Non-autonomous systems: asymptotic behaviour and weak invariance principles

Results pertaining to asymptotic behaviour of solutions of non-autonomous ordinary differential equations with locally integrably bounded right-hand sides are presented. Ramifications for weakly asymptotically autonomous systems and adaptively controlled systems are highlighted.     

Stability and asymptotic behaviour of a class of discrete systems

Summary This paper is concerned with convolution discrete systems on the half-axis. Necessary and sufficient conditions for uniform stability as well as admissibility results similar to that from continuous case (see [1], [4–5], [8–9]) are given. Entrata in Redazione il 24 febbraio 1976.     

Adirectional Markov models for 2D systems

Several deterministic Markovian properties are introduced inan acausal context. We analyse the relationship between theseproperties and investigate to which kind of models they correspondin the scalar case. We find that, in this case, the nearest-neighbourmodel is the natural model describing such systems. In the generalmultivariable case, we need higher-order models even in theID case.     

On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems

In this paper we consider reaction-diffusion systems in which the conditions imposed on the nonlinearity provide global existence of solutions of the Cauchy problem, but not uniqueness. We prove first that for the set of all weak solutions the Kneser property holds, that is, that the set of values attained by the solutions at every moment of time is compact and connected. Further, we prove the existence and connectedness of a global attractor in both the autonomous and nonautonomous cases. The obtained results are applied to several models of physical (or chemical) interest: a model of fractional-order chemical autocatalysis with decay, the Fitz-Hugh-Nagumo equation and the Ginzburg-Landau equation.     

Global existence and asymptotic behaviour for systems of nonlinear hyperbolic equations

We consider the initial-boundary value problem for a class of nonlinear hyperbolic equations system in a bounded domain. Using the potential well theory, the existence of global solutions is investigated. We also established the asymptotic behaviour of global solutions as t?→?+?∞ by applying the multiplier method.     

Stochastic models for two-mode compartmental systems and their asymptotic reduction

We consider a class of population models, which will be named “two-mode” and is suitable to represent phenomena characterized by the interaction of two compartment groups. The two groups, both containing the same number of compartments, may exhibit some difference in their behaviour (“mode”), described in terms of transition rates. We conventionally denote them as “active” and “inactive” compartments. Exchanges between compartments are modelled by a multidimensional jump Markov process. Within both active and inactive subsystems the exchanges of individuals may occur in all possible ways. On the other hand, each active compartment communicates in both directions with just one inactive compartment. Such exchanges have indeed to be considered as transitions of “status” (mode) for the individuals; with this interpretation, a special feature of our model is that they occur much faster than exchanges within each subsystem. Possible non-linear interactions among compartments (for instance due to crowding effects) are included in the model by letting the exchange rates depend on the total numbers of individuals in the pair of corresponding active and inactive compartments. Birth/reproduction and death phenomena are also accounted for. We show that, as the rate of mode transition diverges, an instantaneous relationship (in the mean value sense) links the number of individuals X^akt and X^ikt in each pair of active and inactive compartments. A stronger result is proved to hold for the “pooled variables” X^akt+ X^akt, which converge weakly (in the sense of distributions on the path space) to the solution of a reduced n-dimensional model. In the paper, Section 1 is devoted to discussing the peculiar features of the model and to motivate its relevance in a variety of possible applications fields. Section 2 contains a formal presentation of the original and reduced models, along with an intermediate model (equivalent to the original one) which helps in clarifying connections between the above-mentioned ones. The aim of Section 2 is also to ascertain that these models are well defined. In Section 3 the convergence results are proved.     

The asymptotic behaviour of symbolic generic initial systems of generic points

Consider the ideal I⊆K[x,y,z]$I\subseteq K[x,y,z]$ corresponding to points _p1,…,_pr$p_1,\dots ,p_r$ of P²${\mathbb{P}}^2$. We study the symbolic generic initial system  _{gin_(I(m))}m${\left\{\text{gin}\left(I^{\left(m\right)}\right)\right\}}_m$ of such an ideal and its behaviour as m$m$ gets large. In particular, we describe the limiting shape   of this system explicitly when _p1,…,_pr$p_1,\dots ,p_r$ are generic, assuming that the Uniform SHGH Conjecture holds for r≥9$r\ge 9$. The symbolic generic initial system and its limiting shape reflect information about the asymptotic behaviour of Hilbert functions of fat point ideals.     

Solution and asymptotic/blow-up behaviour of a class of nonlinear dissipative systems

In this paper, we consider a three-parameter class of Liénard type nonlinear dissipative systems of the form . Since such dissipative systems admit an eight-parameter Lie group of point transformations, it follows that there exists a (complex) point transformation mapping such a system into the free particle system . Normally, such an explicit point transformation cannot be found. Here we find such an explicit point transformation through exploiting the group properties of the determining equations that lead to it. Consequently, we obtain the explicit general solution of such dissipative systems. Moreover, we completely characterize the asymptotic and/or finite time blow-up behaviour of such systems in terms of their three parameters and initial data.     

The asymptotic behaviour of attainable sets of singularly-perturbed linear autonomous control systems

The asymptotic properties of attainable sets of singularly-perturbed linear autonomous control systems are investigated. It is shown that if an explicitly given linear scaling operator is applied to the attainable set, the resulting sets converge (as the small parameter tends to zero).     

About positive invariance and asymptotic stability

In this note we consider the differential equation , t0 with initial conditionx(0)G, whereG is a closed pointed convex cone. Our results pertain to positive invariance and asymptotic stability. A result by Stern, [3], will be improved. Subtangentiality will also be discussed as this notion is important in deriving results about positive invariance.     

About the asymptotic behaviour of continuous vector-valued local martingales and application in multiple linear regression models

We study the asymptotic behaviour of the process d M ¢ m 1 M , where M is an R n -continuous vector local martingale and d M ¢ m 1 is the inverse of its predictable quadratic variation (a matrix-valued process). We also give an application (strong consistency and rate of convergence) to multiple linear regression.     

On the asymptotic behaviour of Lee-codes

The asymptotic forms of bounds on the information rate of Lee-codes are derived and their relative strength is discussed. Also it is shown that the covering radius of Lee-codes lies asymptotically on the Varshamov-Gilbert bound.     

On asymptotic behaviour of neutral integrodifferential equations

The necessary and sufficient conditions for all nontrivial solutions of $$\tfrac{d}{{dt}}\left[ {x\left( t \right) - c\int_0^\infty {H\left( s \right)x\left( {t - s} \right)ds} } \right] + a\int_0^\infty {k\left( s \right)x\left( {t - s} \right)ds + bx\left( t \right) = 0} $$ to have ‘zero crossing’ are given. Also, an application to level crossing is given.