Nonautonomous Ordinary Differential Equations Research Articles

Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions

<p style='text-indent:20px;'>We define the notions of impulsive evolution processes and their pullback attractors, and exhibit conditions under which a given impulsive evolution process has a pullback attractor. We apply our results to a nonautonomous ordinary differential equation describing an integrate-and-fire model of neuron membrane, as well as to a heat equation with nonautonomous impulse and a nonautonomous 2D Navier-Stokes equation. Finally, we introduce the notion of tube conditions to impulsive evolution processes, and use them as an alternative way to obtain pullback attractors.

К тензорному анализу решения задачи реализации билинейной системы второго порядка с опозданием

Определены аналитические условия (необходимые/достаточные) решения задачи дифференциальной реализации континуального пучка управляемых траекторных кривых в классе билинейных неавтономных обычных дифференциальных уравнений (с опозданием и без) второго порядка в материальном сепарабельном гильбертовом пространстве. Эта задача относится к типу обратных задач для аддитивной комбинации нестационарных линейных и билинейных операторов эволюционных уравнений в бесконечномерном гильбертовом пространстве. Метамовой данной теории служат конструкции тензорных произведений гильбертовых пространств, структуры решеток с ортодополнением и функциональный аппарат нелинейного оператора Релея-Ритца. При этом показано, что при конечном пучке траекторий наличие свойств типа сублинейности данного оператора позволяет получить достаточные условия для существования таких реализаций. Попутно обосновываются тополого-метрические условия непрерывности проектизации нелинейного функционального опера-тора Релея–Ритца с вычислением фундаментальной группы его образа. Полученные результаты побуждают к развитию теории нелинейной структурной идентификации полилинейных дифференциальных моделей высших порядков (например, для моделирования многоканальных нейроимплантов типа Neuralink).

Numerical integrators based on the Magnus expansion for nonlinear dynamical systems

Explicit numerical integration algorithms up to order four based on the Magnus expansion for nonlinear non-autonomous ordinary differential equations are presented and tested on problems possessing qualitative (very often, geometric) features that is convenient to preserve under numerical discretization. The range of applications covers augmented dynamical systems, highly oscillatory problems and nonlinear non-autonomous partial differential equations of evolution previously discretized in space.

Determination of the blow up point for complex nonautonomous ODE with cubic nonlinearity

We present a method for determination of the blow up point for a complex nonautonomous ordinary differential equation with a cubic nonlinearity. This equation describes stationary nonlinear modes for two important NLS-based models: (i) the Gross–Pitaevskii equation with complex potential and repulsive nonlinearity and (ii) the Lugiato–Lefever equation in the case of normal dispersion. We derive and justify an asymptotic expansion in the vicinity of the blow up point. This expansion is employed for the construction of a numeric procedure for the computation of the coordinate of the blow up point. We illustrate applications of the proposed procedure by two examples, where exact analytical solutions are available. It is shown that it allows one to find the blow up point with high accuracy. The method may be efficiently used for the search of soliton solutions for vector and scalar NLS-type equations within the strategy of ’filtering out’ of blow up solutions (Alfimov et al., Physica D, 394, 39 (2019)).

Global, nonparametric, noniterative optimization of time-averaged quantities under small, time-varying forcing: An application to a thermal convection field

This study demonstrates a global, nonparametric, noniterative optimization of time-mean value of a kind of index vibrated by time-varying forcing. It is based on the fact that the (steady) forced vibration of nonautonomous ordinary differential equation systems is well approximated by an analytical solution when the amplitude of forcing is sufficiently small and its base state without forcing is linearly stable and steady. It is applied to optimize a time-averaged heat-transfer rate on a two-dimensional thermal convection field in a square cavity with horizontal temperature difference, and the globally optimal way of vibrational forcing, i.e. the globally optimal, spatial distribution of vibrational heat and vorticity sources, is first obtained. The maximized vibrational thermal convection corresponds well to the state of internal gravity wave resonance. In contrast, the minimized thermal convection is weak, keeping the boundary layers on both sidewalls thick.

Existence and classification of pseudo-asymptotic solutions for Tolman–Oppenheimer–Volkoff systems

The Tolman–Oppenheimer–Volkoff (TOV) equations are a partially uncoupled system of nonlinear and non-autonomous ordinary differential equations which describe the structure of isotropic spherically symmetric static fluids. Nonlinearity makes finding explicit solutions of TOV systems very difficult and such solutions are very rare. In this paper we introduce the notion of pseudo-asymptotic TOV systems and we show that the space of such systems is at least fifteen-dimensional. We also show that if the system is defined in a suitable domain (meaning the extended real line), then well-behaved pseudo-asymptotic TOV systems are genuine TOV systems in that domain, ensuring the existence of new fourteen analytic solutions for extended TOV equations. The pseudo-asymptotic solutions are classified according to the nature of the matter (ordinary or exotic) and to the existence of cavities and singularities. It is shown that at least three of them are realistic, in the sense that they are formed only by ordinary matter and contain no cavities or singularities.

Ergodicity in nonautonomous linear ordinary differential equations

The weak and strong ergodic properties of nonautonomous linear ordinary differential equations are considered. It is shown that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices.

Mathematical analysis of a generalised model of chemotherapy for low grade gliomas

We study mathematical properties of a model describing growth of primary brain tumours called low-grade gliomas (LGGs) and their response to chemotherapy. The motivation for considering this particular type of cancer is its large impact on society. LGGs affect mainly young adults and eventually result in death, despite the tumour growth rate being slow. The model studied consists of two non-autonomous ordinary differential equations and is a generalised version of the model proposed by Bogdanska et al. (Math. Biosci. 2017). We discuss the stability of stationary states, prove global stability of tumour-free steady state and, in some cases, justify the existence of periodic solutions. Assuming that chemotherapy effectiveness remains constant in time, we provide analytical estimates and calculate minimal doses of the drug that should eliminate the tumour for particular patients with LGGs.

Analyzing and Projection of Future Bangladesh Population Using Logistic Growth Model

Bangladesh is a densely populated country than many other countries of the world. The population growth is termed as alarming, however, knowledge of growth in the years to come would be useful in planning for the development of the country. This article is based on the projection of future population growth of the country. The available actual population census data during 1991-2011 of Bangladesh was applied to the application of a non-linear, non-autonomous ordinary differential equation familiar as Verhulst logistic population model with the maximum environmental capability of Bangladesh. Bangladesh will reach its carrying capacity of 245.09 million population in the next 56 years i.e. the year 2067 and then it decreases as S-shaped curve. The article has provided a focus on the changing trends of the growth of the population of Bangladesh.

Time to change your mind? Modelling transient properties of cortex formation highlights the importance of evolving cell division strategies

The successful development of the mammalian cerebral neocortex is linked to numerous cognitive functions such as language, voluntary movement, and episodic memory. Neocortex development occurs when neural progenitor cells divide and produce neurons. Critically, although the progenitor cells are able to self-renew they do not reproduce themselves endlessly. Hence, to fully understand the development of the neocortex we are faced with the challenge of understanding temporal changes in cell division strategy. Our approach to modelling neuronal production uses non-autonomous ordinary differential equations and allows us to use a ternary coordinate system in order to define a strategy space, through which we can visualise evolving cell division strategies. Using this strategy space, we fit the known data and use approximate Bayesian computation to predict the founding progenitor population sizes, currently unavailable in the experimental literature. Counter-intuitively, we show that humans can generate a larger number of neurons than a macaque’s even when starting with a smaller number of progenitor cells. Accompanying the article is a self-contained piece of software, which provides the reader with immediate simulated results that will aid their intuition. The software can be found at www.dpag.ox.ac.uk/team/noemi-picco.

On symmetric and asymmetric Van der Pol-Duffing oscillators

We investigate numerically the dynamics of both symmetric and asymmetric Van der Pol-Duffing oscillators driven by a periodic force F(t) = f cosωt. Each system is modeled by a different second order nonautonomous nonlinear ordinary differential equation controlled by five parameters. Our investigation takes into account the (ω, f) parameter-space in the two systems, keeping the other three parameters fixed. We verify the existence of parameter regions for which the corresponding trajectories in the phase-space are periodic, quasiperiodic, and chaotic, for the symmetric case. In the asymmetric case we verify the existence only of periodic and chaotic regions in the (ω, f) parameter-space. Finally, we also investigate the organization of the dynamics in the two systems, identifying Fibonacci and period-adding sequences of periodic structures.

Descent Rate Models of the Synchronization of the Quasi-Biennial Oscillation by the Annual Cycle in Tropical Upwelling

Abstract The response of the quasi-biennial oscillation (QBO) to an imposed mean upwelling with a periodic modulation is studied, by modeling the dynamics of the zero wind line at the equator using a class of equations known as descent rate models. These are simple mathematical models that capture the essence of QBO synchronization by focusing on the dynamics of the height of the zero wind line. A heuristic descent rate model for the zero wind line is described and is shown to capture many of the synchronization features seen in previous studies of the QBO. It is then demonstrated using a simple transformation that the standard Holton–Lindzen model of the QBO can itself be put into the form of a descent rate model if a quadratic velocity profile is assumed below the zero wind line. The resulting nonautonomous ordinary differential equation captures much of the synchronization behavior observed in the full Holton–Lindzen partial differential equation. The new class of models provides a novel framework within which to understand synchronization of the QBO, and we demonstrate a close relationship between these models and the circle map well known in the mathematics literature. Finally, we analyze reanalysis datasets to validate some of the predictions of our descent rate models and find statistically significant evidence for synchronization of the QBO that is consistent with model behavior.

A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly, exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.

On the Existence of Nonautonomous ODE with Application to Semilinear Elliptic Equations

We prove an existence result for a second-order nonautonomous nonlinear ordinary differential equation with a particular asymptotic behavior. This solution is then proved to be a limiting profile for a semilinear elliptic equation on the quarter plane. Our results are mainly based upon the maximum principle; the method of sub- and super-solutions and the sliding method.

Asymptotic Representation of the Solutions of Differential Equations with Regularly Varying Nonlinearities

We establish the conditions of existence of some types of power-type solutions of a binomial nonautonomous ordinary differential equation with regularly varying nonlinearities.

Underlying one-step methods and nonautonomous stability of general linear methods

We generalize the theory of underlying one-step methods to strictly stable general linear methods (GLMs) solving nonautonomous ordinary differential equations (ODEs) that satisfy a global Lipschitz condition. We combine this theory with the Lyapunov and Sacker-Sell spectral stability theory for one-step methods developed in [ 34 , 35 , 36 ] to analyze the stability of a strictly stable GLM solving a nonautonomous linear ODE. These results are applied to develop a stability diagnostic for the solution of nonautonomous linear ODEs by strictly stable GLMs.

Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation

We consider a non-autonomous ordinary differential equation over a finite time interval \begin{document}$[T_1,T_2]$\end{document} . The area of exponential attraction consists of solutions such that the distance to adjacent solutions exponentially contracts from \begin{document}$T_1$\end{document} to \begin{document}$T_2$\end{document} . One can use a contraction metric to determine an area of exponential attraction and to provide a bound on the rate of attraction. In this paper, we will give the first method to algorithmically construct a contraction metric for finite-time systems in one spatial dimension. We will show the existence of a contraction metric, given by a function which satisfies a second-order partial differential equation with boundary conditions. We then use meshless collocation to approximately solve this equation, and show that the resulting approximation itself defines a contraction metric, if the collocation points are sufficiently dense. We give error estimates and apply the method to an example.

Mathematical modelling and numerical simulations of the influence of hygiene and seasons on the spread of cholera

Cholera is a bacterial disease, its spread is strongly influenced by environmental factors and some socio-economic factors such as hygiene standards and nutrition of the population. This paper is devoted to the modelling of the impact of climatic factors and human behaviour on the spread of cholera. The mathematical modelling incorporates the direct transmission and the indirect transmission due to environmental knowledge. Taking into account the effect of the intra-annual variation of climatic factors on the transmission of cholera, a non-autonomous ordinary differential equations is proposed to describe the dynamics of the transmission of cholera. When the intra-annual variation of climate is not incorporated into the model, the latter becomes autonomous. The basic reproductive number is calculated and the stabilities of equilibria are investigated. In the non-autonomous case, the disease extinction and uniform persistence of disease are investigated. The results suggest that the transmission and spread of cholera can be affected by climatic factors, the proportion of the undernourished individuals and the proportion of people who respect the hygiene standards. Finally, some numerical simulations are proposed using the parameters values of climatic factors and socio-economic factors of some localities situated in Lake Chad border between Chad, Cameroon and Nigeria.

Multistability and organization of periodicity in a Van der Pol–Duffing oscillator

We investigate the dynamics of a Van der Pol–Duffing forced oscillator, which is modelled by a five-parameter second order nonautonomous nonlinear ordinary differential equation. Firstly we fix three of these parameters, and investigate the dynamics of this system by varying the other two, namely the amplitude and the angular frequency of the external forcing. We also investigate the existence of different attractors, periodic, quasiperiodic, and chaotic. Finally, we investigate the occurrence of multistability in the considered Van der Pol–Duffing forced oscillator, for some fixed sets of parameters.

Dynamics of Nonautonomous Ordinary Differential Equations with Quasi-Periodic Coefficients

We investigate the dynamics of two types of nonautonomous ordinary differential equations with quasi-periodic time-varying coefficients and nonlinear terms. The vector fields for the nonautonomous systems are written as [Formula: see text], [Formula: see text], where [Formula: see text] is the spacial part and [Formula: see text] is the time-varying part, and [Formula: see text] and [Formula: see text] are real parameters. The first type has a polynomial as the nonlinear term, another type has a continuous periodic function as the nonlinear term. The polynomials and periodic functions have simple zeros. Several examples with numerical experiments are given. It is found by numerical calculation that there might exist only one attractor for the systems with polynomials as nonlinear terms and [Formula: see text], and there might exist infinitely many attractors for systems with periodic functions as nonlinear terms and [Formula: see text]. For [Formula: see text] sufficiently small, the parameter regions for [Formula: see text] are roughly divided into three parts: the spacial region ([Formula: see text]), the balance region ([Formula: see text]), and the time-varying region ([Formula: see text]); (i) for [Formula: see text], the orbits approach some planes depending on the zeros of the polynomials or the periodic functions; (ii) for [Formula: see text], there exist attractors with the number no less than the number of zeros of the polynomials or the periodic functions, implying the existence of infinitely many attractors for systems with periodic functions as nonlinear terms; (iii) for [Formula: see text], the orbits wind around some region depending on the choice of the initial position. The shape of the attractors might be strange or regular for different parameters, and we obtain the existence of ball-like (regular) attractors, two-wings (strange) attractors, and other attractors with different shapes. The Lyapunov exponents are negative. These results reveal an intrinsic relationship between the existence of attractors (or strange dynamics) and the parameters [Formula: see text] and [Formula: see text] for nonautonomous systems with quasi-periodic coefficients. These results will be very useful in the understanding of the dynamics of general nonautonomous systems, nonautonomous control theory and other related fields.