Theory Of Differential Equations Research Articles

ABOUT ONE PROPERTY OF THE DISPERSION EQUATION FOR LATITUDINAL ACOUSTIC-GRAVITATIONAL WAVES

Acoustic-gravity waves are an example of processes that largely determine the dynamics of the Earth’s atmosphere. This is due to the fact that the sources of these waves are located throughout the height of the atmosphere, from the very “bottom”, where earthquakes, volcanic emissions, tsunamis, tornadoes, etc., occur, and to the very “top”, where perturbations of the solar wind, magnetic storms, and precipitation of particles in high latitudes are active. All these phenomena lead to the active energy exchange between all layers of the Earth’s atmosphere and the interaction of wave disturbances of significantly different scales — from several thousand kilometers to several hundred meters, and this — to the appearance and development of processes of convection and turbulence in the environment. It seems that only nonlinear processes should dominate under such conditions. To a large extent, it is true, but at the same time, observations indicate that in many cases in the process of propagation of acoustic-gravity waves (AGW), the effects can be comprehensively described within the framework of the linear approximation of perturbation theory and well-developed theory of oscillations. At the same time, when creating models of the process, it turned out to be appropriate to use sufficiently justified physical approximations, such as isothermality of the atmosphere, its unlimitedness in the horizontal direction and compressibility in the vertical direction. Taking into account the real scales of the AGW, it is possible to neglect the curvature of the Earth’s surface and consider it locally flat at any point of the surface and use the Cartesian coordinate system X, Y, Z in the calculations. To describe the environment, it makes sense to use non-dissipative hydrodynamics and in an equilibrium state — the hydrostatic equilibrium equation and barometric equation. The above-mentioned approximations and the mathematical apparatus of the theory of oscillations and the theory of differential equations allow when studying the initial system of equations describing the dynamics of AGW, to obtain a dispersion equation in the form of a polynomial of the fourth degree relative to the angular frequency of rotation as a function of the normalized wave vector of disturbance k  (AGW). AGW spectrum is a spectrum of the atmosphere’s own oscillations in the form ( ) k  , and its obtaining can be considered as the final solution to the initial problem if we ignore the obvious influence on the AGW spectrum of the angular frequency of rotation of the atmosphere , which must necessarily be present in the dispersion equation due to the influence of the Coriolis force. The formal reason for the absence of the components of the vector  in the dispersion equation (DE) is the fact that the | |   is a minimum of two orders of magnitude smaller than the characteristic rotation frequency of the atmosphere 0 , which is equal to the acoustic cutoff frequency. At the same time, the improvement of modern atmospheric observation equipment places increases the requirements for the accuracy of DE model solutions. In this sense, the resolution of DE in the work [Cheremnykh O. K. et al. Kinematics and Phys. Celestial Bodies. 2020. 36, № 2. P. 64—78] can be considered as a “zero-order” solution with a small parameter 0    | |/  . In addition, according to the method of obtaining, this solution is approximate. By definition, the solution obtained in the work [Cheremnykh O. K. et al. Kinematics and Phys. Celestial Bodies, 2022. 38, № 3. P. 121—131] by taking into account terms   0 in the modified DE is more accurate. But it is also approximate, although more accurate. In this work, we study in detail the dispersion equation for latitudinal AGW. The need for such consideration, as will be shown, is a consequence of the structure of this equation, namely the presence of a linear frequency term in it. Preliminary analysis showed that existing mathematical methods do not provide an unambiguous solution to this equation. This suggests the need to study possible solutions of the equation in terms of their coincidence with previously obtained ones for some partial cases. Such research allows us to choose the right decision. In the proposed study, we have shown that the Euler-Lagrange method allows, under certain additional conditions, to obtain an exact solution of the modified equation for AGW in closed analytical form.

An improved fractional Halanay inequality with distributed delays

In this paper, we establish global fractional Halanay‐type inequalities with distributed delays in differential and integral forms and consequently give a simple sufficient condition for Mittag–Leffler stability of solutions to fractional differential equations with this type of delays. Our essential tool is the submultiplicative property of the Mittag–Leffler functions which imitates the multiplicative property of the exponential functions in the classical theory of differential equations. The estimation with explicit decaying rate and amplitude constant is a faithful extension of the classical Halanay inequality.

A novel algorithm for asymptotic stability analysis of some classes of stochastic time-fractional Volterra equations

The article presents a regularization method to study global asymptotic stability of stochastic multi-time scale Volterra equations as an alternative to the algorithms based on Lyapunov functionals. Two different concepts of stability were linked and examined. The central idea of the method is based on a parallelism between the Lyapunov stability and a stochastic version of the input-to-state stability, which is well-known in the control theory of deterministic ordinary differential equations. At the first step the algorithm transforms the given delay equation into a Volterra differential equation with stochastic control, following replacement of the Lyapunov stability of the former with the input-to-state stability of the latter. To estimate the norms of the solutions we use a regularization technique based on the concept of inverse-positive matrices. This algorithm could be extended and applied for stability analysis of new classes of stochastic fractional differential equations and their applications.

BIFURCATION CONTROL STRATEGY FOR A DELAYED FRACTIONAL-ORDER POPULATION DYNAMICS MODEL WITH INCOMMENSURATE ORDERS

This paper excogitates a bifurcation control strategy for a delayed fractional-order population dynamics model with incommensurate orders. First and foremost, by using stability theory of fractional differential equations, the sufficient conditions for the stability of the positive equilibrium are established. It is not difficult to find that the fractional-order system has a wider stability region than the traditional integer-order system. Second, taking time delay as bifurcation parameter, the sufficient criteria for Hopf bifurcation are obtained. In the next place, it is interesting to introduce a delayed feedback controller to guide Hopf bifurcation. The results reveal that the bifurcation dynamics of the model could be effectively controlled as long as the delay or fractional order is carefully adjusted. In conclusion, numerical simulations are carried out to confirm our theoretical results.

Abstraction of Continuous-Time Systems Based on Feedback Controllers and Mixed Monotonicity

In this paper, we consider the problem of the computation of efficient symbolic abstractions for continuous-time control systems. The new abstraction algorithm builds symbolic models with the same number of states but fewer transitions in comparison to the one produced by the standard algorithm. At the same time, the new abstract system is at least as controllable as the standard one. The proposed algorithm is based on the solution of a region-to-region control synthesis problem. This solution is formally obtained using the theory of viscosity solutions of the dynamic programming equation and the theory of differential equations with discontinuous righthand side. In the new abstraction algorithm, the symbolic controls are essentially the feedback controllers that solve this control synthesis problem. The improvement in the number of transitions is achieved by reducing the number of successors for each symbolic control. For a certain class of control systems, with a suitable set of discretization parameters, the new algorithm may even produce deterministic abstract systems or systems with a singleton input alphabet. The approach is illustrated by examples that compare the two abstraction algorithms.

CONSTRUCTION OF THE SOLUTION OF THE SYSTEM OF SINGULARLY PERTURBATED DIFFERENTIAL EQUATIONS WITH A DIFFERENTIAL TURNING POINT

In this work, a system of singularly perturbed differential equations of the 4th order with a small parameter at the highest derivative and a turning point is considered. The turning point x=0 is located in the end of the segment [-l;0] under consideration. The coefficients of a given matrix are infinitely differentiable functions on a given interval. The goal is to construct a uniform solution asymptotics for a system of singularly perturbed differential equations with a differential turning point on the segment [-l;0]. In this case, the spectrum of the limit operator contains multiple and identically zero elements. The uniform asymptotic solution is constructed by the method of essential-singular functions using the Airy functions [0;l] and their derivatives. Studies have shown that the Apparatus of Airy functions and model equations is quite effective for constructing a uniform asymptotic solution of problems with turning points. It was also established that in order to select these functions along with the independent variable it is necessary to introduce a new variable t of the segment [0;l]. As a result, an extended problem will be obtained, which will make it possible to formally find the solution in the form of series with a small parameter. The constructions of the solutions are obtained by sequentially solving the systems of iterative equations, which at a certain step of the iterations make it possible to determine all the components of the sought-after vector functions with an accuracy of two scalar factors that form an arbitrary vector. The regularizing function Bi(t) is chosen in such a way that the function increases indefinitely when t approaches the infinity. The conducted studies showed that for any ratio of the signs of the matrix coefficient near which the inflection point is located, it is not possible to write down a smooth solution in the form of a single analytical expression. The solution of a degenerate vector equation in the general case has a discontinuity of the second kind at the inflection point, so it cannot be used explicitly to construct the asymptotic solution of this problem. For this purpose, a technique from the classical theory of linear differential equations was applied and solutions for a degenerate equation of the 2nd order were obtained.

Mathematical Modeling of Plants, Herbivore and Natural Enemies of Herbivores Interaction with Harvesting

Plant-herbivore-natural enemies of herbivores interaction is one of the basic interactions that drives of the ecosystem yields. In this interaction, plants are the primary food source for herbivores while natural enemies of herbivores depends on herbivores for food and on plants for shelter. Harvesting of every species which is common in many habitat may affect the population densities of the species and even the entire ecosystem. Therefore, conservation and maintenance of the harvested species is critical for ecosystem balance. In this paper, a model of plant-herbivore-natural enemies interactions with the constant effort harvesting of every species was developed and analyzed. The positive invariant set, the conditions of existence and locally asymptotically, stability of the equilibria were determined using the stability theory of ordinary differential equations. The results shows that the species being harvested would become extinct if harvesting effort exceeded a threshold value for the given population. While maintaining the coexistence of populations in the ecosystem requires sensible harvesting practices. Therefore, it is fair to choose a reasonable harvesting effort to allow all species to coexist in order to govern the species’ dynamic behavior. The insights of the solutions of this study are of great essence to ecologists and policy developers in environmental conservation. The authorities to pay attention to the minimum number required based on the area coverage in deciding when to harvest and also be cautious to the amount and effort of harvesting in view of conserving the species and the environment.

Effects of noise on the critical points of Turing instability in complex ecosystems.

Noise is ubiquitous in natural and artificial systems. In a noisy environment, the interactions among nodes may fluctuate randomly, leading to more complicated interactions. In this paper we focus on the effects of noise and network topology on the Turing pattern of ecological networks with activator-inhibitor structure, which may be interpreted as prey-predator interactions. Based on the stability theory of stochastic differential equations, a sufficient condition for the uniform state is derived. The analytical results indicate that noise is beneficial for the uniform state. When the ratio between the diffusion coefficients of the predator and prey increases, the ecosystems can exhibit a transition from a uniform stable state to a Turing pattern, while when the ratio decreases, the ecosystems transit from a Turing pattern to a uniform stable state. There are two crucial critical points in Turing patterns, forward and backward. We find that both forward and backward critical points increase as the noise intensity increases. This means that noise favors a stable homogeneous state compared to a state with a heterogeneous pattern, which is consistent with the analytical results. In addition, noise can weaken the hysteresis phenomenon and even eliminate it in some cases. Furthermore, we report that network topology plays an important role in modulating the uniform state of ecosystems, such as the size of prey-predator systems, the network connectivity, and the strength of interaction.

A general framework for tropical differential equations

We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional differential information, we define tropicalization of differential equations and tropicalization of their solution sets. This framework includes rings of interest in the theory of p-adic differential equations: rings of convergent power series over a non-archimedean normed field. The tropicalization records the norms of the coefficients. This gives a significant refinement of Grigoriev’s framework for tropical differential equations. We then prove a differential analogue of Payne’s inverse limit theorem: the limit of all tropicalizations of a system of differential equations is isomorphic to a differential variant of the Berkovich analytification.

How to combat atmospheric carbon dioxide along with development activities? A mathematical model

Depletion in forest density due to development activities is one of the reasons for the elevated level of carbon dioxide (CO2). In this scenario, the plantation of leafy trees, which absorb more CO2 compared to regular trees, under a planned strategy may help to attain the desired level of atmospheric carbon dioxide. We have formulated a non-linear mathematical model concerning the strategy of maintaining the atmospheric level of CO2 along with development activities and analyzed the proportion of deforested land needed for the plantation of leafy trees. This strategy will ensure that the absorption of carbon dioxide remains at its previous level. To study the long-term behavior of the formulated model system, we employ the qualitative theory of differential equations. We have derived sufficient conditions under which all the considered dynamical variables settle to their equilibrium level. After analyzing the formulated model system, we find that the system undergoes Hopf and transcritical bifurcations with respect to the deforestation rate. Furthermore, numerical simulations have been carried out to support the analytically obtained results. Through simulation, we have determined the proportion of cleared land that should be utilized for the plantation of leafy trees to maintain the level of atmospheric CO2.

Dynamics of a Predator–Prey Model with Impulsive Diffusion and Transient/Nontransient Impulsive Harvesting

Harvesting is one of the ways for humans to realize economic interests, while unrestricted harvesting will lead to the extinction of populations. This paper proposes a predator–prey model with impulsive diffusion and transient/nontransient impulsive harvesting. In this model, we consider both impulsive harvesting and impulsive diffusion; additionally, predator and prey are harvested simultaneously. First, we obtain the subsystems of the system in prey extinction and predator extinction. We obtain the fixed points of the subsystems by the stroboscopic map theories of impulsive differential equations and analyze their stabilities. Further, we establish the globally asymptotically stable conditions for the prey/predator-extinction periodic solution and the trivial solution of the system, and then the sufficient conditions for the permanence of the system are given. We also perform several numerical simulations to substantiate our results. It is shown that the transient and nontransient impulsive harvesting have strong impacts on the persistence of the predator–prey model.

Complex Dynamics Analysis and Chaos Control of a Fractional-Order Three-Population Food Chain Model

The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey’s fear of the predator and its carry-over effects. This study examines the state evolution of fractional-order systems and compares their dynamic behavior with integer-order systems. By utilizing the Routh–Hurwitz condition and the stability theory of fractional differential equations, this paper establishes the local stability conditions of the model through the application of the Jacobi matrix and eigenvalue method. Furthermore, the conditions for the Hopf bifurcation generation are determined. Subsequently, chaos control techniques based on the Lyapunov stability theory are employed to stabilize the unstable trajectory at the equilibrium point. The theoretical findings are validated through numerical simulations. These results enhance our understanding of the stability properties and chaos control mechanisms in fractional-order three-population food chain models.

On the expressibility of the functions of the system x'(t) = A∙x(t), the eigenvalues of the matrix of which are non-multiple in the form of linear combinations of derivatives of one function included in this system

The problem of solving a system of linear ordinary differential equations with constant coefficients is one of the most important problems in both the theory of ordinary differential equations and linear algebra. Therefore, on the one hand, new methods and algorithms are being developed for such systems, and on the other hand, existing methods and algorithms for solving such systems are being improved. One of the most well-known methods for solving a system of linear ordinary differential equations with constant coefficients is the method of reducing a system of linear equations to a single higher-order equation, which makes it possible to find solutions to the original system in the form of linear combinations of derivatives of only one unknown function.In this paper, we consider a refinement of the method for reducing a system of linear ordinary differential equations with constant coefficients to a single higher-order equation, which makes it possible to find a general solution to the original system; namely, we study the expressibility of all functions of the system of linear homogeneous differential equations with constant coefficients x^' (t)= A⋅x(t) in the form of linear combinations of derivatives of only one unknown function x_k (t), which is part of this system. For any matrix A, all of whose eigenvalues are not multiples, a new simple criterion for expressibility in terms of matrix ranks is formulated, and its correctness is proved. The result obtained can also be applied in the study of solutions of the system x^' (t)= A⋅x(t) for periodicity and in the study of linear systems for complete observability.

Stability and Optimal Control of a Fractional SEQIR Epidemic Model with Saturated Incidence Rate

The fractional differential equation has a memory property and is suitable for biomathematical modeling. In this paper, a fractional SEQIR epidemic model with saturated incidence and vaccination is constructed. Firstly, for the deterministic fractional system, the threshold conditions for the local and global asymptotic stability of the equilibrium point are obtained by using the stability theory of the fractional differential equation. If R0<1, the disease-free equilibrium is asymptotically stable, and the disease is extinct; when R0>1, the endemic equilibrium is asymptotically stable and the disease persists. Secondly, for the stochastic system of integer order, the stochastic stability near the positive equilibrium point is discussed. The results show that if the intensity of environmental noise is small enough, the system is stochastic stable, and the disease will persist. Thirdly, the control variables are coupled into the fractional differential equation to obtain the fractional control system, the objective function is constructed, and the optimal control solution is obtained by using the maximum principle. Finally, the correctness of the theoretical derivation is verified by numerical simulation.

Development and justification of the system methodological approach to assessing the investment business project implementation efficiency under conditions of the external market environment factors impact

Evaluating the effectiveness of the implementation of an investment project is a key issue when making management decisions both at the stage of setting up a startup and for expanding an existing business. This paper reports a systematic approach to building a mathematical model to solve the task of forecasting the effectiveness of business projects, taking into account the influence of factors of the external economic environment. Proposed factors include the impact of supply and demand on the price of goods, political and industry risks, the volume of commodity supply and sales. In view of this, a method for calculating the political component of the discount coefficient using the Fourier series has been proposed. Using the theory of differential equations, correlation and regression analysis, a mathematical model for forecasting indicators of efficiency of business project implementation taking into account the influence of factors of the external economic environment has been constructed. Based on it, a generalized algorithm for applying a mathematical model to predict the effectiveness of investment projects in various business sectors has been developed. The results from applying differential equations and variable discount coefficient showed a decrease in NPV by 14 %, and PI by 5.1 %, due to more accurate consideration of the political component in calculating the discount factor. Also, with the influence of supply and demand on the price of goods and nonlinear cash flows, it was found that the payback period does not clearly indicate the effectiveness of the implementation of an investment business project. Determining these factors provides more accurate information to the investor or business owner when forecasting the stability of a business project for making management decisions on its implementation

Measures to sustain endangered languages: A bilingual competition model with sliding mode control.

There are thousands of languages in the world, many of which are in danger of extinction due to language competition and evolution. Language is an aspect of culture, the rise, and fall of a language directly affects its corresponding culture. To preserve languages and prevent their mass extinction, it is crucial to develop a mathematical model of language coexistence. In this paper, we use a qualitative theory of ordinary differential equations to analyze the bilingual competition model, and obtain the trivial and non-trivial solutions of the bilingual competition model without sliding mode control, then analyze the stability of solutions and prove that solutions of the model have positive invariance. In addition, to maintain linguistic diversity and prevent mass extinction of languages, we propose a novel bilingual competition model with sliding control. The bilingual competition model is analyzed by proposing a sliding control policy to obtain a pseudo-equilibrium point. Meanwhile, numerical simulations clearly illustrate the effectiveness of the sliding mode control strategy. The results show that the likelihood of successful language coexistence can be increased by changing the status of languages and the value of monolingual-bilingual interaction, provides theoretical analysis for the development of policies to prevent language extinction.

The cooperatability of the first-order multi-agent systems consisting of a leader and a follower with multiplicative noises under Markov switching topologies

We investigate the cooperatability of the first-order leader-following multi-agent systems consisting of a leader and a follower with multiplicative noises under Markov switching topologies. Each agent exhibits first-order linear dynamics, and there are multiplicative noises along with information exchange among the agents. What is more, the communication topologies are Markov switching topologies. By utilizing the stability theory of the stochastic differential equations with Markovian switching and the Markov chain theory, we establish the necessary and sufficient conditions for the cooperatability of the leader-following multi-agent systems. The conditions are outlined below: (ⅰ) The product of the system parameter and the square of multiplicative noise intensities should be less than 1/2; (ⅱ) The transition rate from the unconnected graph to the connected graph should be twice the system parameter; (ⅲ) The transition rate from the connected graph to the unconnected graph should be less than a constant that is related to the system parameter, the intensities of multiplicative noises, and the transition rate from the unconnected graph to the connected graph. Finally, the effectiveness of our control strategy is demonstrated by the population growth systems.

Kink–antikink interaction forces and bound states in a nonlinear Schrödinger model with quadratic and quartic dispersion

In the present work we explore the competition of quadratic and quartic dispersion in producing kink-like solitary waves in a model of the nonlinear Schrödinger type bearing cubic nonlinearity. We present the first 6 families of multikink solutions and explore their bifurcations as the strength of the quadratic dispersion is varied. We reveal a rich bifurcation structure for the system, connecting two-kink states with states involving 4-, as well as 6-kinks. The stability of all of these states is explored. For each family, we discuss a “lower branch” adhering to the energy landscape of the 2-kink states. We also, however, study in detail the “upper branches” bearing higher numbers of kinks. In addition to computing the stationary states and analyzing their stability within the partial differential equation model, we develop an effective particle ordinary differential equation theory that is shown to be surprisingly efficient in capturing the kink equilibria and normal (as well as unstable) modes. Finally, the results of the bifurcation analysis are corroborated by means of direct numerical simulations involving the excitation of the states in a targeted way in order to explore their instability-induced dynamics.

Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Abstract The oscillation theory of differential equations is an important branch of performance of differential equations, which is widely used in engineering control, vibration mechanics, mechanics, and industry. Therefore, the vibration performance of different parts has attracted people’s attention, and a lot of research work has been done. For a special class of delay differential equations - advanced piecewise continuous differential equations, the oscillation of numerical solution is discussed. The θ − method is used to discretize the equation, and the numerical method is obtained to keep the oscillation of the analytical solution of the equation, progressive conditions. At the same time, four different states of the dynamic behavior are discussed in detail for the analytical solution and the numerical solution respectively. Some numerical examples further verify the corresponding conclusions.

Study on the Multi-Point Boundary Value Problem for Second-Order Nonlinear Impulsive Integro-Differential Equation

In the field of biological control, there are a large number of systems that gradually evolve over a certain period of time. However, due to some natural or human intervention behavior, the system state will be subjected to some relatively short time interference, so that the system state changes in an instant. This sudden change of state makes the system not simply described by continuous or discrete dynamical systems, but by means of impulse dynamical systems. In this paper, the multi-point boundary value problem of a class of second-order nonlinear impulsive integral differential equations is studied on the basis of impulsive differential equation theory. The main results of this kind of equation are obtained by using the fixed point theorem of strict set contraction operator. Under certain assumptions, the existence of the solution of the equation is proved by constructing the operator on the special cone. Finally, combined with the practical application, the theory was applied to the stability prediction of biological ecosystem, and the correctness of the conclusion was verified.