The Motion of Weakly Interacting Pulses in Reaction-Diffusion Systems

The interaction of stable pulse solutions on R ¹ is considered when distances between pulses are sufficiently large. We construct an attractive local invariant manifold giving the dynamics of interacting pulses in a mathematically rigorous way. The equations describing the flow on the manifold is also given in an explicit form. By it, we can easily analyze the movement of pulses such as repulsiveness, attractivity and/or the existence of bound states of pulses. Interaction of front solutions are also treated in a similar way.     

Wave equations and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method.The invariance of some sets under the flow of these problems and the vac- uum isolation of solutions are obtained by introducing a family of potential wells.Then the threshold result of global existence and nonexistence of solutions are given.Finally, the problem with critical initial conditions are discussed.     

Hopf bifurcation in general Brusselator system with diffusion

The general Brusselator system is considered under homogeneous Neumann boundary conditions. The existence results of the Hopf bifurcation to the ordinary differential equation (ODE) and partial differential equation (PDE) models are obtained. By the center manifold theory and the normal form method, the bifurcation direction and stability of periodic solutions are established. Moreover, some numerical simulations are shown to support the analytical results. At the same time, the positive steady-state solutions and spatially inhomogeneous periodic solutions are graphically shown to supplement the analytical results.     

A new model reduction method for nonlinear dynamical systems

The present research work proposes a new systematic approach to the problem of model-reduction for nonlinear dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear invariance partial differential equations (PDEs), and a rather general explicit set of conditions for solvability is derived. In particular, within the class of analytic solutions, the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution to the above system of singular PDEs is then proven to represent the slow invariant manifold of the nonlinear dynamical system under consideration exponentially attracting all dynamic trajectories. As a result, an exact reduced-order model for the nonlinear system dynamics is obtained through the restriction of the original system dynamics on the aforementioned slow manifold. The local analyticity property of the solution’s graph that corresponds to the system’s slow manifold enables the development of a series solution method, which allows the polynomial approximation of the system dynamics on the slow manifold up to the desired degree of accuracy and can be easily implemented with the aid of a symbolic software package such as MAPLE. Finally, the proposed approach and method is evaluated through an illustrative biological reactor example.     

Exponential attractors of reaction-diffusion systems in an unbounded domain

We consider reaction-diffusion systems in unbounded domains, prove the existence of expotential attractors for such systems, and estimate their fractal dimension. The essential difference with the case of a bounded domain studied before is the continuity of the spectrum of the linear part of the equations. This difficulty is overcome by systematic use of weighted Sobolev spaces.     

Sine-Gordon方程的截断系统的同宿轨道

研究Ｓｉｎｅ Ｇｏｒｄｏｎ方程的广义渐近惯性流形上的常微分方程组，证实了在一定参数条件下存在Ｗｉｇｇｉｎｓ［１］意义下的同宿轨道．计算表明，与Ｂｉｓｈｏｐ［２］用数值计算得到的Ｓｉｎｅ Ｇｏｒｄｏｎ方程产生混沌的参数值尚有差别，考虑到同宿出现参数值往往低于混沌出现参数值，故结果在定性上正确，而且改进了文［１］中的结果．     

A model-reduction method for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty

The present research work proposes a new systematic approach to the problem of model reduction for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty. The problem of interest is addressed within the context of functional equation theory, and in particular, through a system of invariance functional equations for which a general set of conditions for solvability is provided. Within the class of analytic solutions, this set of conditions guarantees the existence and uniqueness of a locally analytic solution which represents the system’s slow invariant manifold attracting all dynamic trajectories in the absence of model uncertainty. An exact reduced-order model is then obtained through the restriction of the original discrete-time system dynamics on the slow manifold. The analyticity property of the solution to the invariance functional equations enables the development of a series solution method that can be easily implemented using MAPLE leading to polynomial approximations up to the desired degree of accuracy. Furthermore, the aforementioned attractivity property and the system’s transition towards the above manifold is analyzed and characterized in the presence of model uncertainty. Finally, the proposed method is evaluated through an illustrative biological reactor example.     

Global Existence for the Vlasov–Poisson System in Bounded Domains

In this paper we prove global existence for solutions of the Vlasov–Poisson system in convex bounded domains with specular boundary conditions and with a prescribed outward electrical field at the boundary.     

Front-Like Entire Solutions for Monostable Reaction-Diffusion Systems

This paper is concerned with front-like entire solutions for monostable reaction-diffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Further, combining a SIS and traveling fronts with different wave speeds and propagation directions, the existence and various qualitative properties of entire solutions are established by using the comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish a comparison theorem for the Cauchy problems of the three systems, and then prove the existence of entire solutions via using the comparison theorem, the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems.     

Robustness of Global Attractors for Reversible Gray–Scott Systems

Global asymptotic dynamics of a typical cubic-autocatalytic reaction-diffusion system, the reversible Gray?CScott system of three components, are investigated. The upper semicontinuity (robustness) of the global attractors in the H ¹ product space for the solution semiflows with respect to the reverse reaction rates as they converge to zero is proved. Through an approach of transformative decomposition, the hurdle of the perturbed singularity between the reversible and non-reversible systems is overcome by showing the uniform dissipation, the uniformly bounded evolution of the union of global attractors, and the uniform convergence property of the bundle of reversible and non-reversible semiflows.     

On 3D Lagrangian Navier–Stokes α Model with a Class of Vorticity-Slip Boundary Conditions

This paper concerns the 3-dimensional Lagrangian Navier–Stokes α model and the limiting Navier–Stokes system on smooth bounded domains with a class of vorticity-slip boundary conditions and the Navier-slip boundary conditions. It establishes the spectrum properties and regularity estimates of the associated Stokes operators, the local well-posedness of the strong solution and global existence of weak solutions for initial boundary value problems for such systems. Furthermore, the vanishing α limit to a weak solution of the corresponding initial-boundary value problem of the Navier–Stokes system is proved and a rate of convergence is shown for the strong solution.     

Ordinary differential equations (ODEs) on inertial manifolds for reaction-diffusion systems in a singularly perturbed domain with several thin channels

Dynamics of solutions to a reaction-diffusion system in a domain of specific shape is investigated under the homogeneous Neumann boundary conditions. It is assumed that the domain hasN large regionsD _i ,i=1,...,N, and thin channelsQ _i,j () connectingD _i andD _j , which approach a line segment as 0 in some sense. In such a domain the firstN eigenvalues of – with the Neumann boundary conditions tend to zero as 0, while the (N + 1)-th eigenvalue is bounded away from zero. By virtue of this gap of the eigenvalues, an inertial manifold which is invariant and attracts every solution exponentially can be constructed under a certain condition. Moreover, the ODE describing the dynamics on the inertial manifold can be given in quite an explicit form through the analysis of the limit of the manifold as 0.     

Coagulation-Fragmentation Processes

We study the well-posedness of coagulation-fragmentation models with diffusion for large systems of particles. The continuous and the discrete case are considered simultaneously. In the discrete situation we are concerned with a countable system of coupled reaction-diffusion equations, whereas the continuous case amounts to an uncountable system of such equations. These problems can be handled by interpreting them as abstract vector-valued parabolic evolution equations, where the dependent variables take values in infinite-dimensional Banach spaces. Given suitable assumptions, we prove existence and uniqueness in the class of volume preserving solutions. We also derive sufficient conditions for global existence. Accepted: (August 18, 1999)     

Dynamical Properties of Models for the Calvin Cycle

Modelling the Calvin cycle of photosynthesis leads to various systems of ordinary differential equations and reaction-diffusion equations. They differ by the choice of chemical substances included in the model, the choices of stoichiometric coefficients and chemical kinetics and whether or not diffusion is taken into account. This paper studies the long-time behaviour of solutions of several of these systems, concentrating on the ODE case. In some examples it is shown that there exist two positive stationary solutions. In several cases it is shown that there exist solutions where the concentrations of all substrates tend to zero at late times and others (runaway solutions) where the concentrations of all substrates increase without limit. In another case, where the concentration of ATP is explicitly included, runaway solutions are ruled out.     

Global solution to the viscous compressible barotropic multipolar gas

The global existence of strong solutions of the initial boundary-value problem in bounded domains to the system of partial differential equations for viscous compressible polytropic multipolar fluids is proved. Some other properties such as uniqueness and cavitation are discussed.     

Relative Entropy and the Stability of Shocks and Contact Discontinuities for Systems of Conservation Laws with non-BV Perturbations

We develop a theory based on relative entropy to show the uniqueness and L ² stability (up to a translation) of extremal entropic Rankine?CHugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness conditions. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuums.     

Periodic Solutions of an Autonomous Reaction-Diffusion System—A Model System

We describe the dynamics of an autonomous system of two reaction-diffusion equations which can be looked at as a model system for more general reaction-diffusion systems. In our system all solutions tend to zero or to (finitely many) periodic orbits which can be fully described—including their stability properties. Furthermore, we construct invariant sets for the period map and show how a new invariant called torsion number is related to our model system.     

Diffusion-Induced Blowup in a Nonlinear Parabolic System

A two-component semilinear parabolic system on a bounded domain with Neumann boundary conditions is studied. It is shown that for a certain kind of nonlinearity, the blowup of solutions may occur when the diffusion coefficients are not equal, though the corresponding ODE possesses a globally stable equilibrium.     

A generalization of the method of averaging for systems with two time scales

In this paper we shall consider systems of the form x = ? f(t, ?t, x, y, ?), y = g(t,?t, x, y,?), where x and y are vectors of finite dimensions, f and g are assumed to be bounded for all t, and ? is a real parameter. Sufficient conditions are obtained for the existence of certain solutions which are bounded for all t. These solutions are shown to approach special solutions of a derived simpler averaged system of equations as ? → 0. Moreover, it is shown that there exists only one such bounded solution in the neighborhood of each special solution. In the special case when y is not present, it is shown that if a special solution is stable, solutions starting in nonlocal neighborhoods of this special solution approach the bounded solutions adjacent to it as t → ∞. These results generalize most of the existing work for systems of the type discussed here. Finally, we employ our results to study some problems of physical importance.     

Infinite dimensional slow modulations in a well known delayed model for cutting tool vibrations

We apply the method of multiple scales (MMS) to a well-known model of regenerative cutting vibrations in the large delay regime. By “large” we mean the delay is much larger than the timescale of typical cutting tool oscillations. The MMS up to second order, recently developed for such systems, is applied here to study tool dynamics in the large delay regime. The second order analysis is found to be much more accurate than the first order analysis. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy in that plotted solutions of moderate amplitudes are visually near-indistinguishable. The advantages of the present analysis are that infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space; lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS; the strong sensitivity of the slow modulation dynamics to small changes in parameter values, peculiar to such systems with large delays, is seen clearly; and though certain parameters are treated as small (or, reciprocally, large), the analysis is not restricted to infinitesimal distances from the Hopf bifurcation.