Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications

The paper concerns a class of n-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.     

Asymptotic Behavior of Quasi-Autonomous Expansive Type Evolution Systems in a Hilbert Space

We introduce the notion of almost expansive sequences and curves and study their ergodic and asymptotic properties in a Hilbert space H. We apply our results to study the asymptotic behavior of solutions to the quasi-autonomous expansive type evolution system (du/dt)(t) + f(t) ∈ Au(t) on [0, ∞).     

Instability of Dual Eigenvalue Fourth-Order Systems

Investigations of the postbuckling behavior of an elastic structure at a twofold branching point generally involve a potential of the cubic type. However, when the cubic part vanishes identically (for example, when there is symmetry in both active coordinates), the potential becomes of the quartic type. Here, quartic potentials in normalized coordinates are considered and formulas for limit points and bifurcations on imperfect paths are given in terms of trigonometric polynomials. These are used in certain structural examples to show that a theorem of Ho is invalid unless extra conditions are placed on the potential. They are also used in a proof of the modified Ho theorem.     

Gradient Estimates for Parabolic and Elliptic Systems from Linear Laminates

We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be Hölder or Dini continuous in the time variable and all but one spatial variable. This type of system arises from the problems of linearly elastic laminates and composite materials. For the proof, we use Campanato’s approach in a novel way. Non-divergence type equations under similar conditions are also discussed.     

Uniform Hölder Estimates in a Class of Elliptic Systems and Applications to Singular Limits in Models for Diffusion Flames

The main result of this paper is a general Hölder estimate in a class of singularly perturbed elliptic systems. This estimate is applied to the study of the well-known Burke–Schuman approximation in flame theory. After reviewing some classical cases (equidiffusional case; high activation energy approximation) we turn to the non-equidiffusional case, and to nonlinear diffusion models. The limiting problems are nonlinear elliptic equations; they have Hölder or Lipschitz maximal global regularity. A natural question is then whether this regularity is kept uniformly throughout the approximation process, and we show that this is true in general.     

Principal Trajectories of Forced Vibrations for Discrete and Continuous Systems

Principal trajectories of forced vibration of linear and nonlinear continuous systems are introduced as such motions in which the system is equivalent to a Newtonian particle in the function space of the system configurations. The corresponding 'effective mass' of the particle gives physical characteristics of the system response, so that zero effective mass is associated with resonance. The methodology can be viewed as a complementary tool to the method of normal modes, when considering the class of forced vibrating systems, since the related basis accounts for the system physical properties as well as the external forcing factor. In particular, it is shown that a two degrees of freedom system can possess an infinite discrete set of in-phase and out-of-phase forced vibrations of the normal modes type. The corresponding forcing vector-functions obey the second Newton law due to the definition of principal trajectories.     

Global Boundedness of the Gradient for a Class of Nonlinear Elliptic Systems

Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.     

On the Implementation of the Eigenvalue Method for Limit Cycle Determination in Nonlinear Systems

In many practical systems, limit cycles can be predicted with suitable precision by frequency domain methods using describing functions. Within such an approach, limit cycles can be predicted using the “eigenvalue method” [Somieski, G., Nonlinear Dynamics 26(1), 2001, 3–22]. This contribution presents a novel and advantageous implementation of this method, using singular value instead of eigenvalue calculations, and enhancing computational efficiency by avoiding a so called “frequency iteration”. An erratum to this article is available at .     

The Asymptotic Perturbation Method for Nonlinear Continuous Systems

We apply the asymptotic perturbation (AP) method to the study of the vibrations of Euler--Bernoulli beam resting on a nonlinear elastic foundation. An external periodic excitation is in primary resonance or in subharmonic resonance in the order of one-half with an nth mode frequency. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing the harmonic terms with a simple iteration. We obtain amplitude and phase modulation equations and determine external force-response and frequency-response curves. The validity of the method is highlighted by comparing the approximate solutions with the results of the numerical integration and multiple-scale methods.     

Strong Convergence Results for the Asymptotic Behavior of the 3D-Shell Model

We revisit the asymptotic convergence properties—with respect to the thickness parameter—of the earlier-proposed 3D-shell model. This shell model is very attractive for engineering applications, in particular due to the possibility of directly using a general 3D constitutive law in the corresponding finite element formulations. We establish strong convergence results for the 3D-shell model in the two main types of asymptotic regimes, namely, bending- and membrane-dominated behavior. This is an important achievement, as it completely substantiates the asymptotic consistency of the 3D-shell model with 3D linearized isotropic elasticity.     

An Eigenvalue Method for Calculation of Stability and Limit Cycles in Nonlinear Systems

In the calculation of periodic oscillations of nonlinear systems –so-called limit cycles – approximative and systematic engineeringmethods of linear system analysis are known. The techniques, working inthe frequency domain, perform a quasi-linearization of the nonlinear system,replacing nonlinearities by amplitude-dependent describing functions.Frequently, the resulting equations for the amplitude and frequency ofpresumed limit cycles are solved directly by a graphical procedure in aNyquist plane or by solving the nonlinear equations or a parameteroptimization problem. In this paper, an indirect numerical approach isdescribed which shows that, for a system of nonlinear differentialequations, the eigenvalues of the quasi-linear system simply indicateall limit cycles and, additionally, yield stability regions for thelinearized case. The method is applicable to systems with multiplenonlinearities which may be static or dynamic. It is demonstrated foran example of aircraft nose gear shimmy dynamics in the presence ofdifferent nonlinearities and the results are compared with those fromsimulation.     

Singular Solutions of Fully Nonlinear Elliptic Equations and Applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of , and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragmén–Lindel?f result as well as a principle of positive singularities in certain Lipschitz domains.     

A General Approach to the Asymptotic Behavior of Traveling Waves in a Class of Three-Component Lattice Dynamical Systems

In this paper, we develop a general approach to deal with the asymptotic behavior of traveling wave solutions in a class of three-component lattice dynamical systems. Then we demonstrate an application of these results to construct entire solutions which behave as two traveling wave fronts moving towards each other from both sides of x-axis for a three-species competition system with Lotka–Volterra type nonlinearity in a lattice.     

Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem $$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu|\nabla_{x} u|^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$ where ${\Omega(t) \subset \mathbb{R}^{n}}$ ( ${n \geqq 2}$ ) is bounded by the free boundary ${\Gamma(t)}$ , with ${\Omega(0) = \Omega_0}$ , μ and d are given positive constants. The initial function u ₀ is positive in ${\Omega_0}$ and vanishes on ${\partial \Omega_0}$ . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary ${\Gamma(t)}$ is smooth outside the closed convex hull of ${\Omega_0}$ , and as ${t \to \infty}$ , either ${\Omega(t)}$ expands to the entire ${\mathbb{R}^n}$ , or it stays bounded. Moreover, in the former case, ${\Gamma(t)}$ converges to the unit sphere when normalized, and in the latter case, ${u \to 0}$ uniformly. When ${g(u) = au - bu^2}$ , we further prove that in the case ${\Omega(t)}$ expands to ${{\mathbb R}^n}$ , ${u \to a/b}$ as ${t \to \infty}$ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists ${\mu^* \geqq 0}$ such that ${\Omega(t)}$ expands to ${{\mathbb{R}}^n}$ exactly when ${\mu > \mu^*}$ .     

Asymptotic Behaviour of a Semilinear Elliptic System with a Large Exponent

Consider the problem where Ω is a bounded convex domain in , N > 2, with smooth boundary . We study the asymptotic behaviour of the least energy solutions of this system as . We show that the solution remain bounded for p large. In the limit, we find that the solution develops one or two peaks away from the boundary, and when a single peak occurs, we have a characterization of its location.This research was supported by FONDECYT 1061110 and 3040059.     

Principal Lyapunov Exponents and Principal Floquet Spaces of Positive Random Dynamical Systems. III. Parabolic Equations and Delay Systems

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors’ paper Mierczyński and Shen (Trans Am Math Soc 365(10):5329–5365, 2013), to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents.     

Asymptotic Behavior of Minimizers for the Ginzburg-Landau Functional with Weight. Part I

(Accepted July 23, 1996)