First-order Nonlinear Differential Equations Research Articles

Asymptotic Stability of Periodic Solutions for a Nonlinear Neutral First-order Differential Equation with Functional Delay

We consider a nonlinear neutral differential equation and use the contraction mapping principle to show the asymptotic stability of the zero solution provided that 𝑄(𝑡, 0) = 𝑓(0) = 0.

DC operating points of Mott neuristor circuits

In 1960 Hewitt Crane conceptualized neuristors as electronic logic networks that could mimic action potential generation by biological neurons. In 2012 Mott memristors were presented as nano-scale electronic devices for physically constructing neuristor networks. Both the original Mott memristor model and the simplified model presented in 2020 are stiff nonlinear first-order ordinary differential equations (ODEs) that describe the voltage controlled threshold switching and current controlled negative differential resistance, attributed to a Mott insulator-to-metal phase transition. By design, a Mott neuristor contains two identical Mott memristors plus linear resistors and capacitors (no inductors), which enables alternating threshold switching by the memristors to generate periodic (AC) output voltage spikes in response to a DC current or voltage input. This paper presents a steady-state analysis of dynamic neuristor models, including four-, five-, and six-state variables (each of which is a system of first-order ODEs involving the states) based on the original phenomenological model of the Mott memristor. Specifically, it reveals the unique stable operating points that occur for applied currents or voltages below the amounts needed to induce resistive switching. These DC operating points represent the physically meaningful “OFF” states for the circuit when the neuristor does not output voltage spikes. Jacques Hadamard's mathematical criteria for a well-posed physical model (existence, uniqueness, continuity, stability) motivated the search for these DC operating points (OFF states), which are needed for efficiently initializing neuristor models before conducting dynamic neuristor simulations. The key result of our analysis is the neuristor steady-state eq. (29) which is useful for understanding neuristor OFF states as well as the role of negative-differential resistance (NDR) in the operation of the Mott memristor model and neuristor circuits. In addition to the steady-state analysis, three numerical examples of dynamic neuristor circuit operation are also given and MATLAB code for the dynamic four-state model is available for interested readers.

Sensitivity Analysis-Based Validation of the Modified NERA Model for Improved Performance

Marijuana is an illicit narcotic with multiple negative health effects as it continues to pose a severe danger to the health of people in emerging regions. Marijuana is spread through interactions between users and non-users. This study utilizes first-order non-linear ordinary differential equations to modify the Non-user, Experimental, Recreational, and Addicted (NERA) users’ model by adding a new class known as the hospitalized class of marijuana smokers. Moreover, this paper focuses on validating the NERA model using the sensitivity analysis concept. The NERA model presented in a previous study for marijuana usage among the target population consists of four distinct stages: the first stage is known as the non-user stage and is represented by the letter N; the second stage is known as the experimental stage of marijuana smoking and is represented by the letter E; the third stage is known as the recreational stage of marijuana smoking and is represented by the letter R; the fourth stage is represented by the letter A; and this last stage is known as the addicted stage of marijuana smoking. The suggested mathematical method achieved the reproduction number () for marijuana use. The results of the sensitivity tests indicate how crucial a variety of variables are to the spread of marijuana. The parameters that have a key impact on marijuana consumption were discovered based on sensitivity analysis. All these procedures, including the reproductive number and invariant region, using the sensitivity analysis concept were utilized to validate the updated model. In this study, the NERA model is modified by utilizing first-order non-linear ordinary differential equations. Furthermore, as demonstrated by the graphical and technique sections, the proposed modified models were successfully validated for improved performance.

Dynamical Modelling of a Centrifugal Fan Driven by an Induction Motor and Experimental Validation

The application of electrical drives in the control of centrifugal fans and pumps is a well-established area that leads to substantial energy savings. It requires electrical and automation engineers to have some knowledge relevant to drives about fan and pump modelling since ignoring or oversimplifying fan/pump modelling for the intended drive design compromises the required control quality. This paper improves the existing dynamical models of fans and pumps integrated with induction motors via neural network estimation of overall fan/pump efficiency; this estimation is based on the voltage frequency of the driving induction motor, pressure, and flow rate, followed by the separation of the fan’s or pump’s own efficiency for the motor load torque computation, enhancing the accuracy due to the correct reflection of the power balance. The model is developed with a view to being convenient for control design applications. For the first time for dynamical models, it is verified experimentally, justifying the necessity of the first-order nonlinear differential equation for the flow rate. The validation includes a direct approach based on analysis of transient behaviour caused by a small-step perturbation of the frequency of the motor voltages and an indirect approach based on the introduced concept of the dynamical flow rate and pressure estimators.

APPLICATION OF NON-STANDARD FINITE DIFFERENCE METHOD ON COVID-19 MATHEMATICAL MODEL WITH FEAR OF INFECTION

This study presents a novel application of Non -Standard Finite Difference (NSFD) Method to solve a COVID-19 epidemic mathematical model with the impact of fear due to infection. The mathematical model is governed by a system of first-order non-linear ordinary differential equations and is shown to possess a unique positive solution that is bounded. The proposed numerical scheme is used to obtain an approximate solution for the COVID-19 model. Graphical results were displayed to show that the solution obtained by NSFD agrees well with those obtained by the Runge-Kutta-Fehlberg method built-in Maple 18.

Extended Runge-Kutta Scheme and Neural Network Approach for SEIR Epidemic Model with Convex Incidence Rate

For solving first-order linear and nonlinear differential equations, a new two-stage implicit–explicit approach is given. The scheme’s first stage, or predictor stage, is implicit, while the scheme’s second stage is explicit. The first stage of the proposed scheme is an extended form of the existing Runge–Kutta scheme. The scheme’s stability and consistency are also offered. In two phases, the technique achieves third-order accuracy. The method is applied to the SEIR epidemic model with a convex incidence rate. The local stability is also examined. The technique is evaluated compared to existing Euler and nonstandard finite difference methods. In terms of accuracy, the produced plots show that the suggested scheme outperforms the existing Euler and nonstandard finite difference methods. Furthermore, a neural network technique is being considered to map the relationship between time and the amount of susceptible, exposed, and infected people.

Ramanujan systems of Rankin–Cohen type and hyperbolic triangles

Abstract In the first part of the paper, we characterize certain systems of first-order nonlinear differential equations whose space of solutions is an s ⁢ l 2 ⁢ ( C ) \mathfrak{sl}_{2}(\mathbb{C}) -module. We prove that such systems, called Ramanujan systems of Rankin–Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin–Cohen structure. In the second part of the paper, we consider triangle groups Δ ⁢ ( n , m , ∞ ) \Delta(n,m,\infty) . By means of modular embeddings, we associate to every such group a number of systems of nonlinear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on Δ ⁢ ( n , m , ∞ ) \Delta(n,m,\infty) are generated by the solutions of one such system (which is of Rankin–Cohen type). As a corollary, we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non-classical setting, we construct the space of integral weight twisted modular form on Δ ⁢ ( 2 , 5 , ∞ ) \Delta(2,5,\infty) from solutions of systems of nonlinear ODEs.

A Method of Qualitative Analysis for Determining Monotonic Stability Regions of Particular Solutions of Differential Equations of Dynamic Systems

Developing stability analysis methods for modern dynamical system solutions has been a significant challenge in the field. This study aims to formulate a qualitative analysis approach for the monotone stability region of a specific solution to a single differential equation within a dynamical system. The system in question comprises two first-order nonlinear ordinary differential equations of a particular kind. The method proposed hinges on applying elements of combinatorics to the traditional mathematical investigation of a function with a single independent variable. This approach enables the exact determination of the different qualitative scenarios in which the desired solution changes, under the assumption that the function values monotonically diminish from a specified value down to a finite zero. This paper outlines the creation and decomposition of the monotone stability region associated with the solution under consideration.

Real-time adaptive sparse-identification-based predictive control of nonlinear processes

This study introduces a sparse identification-based model predictive control (MPC) framework that incorporates on-line updates of the sparse-identified model to account for nonlinear dynamics and model uncertainty in process systems. The methodology involves obtaining a nonlinear first-order ordinary differential equation model using sparse identification for nonlinear dynamics (SINDy), which is integrated into two control schemes: Lyapunov-based MPC (LMPC) for achieving steady-state operation and Lyapunov-based economic MPC (LEMPC) for achieving both closed-loop stability and optimal economic performance. To improve prediction accuracy, an on-line model update scheme is proposed for the SINDy models. Specifically, an error-trigger mechanism that utilizes prediction errors and then uses the most recent process data to update the parameters of the SINDy model in real-time is designed. By incorporating the error-triggered on-line model updates in the SINDy-based LMPC and LEMPC, the dynamic performance of the process is enhanced, ensuring closed-loop stability, optimality, and smooth control actions. Following theoretical results on the boundedness of the closed-loop states and detailed discussions on the selection criteria for parameters of the error-triggered SINDy update scheme, the effectiveness of the proposed methodology is demonstrated through a chemical process example with time-varying disturbances under the LEMPC framework.

Nonlinear dynamics of Kerr optical microresonators with spatially fluctuating loss

Dissipative soliton crystals (the so-called soliton combs) form in Kerr microresonators as a result of the competition between the group-velocity dispersion and the Kerr nonlinearity on one hand, and the balance of cavity loss by an external pump on the other hand. In some physical contexts, the loss can fluctuate within the microresonator cavity, inducing a saturable-absorption process which impacts the dynamics of the optical field. In this study, dissipative soliton crystals are investigated in a Kerr optical microresonator with spatially fluctuating loss. The underlying mathematical model consists of a modified Lugiato–Lefever equation with a space-dependent loss, coupled to a rate equation for the fluctuating loss. Adopting an ansatz that describes the optical-field envelope as a complex function of real amplitude and real phase with a characteristic modulation frequency, the mathematical model is reduced to a set of first-order nonlinear ordinary differential equations which are solved numerically. Simulations suggest that when the homogeneous cavity loss is small enough, the impact of loss fluctuation on the soliton-comb profile is rather moderate. The effect of loss fluctuations becomes noticeable when the homogeneous loss is sizable, with the recovery time of the induced saturable-absorption process being reasonably long to promote a slow saturable absorption. An analysis of the influence of the detuning on the amplitude and phase of the dissipative soliton crystal, as well as on the spatial variation of the loss for a fixed value of the characteristic frequency, is taken into consideration in the study.

Internal ring waves in a three-layer fluid on a current with a constant vertical shear

Oceanic internal waves often have curvilinear fronts and propagate over vertically sheared currents. We present the first study of long weakly-nonlinear internal ring waves in a three-layer fluid in the presence of a horizontally uniform background current with a constant vertical shear. The leading order of this theory leads to the angular adjustment equation—a nonlinear first-order ordinary differential equation describing the dependence of the linear long-wave speed on its angle to the direction of the current. The compact ring waves, well studied in the absence of a current, correspond to the singular solution (envelope of the general solution) of this equation, and they can exist only under certain conditions. The constructed solutions reveal qualitative differences in the shapes of the wavefronts of the two baroclinic modes: the wavefront of the faster mode is elongated in the direction of the current, while the wavefront of the slower mode is squeezed. Moreover, depending on the vorticity strength, several different regimes have been identified. When the vertical shear is weak, part of the wavefront is able to propagate upstream, while when the shear is strong enough, the whole wavefront propagates downstream. A richer pattern of behaviour is observed for the slower mode. As the shear increases, singularities of the swallowtail-type may arise and, eventually, solutions with compact wavefronts crossing the downstream axis cease to exist. We show that the latter is related to the long-wave instability of the base flow. We obtain the cKdV-type amplitude equation and examine analytical expressions for its coefficients. Using this cKdV-type equation we numerically model the evolution of the waves for both modes. The initial stage of the evolution is in agreement with the leading-order predictions for the deformations of the wavefronts. Then, as the wavefronts expand, strong dispersive effects occur in the upstream direction. Moreover, when nonlinearity is enhanced, fission of waves is observed in the upstream part of the ring waves.

Positive Periodic Solutions for a First-Order Nonlinear Neutral Differential Equation with Impulses on Time Scales

In this article, we discuss the existence of a positive periodic solution for a first-order nonlinear neutral differential equation with impulses on time scales. Based on the Leggett–Williams fixed-point theorem and Krasnoselskii’s fixed-point theorem, some sufficient conditions are established for the existence of positive periodic solution. An example is given to show the feasibility and application of the obtained results. Since periodic solutions are solutions with symmetry characteristics, the existence conditions for periodic solutions also imply symmetry.

Soliton trains induced by femtosecond laser filamentations in transparent materials with saturable nonlinearity

Femtosecond laser inscriptions in optical media current offer the most reliable optical technology for processing of transparent materials, among which is the laser micromachining technology. In this process, the nonlinearity of the transparent medium can be either intrinsic or induced by multiphoton ionization processes. In this work, a generic model is proposed to describe the dynamics of femtosecond laser inscription in transparent materials characterized by a saturable nonlinearity. The model takes into account multiphoton ionization processes that can induce an electron plasma of inhomogeneous density and electron diffusions. The mathematical model is represented by a one-dimensional complex Ginzburg–Landau equation with a generalized saturable nonlinearity term in addition to the residual nonlinearity related to multiphoton ionization processes, coupled to a rate equation for time evolution of the electron plasma density. Dynamical properties of the model are investigated focusing on the nonlinear regime, where the model equations are transformed into a set of coupled first-order nonlinear ordinary differential equations, which are solved numerically with the help of a sixth-order Runge–Kutta algorithm with a fixed time step. Simulations reveal that upon propagation, spatiotemporal profiles of the optical field and of the plasma density are periodic pulse trains, the repetition rates and amplitudes of which are increased with an increase of both the multiphoton ionization order and the saturable nonlinearity. When electron diffusions are taken into account, the system dynamics remains qualitatively unchanged; however, the electron plasma density gets strongly depleted, leaving almost unchanged the amplitude of pulses composing the femtosecond laser soliton crystals.

A Parametric Study of the Bouc–Wen Model for Bolted Joint Dynamics

AbstractBuilt-up structures exhibit nonlinear dynamic phenomena due to friction at the surfaces that are held together using mechanical fasteners. This nonlinearity is hysteretic or history dependent. Additionally, interfacial slip results in stiffness and damping variations that are dependent on the vibration amplitude. In the microslip regime, the dissipation varies as a power of the amplitude. The four-parameter Iwan model can capture both the hysteretic and power-law dissipation behavior that is characteristic of many bolted joints. However, simulating the dynamic response of this model is computationally expensive since the states of several slider elements must be tracked implicitly, necessitating the use of fixed-step integration schemes with small time-steps. The Bouc–Wen model is an alternative hysteretic model in which the restoring force is given by a first-order nonlinear differential equation. Numerical integration of this model is much faster because it consists of just one additional state variable, i.e., the hysteretic variable. Existing literature predominantly focuses on studying the steady-state behavior of this model. This paper tests the effectiveness of the Bouc–Wen model in capturing power-law dissipation by comparing it to four-parameter Iwan models with various parameters. Additionally, the effect of each Bouc–Wen parameter on the overall amplitude-dependent damping is presented. The results show that the Bouc–Wen model cannot capture power-law behavior over the entire microslip regime, but it can be tuned to simulate the response over a smaller amplitude range.

Data-based modeling and control of nonlinear process systems using sparse identification: An overview of recent results

This paper discusses recent developments in the data-based modeling and control of nonlinear chemical process systems using sparse identification of nonlinear dynamics (SINDy). SINDy is a recent nonlinear system identification technique that uses only measurement data to identify model dynamical systems in the form of first-order nonlinear differential equations. In this work, the challenges of handling time-scale multiplicities and noisy sensor data when using SINDy are addressed. Specifically, a brief overview of novel methods devised to overcome these challenges are described, along with modeling guidelines for using the proposed techniques for process systems. When applied to two-time-scale systems, to overcome model stiffness, which leads to ill-conditioned controllers, a reduced-order modeling approach is proposed where SINDy is used to model the slow dynamics, and nonlinear principal component analysis is used to algebraically “slave” the fast states to the slow states. The resulting model can then be used in a Lyapunov-based model predictive controller with guaranteed closed-loop stability provided the separation of fast and slow dynamics is sufficiently large. To handle high levels of sensor noise, SINDy is combined with subsampling and co-teaching to improve modeling accuracy. The challenges of modeling and controlling large-scale systems using noisy industrial data are then addressed by using ensemble learning with SINDy. After summarizing the advances, a nonlinear chemical process is used to provide an end-to-end demonstration of process modeling using sparse identification with guidelines for chemical engineering practitioners. Finally, several future research directions for the incorporation of SINDy into process systems engineering are proposed.

Mechanistic Modelling of Biomass Growth, Glucose Consumption and Ethanol Production by Kluyveromyces marxianus in Batch Fermentation.

This paper presents results concerning mechanistic modeling to describe the dynamics and interactions between biomass growth, glucose consumption and ethanol production in batch culture fermentation by Kluyveromyces marxianus (K. marxianus). The mathematical model was formulated based on the biological assumptions underlying each variable and is given by a set of three coupled nonlinear first-order Ordinary Differential Equations. The model has ten parameters, and their values were fitted from the experimental data of 17 K. marxianus strains by means of a computational algorithm design in Matlab. The latter allowed us to determine that seven of these parameters share the same value among all the strains, while three parameters concerning biomass maximum growth rate, and ethanol production due to biomass and glucose had specific values for each strain. These values are presented with their corresponding standard error and 95% confidence interval. The goodness of fit of our system was evaluated both qualitatively by in silico experimentation and quantitative by means of the coefficient of determination and the Akaike Information Criterion. Results regarding the fitting capabilities were compared with the classic model given by the logistic, Pirt, and Luedeking-Piret Equations. Further, nonlinear theories were applied to investigate local and global dynamics of the system, the Localization of Compact Invariant Sets Method was applied to determine the so-called localizing domain, i.e., lower and upper bounds for each variable; whilst Lyapunov's stability theories allowed to establish sufficient conditions to ensure asymptotic stability in the nonnegative octant, i.e., R+,03. Finally, the predictive ability of our mechanistic model was explored through several numerical simulations with expected results according to microbiology literature on batch fermentation.

A harmonic balance solution for the intrinsic 1D nonlinear equations of the beams

In this paper, a harmonic balance method is introduced to solve the intrinsic 1D nonlinear equations presented by Hodges for analyzing initially curved and twisted anisotropic rotating beams. First, the nonlinear first-order partial differential equations are discretized in space domain using the Galerkin approach. Then, the time response of the system of equations is approximated by using first and the second harmonic terms under the harmonic excitations applied to the structure. The harmonic balance algorithm has been developed for the linearized system of equations about the steady-state solution and for fully nonlinear equations. The differences between these two systems have been addressed under consideration of different levels of external loads. Fully analytical formulae have been presented for the HB solution of the linearized equations in this work. In the proposed approach for the nonlinear system, the Jacobian matrix utilized in the numerical solution part is obtained analytically to increase the time efficiency of the solution method. The results of the proposed approach for an initially twisted and curved blade are compared with a different algorithm based on a finite difference approach to discretize the equations and a time marching method to solve the system of equations in time domain. The comparison between the outputs of the harmonic balance method and the time marching method demonstrate high correlations between these distinct methods. It should be noted that the accuracy of the time marching method solution has been evaluated in another publication by the authors using the output of FLIGHTLAB for a blade with different types of boundary conditions. The proposed harmonic balance-based method delivers a very time-efficient approach for time analysis of twisted/curved beams and blades under harmonic loads in comparison with other employed methods. The developed approach can be a powerful computational tool for design optimization of composite blades.

Analytical approaches for growth models in economics

This study deals with the group-theoretical analysis of nonlinear optimal control problems called the optimal growth model with the environmental asset and capitalist decision model of endogenous growth, which are expressed in terms of the current and present value Hamiltonian functions. The first-order conditions for optimal control problems based on Pontryagin’s maximum principle are considered. In addition, Lie point symmetries and their relation with the Prelle-Singer, λ-symmetries, adjoint symmetries, Darboux polynomials, and Jacobi’s last multiplier approaches are studied by analyzing the coupled nonlinear first-order ordinary differential equations corresponding to the first-order conditions. For both current and present Hamiltonian cases, the first integrals and the corresponding analytical (closed-form) solutions and graphical representations for the optimal control problems are represented.

Orbital Decay in the Classroom

The objective of this paper is to provide a pedagogical framework for the phenomenon of orbital decay of satellites in low Earth orbit. The dynamics of orbital decay are derived considering atmospheric drag as the only dissipative mechanism and using an educational approach suitable for undergraduate calculus-based physics and engineering courses. The resulting nonlinear first-order differential equation for the altitude as a function of time is solved numerically for the isothermal-barotropic atmospheric model with a fixed scale height. The model is validated using the uncontrolled reentry data of the Chinese space station Tiangong-1.

Investigation of the self-propulsion of a wetting/nonwetting ganglion in tapered capillaries with arbitrary viscosity and density contrasts

The movement of a meniscus inside a capillary tube has been extensively studied in the context of displacing one fluid with another immiscible one. This phenomenon exists in many applications including pharmaceutical, oil production, filtration and separation processes, and others. When one of the phases is entrapped inside a capillary tube, it forms what is called a ganglion with two menisci between the two fluids. In a straight uniform capillary tube, a stagnant entrapped ganglion is symmetric. The situation is different if the capillary tube is tapered in which case the two menisci assume different curvatures. Such inhomogeneity of the capillary pressure self-propels the ganglion to move. The fate of the ganglion inside the tapered tube depends on whether it is wetting or nonwetting to the tube wall. That is, after the initial movement, a wetting ganglion accelerates towards the tapered end of the tube while a nonwetting one decelerates towards the wider end before reaching a terminal configuration. Such fates are linked to the variations of the capillary pressure, which continuously increases for a wetting ganglion and decreases for the nonwetting one. In this work, a generalized model is developed that not only describes capillary-driven dynamics over a wide range of viscosity and density contrasts but also pressure-driven scenarios with/without gravity. The model, however, neglects the inertial effect of the two fluids on account of the fact that it is confined to the very early time of the movement process. A first-order nonlinear ordinary differential equation is developed that describes the dynamic behavior of both the wetting and nonwetting ganglions. A fourth-order Runge-Kutta algorithm is developed to solve the model equations. Furthermore, a computational fluid dynamics (CFD) analysis was used to provide a comparison and verification framework.