Theory Of Ordinary Differential Equations Research Articles

The non-exponential decay characteristics of HTS NI-class coils in sudden discharge experiments

Abstract The decay of the voltage and magnetic field of the high-temperature superconductivity (HTS) no-insulation-class (NI-class) coil in the sudden discharge experiment is usually considered to be exponential. However, our experiment found that it is not accurate to describe the decay characteristics of the NI-class coil with a simple exponential function. Combined with equivalent circuit models of the NI-class coil, we use the theory of ordinary differential equations to explain the non-exponential decay characteristics and point out that it is closer to the matrix exponential characteristics. Based on this finding, we believe that the non-uniform turn-to-turn contact resistivity (TTCR) is an important cause of the non-exponential decay characteristics of the NI-class coil. Reverse thinking leads us to realize that the voltage decay curve of the NI-class coil in the sudden discharge experiment contains the distribution information of its TTCR. Therefore, we propose a method for measuring the TTCR of NI-class coils based on the matrix exponential model. The co-simulation combining the finite element model and the equivalent circuit model shows that the non-uniformly distributed TTCR has a great influence on the transient process of the NI-class coil, which makes the current and current density in the transient process vary greatly in different regions of the NI-class coil. Those findings mean that the simulation study of the NI-class coils and magnets with uniform TTCR usually used in the past may be far from the actual situation.

Group Classification of Second-Order Linear Neutral Differential Equations

In this paper, we shall extend the method of obtaining symmetries of ordinary differential equations to second-order non-homogeneous functional differential equations with variable coefficients. The existing research for delay differential equations defines a Lie-Bäcklund operator and uses the invariant manifold theorem to obtain the infinitesimal generators of the Lie group. However, we shall use a different approach that requires Taylor’s theorem for a function of several variables to obtain a Lie invariance condition and the determining equations for second-order functional differential equations. Certain standard results from the theory of ordinary differential equations have been employed to simplify the equation under study. The symmetry analysis of this equation was found to be non-trivial for arbitrary variable coefficients. In such cases, by selecting certain specific functions, arising in most practical models, we find the symmetries that are seen to be in terms of Bessel’s functions, Mathieu functions, etc. We then make a complete group classification of the second-order linear neutral differential equation, for which there is no existing literature.

Dynamical properties in the axiomatic theory of ordinary differential equations

The axiomatic theory of ordinary differential equations, owing to its simplicity, can provide a useful framework to describe various generalizations of dynamical systems. In this study, we consider how dynamical properties can be generalized to this setting. First, we study the generalizations of dynamical properties, such as invariance and limit sets and show that compact weakly invariant sets can be characterized in terms of the corresponding shift-invariant sets in the function space. This result enables us to consider weakly invariant sets in terms of shift-invariant sets. Next, we compare the axiomatic theory of ODE with other formalisms of generalized dynamical systems, namely generalized semiflow and multivalued semigroup. Finally, we introduce the notion of invariant measures and show that we can generalize classical results, such as Poincaré recurrence theorem.

Approximate Damped Oscillatory Traveling Wave in Viscous Compressible Fluid With Capillary Effect

We convert the viscous compressible fluid equations with capillary effect into the equivalent planar dynamical system and illustrate the global qualitative phase diagrams corresponding to its bounded orbits by the qualitative theory of ordinary differential equations. From the evolutionary relationship of bounded orbits in global qualitative phase diagrams and the correspondence between the bounded orbit and traveling wave, we discover the evolutionary rule of the profile of the traveling wave as the viscous coefficient varying and the forming reason for the damped oscillatory traveling wave. Further, we construct the analytical approximate explicit expression of the damped oscillatory traveling waves in accordance with the forming reason and prove its error decreases in exponential form. Finally, we compare the approximate damped oscillatory traveling wave with the numerical solution. The result shows they fit well.

Mathematical Modeling of the Transmission Dynamics of Gumboro Disease

Gumboro disease is a viral poultry disease that causes immune suppression on the infected birds leading to poor production, mortality, and exposure to secondary infections, hence a major threat in the poultry industry worldwide. A mathematical model of the transmission dynamics of Gumboro disease is developed in this paper having four compartments of chicken population and one compartment of Gumboro pathogen population. The basic reproduction number Rog is derived, and the dynamical behaviors of both the disease‐free equilibrium (DFE) and endemic equilibrium are analyzed using the ordinary differential equation theory. From the analysis, we found that the system exhibits an asymptotic stable DFE whenever Rog < 1 and an asymptotic stable EE whenever Rog > 1. The numerical simulation to verify the theoretical results was carried out using MATLAB ode45 solver, and the results were found to be consistent with the theoretical findings.

Properties of the Wrońskian determinant of a system of solutions to a linear homogeneous equation: The case when the number of solutions is less than the order of the equation

Objectives. The work sets out to study the properties of the Wrońskian determinant of the system of solutions to a linear homogeneous equation in cases when the number of solutions is less than the order of the equation, comparing them with the known properties of the same determinant when the number of solutions is equal to the order of the equation.Methods. The work uses the methods of linear algebra according to the theory of ordinary differential equations, as well as mathematical and complex analysis.Results. It is shown that the vanishing of a considered determinant on an arbitrarily small interval implies its vanishing on the entire domain of definition; the solutions turn out to be linearly dependent. A stronger result is obtained in three cases: (1) if the coefficients of the equation are analytic functions; (2) if the number of solutions is equal to one; (3) if the number of solutions is one less than the order of the equation. Namely, if the set of zeros of the considered Wrońskian has a limit point belonging to the domain of definition of solutions, then the determinant is identically equal to zero and the solutions are linearly dependent.Conclusions. According to the obtained results, the Wrońskian of a system of solutions of a linear homogeneous equation can serve as an indicator of the linear dependence or independence of this system in cases where the number of solutions is lower than the order of the equation; here, the solutions are linearly dependent if and only if their Wrońskian is identically equal to zero. In this case, there is no need to check whether the determinant vanishes over the entire domain of definition, since it is sufficient to do this on an arbitrarily chosen interval or even (in the special cases listed above) on an arbitrarily chosen set having a limit point.

On the periodic and antiperiodic aspects of the Floquet normal form

Floquet's Theorem is a celebrated result in the theory of ordinary differential equations. Essentially, the theorem states that, when studying a linear differential system with T-periodic coefficients, we can apply a, possibly complex, T-periodic change of variables that transforms it into a linear system with constant coefficients. In this paper, we explore further the question of the nature of this change of variables. We state necessary and sufficient conditions for it to be real and T-periodic. Failing those conditions, we prove that we can still find a real change of variables that is “partially” T-periodic and “partially” T-antiperiodic. We also present applications of this new form of Floquet's Theorem to the study of the behavior of solutions of nonlinear differential systems near periodic orbits.

Dynamical analysis of a Lotka–Volterra competition model with both Allee and fear effects

Population ecology theory is replete with density-dependent processes. However, trait-mediated or behavioral indirect interactions can both reinforce or oppose density-dependent effects. This paper presents the first two species competitive ODE and PDE systems, where the non-consumptive behavioral fear effect and the Allee effect, a density-dependent process, are both present. The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations. It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points, but they do not affect the stability of the positive equilibria. We also observe standard co-dimension one bifurcation in the system by varying the Allee or fear parameter. Interestingly, we find that the Allee effect working in conjunction with the fear effect can bring about several dynamical changes to the system with only fear. There are three parametric regimes of interest in the fear parameter. For small and intermediate amounts of fear, the Allee + fear effect opposes dynamics driven by the fear effect. However, for large amounts of fear the Allee + fear effect reinforces the dynamics driven by the fear effect. The analysis of the corresponding spatially explicit model is also presented. To this end, the comparison principle for parabolic PDE is used. The conclusions of this paper have strong implications for conservation biology, biological control as well as the preservation of biodiversity.

Modelling the Impact of Decomposed Disease-Induced Dead Cashew Plants on Fusarium Wilt Dynamics in South-Eastern Tanzania

Cashew Fusarium wilt disease transmitted by Fusarium oxysporum is dangerous and destroys many cashew plants in Tanzania. The effect of this disease leads households and the government to experience a decrease in overall cashew production and income due to this disease’s capacity to harm cashew plants. This study aimed to ascertain the role of Fusarium wilt-decayed disease-induced dead plants in spreading the disease among cashew plants. A mathematical model was created based on the dynamics of the disease and a stability analysis was conducted using theories of ordinary differential equations. Data from two regions in Tanzania, Lindi, and Mtwara, were used in model fitting and parameter estimation. Additionally, the parameters were estimated using maximum likelihood estimation (MLE). The outcome suggests that the disease’s prevalence and spread increase during an outbreak as decomposed diseased-induced dead plants contribute to the saturation of chlamydospores in the soil.

A Mathematical Model of the Dynamics of Coffee Berry Disease

Coffee berry disease (CBD) is a fungal disease caused by Colletotrichum kahawae. CBD is a major constraint to coffee production to Kenya and Africa at large. In this research paper, we formulate a mathematical model of the dynamics of the coffee berry disease. The model consists of coffee plant population in a plantation and Colletotrichum kahawae pathogen population. We derived the basic reproduction number R k 0 , and analyzed the dynamical behaviors of both disease-free equilibrium and endemic equilibrium by the theory of ordinary differential equations. Using the MATLAB ode45 solver, we carried out numerical simulation, and the findings are consistent with the theoretical results.

Formulation and mathematical analysis of a three-step model of anaerobic digestion with inhibition by ammonia

In this work, we are interested in the construction and mathematical analysis of a three-step anaerobic digestion model (Hydrolysis–Acidogenesis–Methanogenesis). Our aim is to investigate the qualitative properties of the system. In addition to the hypotheses used for formulating the well-known baseline two-step anaerobic model, we take into account the inhibition by ammonia together with the decay rate of acidogenic and methanogenic microorganisms, and consider the hydrolysis step to determine its effect on the growth of the methanogens. By so doing, we obtain a system of six ordinary differential equations describing the whole dynamics of the aforementioned phenomenon. As far as the mathematical analysis of this model is concerned, we first prove the existence of a unique positive and bounded global solution. Then we derive local and global stability results. The proofs of the main results rely essentially on some tools from the theory of ordinary differential equations. Finally, by performing some numerical simulations, we illustrate the theoretical results obtained and highlight the key parameters acting on the model’s behavior.

Stability of the Epidemiological Model SIR with Loss of Immunity

This study approaches the analysis of the stability of the epidemiological model SIR with loss of immunity. This is a model given by a system of ordinary differential equations. Initially, we present the model and its interpretation. Then we define the constants and elements that compose the model, so we present the results obtained using the qualitative theory of ordinary differential equations, especially the theory of planar systems related to the dynamics of fixed points. Finally, we show that the system representing the SIR model is globally stable and they have two types of dynamic that {depend on model constants}, and their meaning for epidemiology.

Технология построения имитационных моделей силовых трансформаторов с произвольной конструкцией активной части

Existing simulation models of power transformers that operate, for example, in the MATLAM Simulink SimPowerSystem environment, are based on an ideal transformer model. It allows using this model as an autonomous link in a complex electrical circuit. The main disadvantage of such models is the difficulty to consider non-standard versions of the magnetic circuit and special schemes for connecting winding elements when modeling. Particularly significant problems arise when modeling special transformers of the certain classes. At the same time, traditionally, when modeling the transient modes of transformers, a different approach is used. It is based on the use of inductance matrices which allows considering all the design features of the active part of the transformer. The disadvantage of this approach is the need to describe an external electrical circuit in addition to the transformer. Therefore, the problem to develop a modern technology for simulation models of transformers with an arbitrary design of the active part is topical. This model meets the requirement for the autonomy of the transformer model from the external circuit model. The authors have used the methods of modeling electrical and magnetic circuits based on the theory of ordinary differential equations, and simulation method using the MatLab Simulink SimPowerSystems package. A technology has been developed for simulation models of transformers with an arbitrary design of the active part based on the use of typical subsystems. An algorithm for matrix of inductances based on the main magnetic field of the transformer with the existing equivalent circuit of the magnetic circuit is given. The authors have presented a diagram of a three-phase transformer model developed using the proposed technology, as well as the results of comparing the current curves in the primary and secondary windings of the transformer when it is turned on at idle and with a resistive load. The results are obtained using the existing and new models. The results of the study can be used to design general industrial and special transformers in design companies and in manufacturing environment. The developed technology can give a special effect when it is used at the R&D stage to study the operating modes of transformers in case the enterprise has no experience to design and manufacture them.

Mathematical analysis of epidemic model to assess the impact of lockdown on COVID-19

Covid-19 and its variants, have been a worst pandemic, the entire world has witnessed. Tens of millions of cases have been recorded in over 210 countries and territories as part of the ongoing global pandemic that is still going on today. In this paper, we propose a SEI mathematical model to investigate the impact of lockdown to the controlling and spreading of infectious disease COVID-19. The epidemic model incorporates constant recruitment, experiencing infectious force in the latent period and the infected period. The equilibrium states are computed. Under some conditions, results for local asymptotic stability and global stability of disease-free and endemic equilibrium are established by using the stability theory of ordinary differential equations. It is seen that when the basic reproduction number , the dynamical system is stable and diseases die out from the system and when , the disease persists in the dynamical system. When , trans critical bifurcation is appeared. The numerical simulations are carried out to validate the analytical results.

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.

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.

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.

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.

Floquet Theory for Linear Time-Periodic Delay Differential Equations Using Orthonormal History Functions

Abstract In the usual approach to determining the stability of a time-periodic delay differential equation (DDE), the DDE is converted into an approximate system of time-periodic ordinary differential equations (ODEs) using Galerkin approximations. Later, Floquet theory is applied to these ODEs. Alternatively, semidiscretization-like approaches can be used to construct an approximate Floquet transition matrix (FTM) for a DDE. In this paper, we develop a method to obtain the FTM directly. Our approach is analogous to the Floquet theory for ODEs: we consider one polynomial basis function at a time as the history function and stack the coefficients of the corresponding DDE solutions to construct the FTM. The largest magnitude eigenvalue of the FTM determines the stability of the DDE. Since the obtained FTM is an approximation of the actual infinite-dimensional FTM, the criterion developed for stability is approximate. We demonstrate the correctness, efficacy and convergence of our method by studying several candidate DDEs with time-periodic parameters and/or delays, and comparing the results with those obtained from the Galerkin approximations.

A class of natural pinus koraiensis population system with time delay and diffusion term

In this paper, we consider the long-term sustainability of the northeast Korean pine. We propose a class of natural Korean pine population system with time delay and diffusion term. First, by analyzing the roots distribution of the characteristic equation, we study the stability of the model system with diffusion terms and prove the occurrence of Hopf bifurcation. Second, we introduce lactation time delay into a population model with a diffusion term, based on stability theory of ordinary differential equation, norm form methods and center manifold theorem, the stability of bifurcating periodic solutions and the relevant formula for the direction of Hopf bifurcation are given. Finally, some numerical simulations are given.