluids and matches excellently with the numerical solution. Furthermore, some new exact solutions are found for particular combination of m (power-law stretching index) and n (power-law fluid index). This leads to generalize the case of linear stretching to nonlinear stretching surface. The effects of fluid index n on the boundary layer thickness and the skin friction for nonlinear stretching surface is analyzed and discussed. It is observed that the boundary layer thickness and the skin friction coefficient increase as non-linear parameter n decreases. This study gives a new dimension to obtain analytical solutions asymptotically for highly nonlinear problems which to the best of our knowledge has not been examined so far.     

一类非线性微分方程周期解的研究

本文研究了一类最普遍的四阶非线性非自治系统的周期解的存在唯一性与渐近稳定性。我们采用了类比缓交系数的方法,作出了相应的Liapunov函数,对缓交系数作了较为精确的估计,得到了存在唯一渐近稳定的周期解的充分条件。     

Analysis of competitive chemostat models with the Beddington–DeAngelis functional response and impulsive effect

In this paper, we introduce and study a competitive system with Beddington–DeAngelis type functional response in periodic pulsed chemostat conditions. We investigate the subsystem with substrate and one of the microorganisms and study the stability of the periodic solutions, which are the boundary periodic solution of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate and one of the microorganism. Further, we prove that the system is permanent if the impulsive period less than some critical value. Therefore, our results are valuable for the manufacture of products by genetically altered organisms.