Global stability analysis of epidemiological models based on Volterra–Lyapunov stable matrices

In this paper, we study the global dynamics of a class of mathematical epidemiological models formulated by systems of differential equations. These models involve both human population and environmental component(s) and constitute high-dimensional nonlinear autonomous systems, for which the global asymptotic stability of the endemic equilibria has been a major challenge in analyzing the dynamics. By incorporating the theory of Volterra–Lyapunov stable matrices into the classical method of Lyapunov functions, we present an approach for global stability analysis and obtain new results on some three- and four-dimensional model systems. In addition, we conduct numerical simulation to verify the analytical results.     

A STRONG ERGODIC THEOREM FOR SOME NONLINEAR MATRIX MODELS FOR THE DYNAMICS OF STRUCTURED POPULATIONS

A general class of matrix difference equation models for the dynamics of discrete class structured populations in discrete time which possess a certain general type of nonlinearity introduced by Leslie for age-structured populations is considered. Arbitrary structuring is allowed in that transitions between any two classes are permitted. It is shown that normalized class distributions for such nonlinear models globally approach a “stable class distribution” and thus possess a strong ergodic property exactly like that of the classical linear theory of demography. However, unlike in the linear theory according to which the total population size grows or dies exponentially, the dynamics of total population size in these nonlinear models are shown to be governed by a nonlinear, nonautonomous scalar difference equation. This difference equation is asymptotically autonomous, and theorems which relate the dynamics of total population size to those of this limiting equation are proved. Examples in which the results are applied to some nonlinear age-structure models found in the literature are given.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

一类具有分离变量的非线性离散系统的镇定控制器

在控制理论和控制工程中，镇定控制器的设计是一个经典问题。许多有关这个问题的结论一般都是针对线性系统。对于非线性系统，很少见到有构造性结果能用于控制工程中。本针对一类广泛的非线性控制系统，我们构造了一些控制器，这些判据在工程实际问题中将具有一定的指导意义。     

Systems of elliptic boundary value problems and applications to competition models

We generalize the main result on existence of nonzero nonnegative solutions of systems of second order elliptic boundary value problems obtained by Lan (2011). The motivation for the generalization is to propose and study the competition models of Ricker and Beverton–Holt types governed by such systems. To the best of our knowledge, there is little study on such continuous competition models although such difference equation and first order ordinary differential equation competition models have been widely studied. Our results enrich and develop the connections among the classical fixed point index theory, systems of second order elliptic boundary value problems and population dynamics.     

Classical social processes: Attractor and computational models*

Attractor models provide a generalized way to represent processes found throughout science. A fuller articulation of the attractor framework requires that it be addressed qualitatively and conceptually as a nonlinear mathematical order residing between cyclical and random processes. Many significant nonlinear social processes have been identified and analyzed in classical social theory. These include the circulation of the elites (Pareto), cultural dynamics (Sorokin), social differentiation (Durkheim) and rationalization in modern institutions (Weber). The present discussion develops a qualitative consideration of such classical social processes as attractor systems, and discusses possible applications of such models in computational social science.     

Method of “transition into space of derivatives”: 40 years of evolution

In 1975, the “method of transition into space of derivatives” was proposed. It is an efficiently verifiable frequency criterion for the existence of a nontrivial periodic solution in multidimensional models of automatic control systems with one differentiable nonlinear term. The method used the classical torus principle and refrained from any constructions in the phase space of the system under study. Moreover, the method allowed researchers to broaden the class of systems to which it could be applied. In this work, we give a survey of the results presenting generalization and expansion of the method. We also show the connection between the method of transition into space of derivatives, the well-known generalized Poincaré–Bendixson principle proposed by R. A. Smith, and the results of contemporary authors who are active in the theory of oscillations in multidimensional systems. In the recent years, the author obtained frequency criteria for the existence of orbitally stable cycles in multiinput multioutput (MIMO) control systems and methods for the construction of multidimensional systems having a unique equilibrium and an arbitrarily prescribed number of orbitally stable cycles, which are described in the paper. The extension of the generalized Poincaré–Bendixson principle to multidimensional systems with angular coordinate is presented. We show the application of described methods of investigation of oscillation processes in multidimensional dynamical systems to solving S. Smale’s problem in the chemical kinetics theory of biological cells and also to finding hidden attractors of the generalized Chua system and the minimal global attractor of a system with a polynomial nonlinear term. The publication is illustrated by numerous examples.     

Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.     

Shadowing Pseudo-Orbits and Gradient Descent Noise Reduction

Shadowing trajectories are one of the most powerful ideas of modern dynamical systems theory, providing a tool for proving some central theorems and a means to assess the relevance of models and numerically computed trajectories of chaotic systems. Shadowing has also been seen to have a role in state estimation and forecasting of nonlinear systems. Shadowing trajectories are guaranteed to exist in hyperbolic systems, but this is not true of nonhyperbolic systems, indeed it can be shown there are systems that cannot have long shadowing trajectories. In this paper we consider what might be called shadowing pseudo-orbits. These are pseudo-orbits that remain close to a given pseudo-orbit, but have smaller mismatches between forecast state and verifying state. Shadowing pseudo-orbits play a useful role in the understanding and analysis of gradient descent noise reduction, state estimation, and forecasting nonlinear systems, because their existence can be ensured for a wide class of nonhyperbolic systems. New theoretical results are presented that extend classical shadowing theorems to shadowing pseudo-orbits. These new results provide some insight into the convergence behaviour of gradient descent noise reduction methods. The paper also discusses, in the light of the new results, some recent numerical results for an operational weather forecasting model when gradient descent noise reduction was employed.     

On problems of Aizerman and Kalman

The problems of the stability of nonlinear control systems posed by Aizerman and Kalman have stimulated the development of methods for detecting hidden periodic oscillations in multidimensional dynamical systems. In the 1950s, Pliss developed an analytical method for detecting periodic oscillations in third-order systems satisfying the generalized Routh-Hurwitz conditions. It has turned out that this generalized method of Pliss can be regarded as a special version of the describing function method in the critical case. Being combined with computational procedures based on applied bifurcation theory, this method makes it possible to obtain new classes of systems for which the conjectures of Aizerman and Kalman are false. The known approaches to constructing counterexamples to Aizerman’s and Kalman’s conjectures proposed by Fitts, Barabanov, and Llibre are reviewed. A new effective analytical-numerical method for constructing such counterexamples is presented. The method is based on combining the classical theory of small parameter, bifurcation theory, and the method of harmonic linearization. It is applied to numerically construct a series of counterexamples to Aizerman’s and Kalman’s conjectures.     

Model Selection of Dnamical Systems via Entropic Regression and Bayesian Information Criteria

Recovering system model from noisy data is a key challenge in the analysis of dynamical systems. Based on a data-driven identification approach, we develop a model selection algorithm called Entropy Regression Bayesian Information Criterion (ER-BIC). First, the entropy regression identification algorithm (ER) is used to obtain candidate models that are close to the Pareto optimum and combine as a library of candidate models. Second, BIC score in the candidate models library is calculated using the Bayesian information criterion (BIC) and ranked from smallest to largest. Third, the model with the smallest BIC score is selected as the one we need to optimize. Finally, the ER-BIC algorithm is applied to several classical dynamical systems, including one-dimensional polynomial and RC circuit systems, two-dimensional Duffing and classical ODE systems, three-dimensional Lorenz 63 and Lorenz 84 systems. The results show that the new algorithm accurately identifies the system model under noise and time variable $t$, laying the foundation for nonlinear analysis.     

On the motion of an oscillator with a periodically time-varying mass

The stability of the motion of an oscillator with a periodically time-varying mass is under consideration. The key idea is that an adequate change of variables leads to a newtonian equation, where classical stability techniques can be applied: Floquet theory for the linear oscillator, KAM method in the nonlinear case. To illustrate this general idea, first we have generalized the results of [W.T. van Horssen, A.K. Abramian, Hartono, On the free vibrations of an oscillator with a periodically time-varying mass, J. Sound Vibration 298 (2006) 1166–1172] to the forced case; second, for a weakly forced Duffing’s oscillator with variable mass, the stability in the nonlinear sense is proved by showing that the first twist coefficient is not zero.     

New results on the parametric stability of nonlinear systems

In this paper, we derive some new results on the parametric stability of nonlinear systems. Explicitly, we derive a necessary and sufficient condition for a nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. Next, we derive some new results on the parametric stability of discrete-time nonlinear systems. As in the continuous case, we derive a necessary and sufficient condition for a discrete-time nonlinear system to be locally parametrically exponentially stable at an equilibrium point. We also derive a necessary condition for the discrete-time nonlinear system to be locally parametrically asymptotically stable at an equilibrium point. We illustrate our results with some classical examples from the bifurcation theory.     

Necessary and Sufficient Conditions of Existence for Positive Solution to a Class of Quasilinear Elliptic Systems in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript>

In this paper, we establish the necessary and sufficient conditions of existence for a positive solution to a class of non-variational quasilinear elliptic systems in R ^N . The sufficient condition of existence result bases on the Mountain Pass Lemma and the sub-super solution methods, and the necessary condition is a consequence of a Picone’s identity. The system models some phenomena in different physical and other natural sciences: non-Newtonian mechanics, nonlinear elasticity and glaciology, combustion theory, population biology and so on.     

Mathematical Genetics: A Hybrid Seed for Educators to Sow

A principal area of applications of mathematical ideas is the study of population genetic systems. Such investigations have inspired the development of mathematical statistics, insights for certain classes of stochastic processes of diffusion type and intriguing algebraic structures. In this paper a series of simple mathematical models for evolutionary theory and inbreeding systems are set forth. Included are the Hardy‐Weinberg law and models of selection balance. We also treat several special inbreeding processes for selfing (self fertilization), sib mating, incompatibility systems and imprinting. Calculations of inbreeding coefficients using the notion of ‘identity by descent’ are made. Some classical simple stochastic models exposing the concept of genetic drift are highlighted.     

Modified Newton’s method for systems of nonlinear equations with singular Jacobian

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method.     

On Russell's method of controllability via stabilizability

Russell has observed that a linear system is controllable provided it is stabilizable in both positive and negative time. We give a version of this result valid for nonlinear systems, and illustrate its use by giving new proofs of two classical results from control theory, the first involving bounded perturbations of controllable linear systems, and the second involving controllability of linear systems by bounded controls.This research was supported by the Natural Sciences and Engineering Research Council of Canada.     

Nonlinear generalized functions and jump conditions for a standard one pressure liquid–gas model

The mixture of a liquid and a gas is classically represented by one pressure models. These models are a system of PDEs in nonconservative form and shock wave solutions do not make sense within the theory of distributions: they give rise to products of distributions that are not defined within distribution theory. But they make sense by applying a theory of nonlinear generalized functions to these equations. In contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori. Jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations. The physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible, and so they do not satisfy equally well their respective state laws on the shock waves. This modelization gives well defined explicit jump conditions. The great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results. The jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions, which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results. We expose with care the mathematical originality of the theory of nonlinear generalized functions (an original abstract analysis issued by the Leopoldo Nachbin team on infinite dimensional holomorphy) that permits to state mathematically physical facts that cannot be formulated within distribution theory, and are the key for the removal of “ambiguities” that classically appear when one tries to calculate on “multiplications of distributions” that occur in the differential equations of physics.     

Populations with individual variation in dispersal in heterogeneous environments: Dynamics and competition with simply diffusing populations

We consider a model for a population in a heterogeneous environment, with logistic-type local population dynamics, under the assumption that individuals can switch between two different nonzero rates of diffusion. Such switching behavior has been observed in some natural systems. We study how environmental heterogeneity and the rates of switching and diffusion affect the persistence of the population. The reactiondiffusion systems in the models can be cooperative at some population densities and competitive at others. The results extend our previous work on similar models in homogeneous environments. We also consider competition between two populations that are ecologically identical, but where one population diffuses at a fixed rate and the other switches between two different diffusion rates. The motivation for that is to gain insight into when switching might be advantageous versus diffusing at a fixed rate. This is a variation on the classical results for ecologically identical competitors with differing fixed diffusion rates, where it is well known that "the slower diffuser wins".