Nonautonomous Differential Equations Research Articles

On Oscillations of Nonautonomous Linear Impulsive Differential Equations with General Piecewise Constant Deviating Arguments

On Oscillations of Nonautonomous Linear Impulsive Differential Equations with General Piecewise Constant Deviating Arguments

An optimal control for non-autonomous second-order stochastic differential equations with delayed arguments

Abstract Optimal control of non-autonomous second-order stochastic differential equations with delayed arguments is indispensable for managing systems exposed to uncertainty, time-dependent dynamics, and historical influences. These equations underpin a wide range of applications, including finance, engineering, and biology, where it’s imperative to make informed decisions that mitigate risks or maximize returns while considering the inherent randomness, evolving conditions, and the impact of past states. By employing optimal control techniques, we can devise strategies that are resilient to uncertainty, adaptable to changing circumstances, and capable of accounting for the memory effects of previous events. This empowers us to optimize system performance, bolster stability, and attain desired objectives in intricate and dynamic environments. So, the goal of this article is to introduce a novel model of second-order perturbed stochastic differential equations incorporating non-local finite delay and deviated arguments in the setting of Hilbert spaces. Moreover, essential criteria are presented to examine the existence of a mild solution and evaluate the potential for approximate and optimal control of the proposed system. These results have been obtained by using evolution operators, fixed point techniques, random analytic methods, and compact semigroup theory. Further, to support the theoretical results, the optimal controllability of our model was studied by considering the Lagrange problem. Finally, the results were applied to discuss the approximate controllability of a partial differential equation. These models have the potential to advance the understanding and application of optimal control techniques for a wider range of complex systems.

Best Ulam constants for damped linear oscillators with variable coefficients

An associated Riccati equation is used to study the Ulam stability of non-autonomous linear differential equations that model the damped linear oscillator. In particular, the best (minimal) Ulam constants for these equations are derived. These robust results apply to equations with solutions that blow up in finite time, as well as to equations with solutions that exist globally on (−∞,∞). Illustrative, non-trivial examples are presented, highlighting the main results.

On the convergence order of the Euler scheme for scalar SDEs with Hölder-type diffusion coefficients

We study the Euler scheme for scalar non-autonomous stochastic differential equations, whose diffusion coefficient is not globally Lipschitz but a fractional power of a globally Lipschitz function. We analyse the strong error and establish a criterion, which relates the convergence order of the Euler scheme to an inverse moment condition for the diffusion coefficient. Our result in particular applies to Cox-Ingersoll-Ross-, Chan-Karolyi-Longstaff-Sanders- or Wright-Fisher-type stochastic differential equations and thus provides a unifying framework.

Skew-product attractors of non-autonomous Caputo fractional differential equations.

A non-autonomous Caputo fractional differential equation (FDE) of order α∈(0,1) in Rd with a driving system on a compact base space P is shown to generate a skew-product semi-flow on Cα×P, where Cα is the space of continuous functions f:R+→Rd with a weighted norm giving uniform convergence on compact time subsets. This skew-product semi-flow is then shown to have a bounded and closed attractor when the vector field of the Caputo FDE satisfies a uniform dissipativity condition. It attracts bounded sets of constant initial functions f in here Cα. The properties and structure of this attractor in Cα×P are also discussed.

Weakly non-linear stability analysis of g-jitter induced Rayleigh–Bénard convection under inclined surfaces

This study investigated the effect of g-jitter on the Rayleigh–Bénard convection between inclined surfaces whose lower layer is concentrated with a solute C. The problem is formulated using the Ginzburg–Landau (GL) equation, which is time-dependent non-autonomous ordinary differential equation and derived from a weakly non-linear theory and Fredholm solvability condition. The graphical solution of the GL equation is obtained with the help of MATHEMATICA by using the Runge–Kutta fourth-order method. The impact of various flow parameters on the convective rate of heat and mass transport has been discussed. The heat and mass transport is found to be advanced by the non-dimensional parameters viz. modulations’ amplitude (δ), Lewis number (Le), and solute Rayleigh number (RC), but it is suppressed by the angle of inclination ϕ.

Nonautonomous Volterra Series Expansion of the Variable Phase Approximation and its Application to the Nucleon-Nucleon Inverse Scattering Problem

Abstract In this paper, the nonlinear Volterra series expansion is extended and used to describe certain types of nonautonomous differential equations related to the inverse scattering problem in nuclear physics. The nonautonomous Volterra series expansion lets us determine a dynamic, polynomial approximation of the variable phase approximation (VPA), which is used to determine the phase shifts from nuclear potentials through first-order nonlinear differential equations. By using the first-order Volterra expansion, a robust approximation is formulated to the inverse scattering problem for weak potentials and/or high energies. The method is then extended with the help of radial basis function neural networks by applying a nonlinear transformation on the measured phase shifts to be able to model the scattering system with a linear approximation given by the first-order Volterra expansion. The method is applied to describe the ${}^1S_0$ NN potentials in neutron+proton scattering below 200 MeV laboratory kinetic energies, giving physically sensible potentials and below $1\%$ averaged relative error between the recalculated and the measured phase shifts.

Generalized periodicity and applications to logistic growth

Classically, a continuous function f:R→R is periodic if there exists an ω>0 such that f(t+ω)=f(t) for all t∈R. The extension of this precise definition to functions f:Z→R is straightforward. However, in the so-called quantum case, where f:qN0→R (q>1), or more general isolated time scales, a different definition of periodicity is needed. A recently introduced definition of periodicity for such general isolated time scales, including the quantum calculus, not only addressed this gap but also inspired this work. We now return to the continuous case and present the concept of ν-periodicity that connects these different formulations of periodicity for general discrete time domains with the continuous domain. Our definition of ν-periodicity preserves crucial translation invariant properties of integrals over ν-periodic functions and, for ν(t)=t+ω, ν-periodicity is equivalent to the classical periodicity condition with period ω. We use the classification of ν-periodic functions to discuss the existence and uniqueness of ν-periodic solutions to linear homogeneous and nonhomogeneous differential equations. If ν(t)=t+ω, our results coincide with the results known for periodic differential equations. By using our concept of ν-periodicity, we gain new insights into the classes of solutions to linear nonautonomous differential equations. We also investigate the existence, uniqueness, and global stability of ν-periodic solutions to the nonlinear logistic model and apply it to generalize the Cushing–Henson conjectures, originally formulated for the discrete Beverton–Holt model.

Periodic Solutions of a Second Order Discontinuous Nonautonomous Differential Equation

Periodic Solutions of a Second Order Discontinuous Nonautonomous Differential Equation

Uniform stability, boundedness and square integrability for non-autonomous third-order neutral differential equations with delay

Uniform stability, boundedness and square integrability for non-autonomous third-order neutral differential equations with delay

Study of Heat and Mass transports due to modulated rotational flow in a triply diffusive Newtonian liquid saturated porous medium/fluid layer: A comparative study

Rotational modulation and the third diffusive component are investigated in relation to transports in the Rayleigh Bènard convection in the x− direction of an infinitely extended fluid layer or fluid-saturated porous layer using a weakly non-linear theory. The direction of travel for the z-axis is upward. The z-axis is interpreted as pointing upward. Temperature and concentration of both the solutes at lower plate are higher than the upper plate. The time dependent rotational speed of fluid is assumed to be modulated by small parameter ϵ2 with amplitude (δ). A linear differential matrix approach is used to examine the system’s behaviour. Ginzburg–Landau’s equation, a non-autonomous differential equation, has been derived using the Fredholm-solvability condition. The impact of dimensionless parameters on stationary convection is also discussed. The expression of the critical Rayleigh number is obtained for three different cases: clear fluid layer, Darcy porous layer, and sparsely packed porous layer (Darcy–Brinkman model). The Nusselt number (Nu) and Sherwood numbers (Sh1, Sh2) for each solute are used to quantify the convective transports. The impact of modulation and other dimensionless parameters on these convective transports has been discussed using graphical representation Furthermore, among the three models under consideration, it is discovered that the heat and mass transports are highest in the fluid layer and minimum in the densely packed porous layer.

On the Boundedness of Solutions of a Non-autonomous Differential Equation of Second Order

We study the boundedness of the solutions to a non-autonomous differential equation of second order. With this work, we improve some stability results in the literature of boundedness of the solutions. We give two examples to illustrate the theoretical analysis in this work. 2000 Mathematics Subject Classification. 34C11, 34D20

On the stability in variation of non-autonomous differential equations with perturbations

In this paper, we investigate the problem of stability in variation of solutions for nonautonomous differential equations. Some new sufficient conditions for the asymptotic or exponential stability for some classes of nonlinear time-varying differential equations are presented by using Lyapunov functions that are not necessarily smooth. The proposed approach for stability analysis is based on the determination of the bounds that characterize the asymptotic convergence of the solutions to a certain closed set containing the origin. Furthermore, some illustrative examples are given to prove the validity of the main results.

Stabilization and destabilisation of non-autonomous stochastic nonlinear delay differential equations

This paper focuses on a class of non-autonomous stochastic nonlinear delay differential equations, where the time delay functions are no longer required to be bounded or differentiable. The key aim is to study the stabilisation and destabilisation of the non-autonomous nonlinear stochastic delay differential equations by using the function ln ⁡ | x ( t ) | 2 , the improved LaSalle-type theorem, and non-negative semimartingale convergence theorem. In comparison with recent works on stabilisation for stochastic time-varying delay nonlinear systems (see, e.g. [Dong, H., & Mao, X. (2022). Advances in stabilization of highly nonlinear hybrid delay systems. Automatica, 136, 110086. https://doi.org/10.1016/j.automatica.2021.110086; Hu, J., Mao,W., & Mao, X. (2023). Advances in nonlinear hybrid stochastic differential delay equations: Existence, boundness and stability. Automatica, 147, 110682. https://doi.org/10.1016/j.automatica.2022.110682; Xu, H., & Mao, X. (2023). Improved delay-dependent stability of superlinear hybrid stochastic systems with general time-varying delays. Nonlinear Analysis: Hybrid Systems, 50, 101413. https://doi.org/10.1016/j.nahs.2023.101413]), the proposed results exhibit significant improvements and can be applied to a broader class of non-autonomous equations.

Flocking dynamics of agents moving with a constant speed and a randomly switching topology

Abstract In this paper, we present a sufficient framework to exhibit the sample path-wise asymptotic flocking dynamics of the Cucker–Smale model with unit-speed constraint and the randomly switching network topology. We employ a matrix formulation of the given equation, which allows us to evaluate the diameter of velocities with respect to the adjacency matrix of the network. Unlike the previous result on the randomly switching Cucker–Smale model, the unit-speed constraint disallows the system to be considered as a nonautonomous linear ordinary differential equation on velocity vector, which forces us to get a weaker form of the flocking estimate than the result for the original Cucker–Smale model.

Properties of the solutions to periodic conformable non-autonomous non-instantaneous impulsive differential equations

In this article, we study properties of the solutions to periodic non-autonomous conformable non-instantaneous impulsive differential equations. We use a conformable Cauchy matrix and obtain some basic properties of the periodic solution to the homogeneous and non=homogeneous problems. We consider the periodicity of solutions to nonlinear problem via a fixed theorem. For more information see https://ejde.math.txstate.edu/Volumes/2024/30/abstr.html

On the motion stabilization of a three-link robotic manipulator with incomplete measurement

Abstract. This paper considers a mathematical model of a manipulator which consists of a vertical column, two links, connected to it in series, and a gripper with a load. The column resting on a fixed base can rotate around its vertical axis. The links are connected by cylindrical hinges allowing them to rotate in the same vertical plane. The column and the links are modeled as rigid bodies with the links having unequal principal moments of inertia. The position of the manipulator in space is determined by three rotation angles of the column and the links. The manipulator can have several types of steady-state program movements. When gravitational torques are compensated by control torques applied in the cylindrical hinges, the manipulator has a program equilibrium position. The manipulator can also have a program motion when the column rotates at a given constant angular velocity, and the links have given relative equilibrium positions in their plane. The stabilization problem of manipulator motion is investigated by means of control torques with feedback when only the rotation angles of the column and links are measured. The problem posed is solved in the form of a nonlinear proportional-integral controller taking into account the cylindrical phase space of the manipulator's mathematical model. The solution includes construction of a Lyapunov functional with a semi-definite derivative and application of the corresponding theorems on the asymptotic stability of non-autonomous functional differential retarded-type equations. The obtained conditions for the program motion stabilization are robust with respect to the mass-inertial parameters of the manipulator. The numerical simulation results demonstrate global attraction to its given position in cylindrical phase space.

Shadowing, Hyers–Ulam stability and hyperbolicity for nonautonomous linear delay differential equations

It is known that hyperbolic nonautonomous linear delay differential equations in a finite dimensional space are Hyers–Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.

Climate-influenced performance and offspring development of merino sheep in a dry temperate-cold valley

This study aims to explore the effects of climate on the performance and offspring development of aged Merino sheep relocated from an arid, cold environment with harsh grazing conditions to a dry, temperate-cold valley with irrigated pasture production. We utilized time series data from merino sheep in a dry temperate-cold climate in southern Argentina to characterize their growth curves, assess the impact of climate on performance, and compare offspring growth with maternal growth. Our approach involved developing a dynamic model, a non-autonomous differential equation growth curve based on the widely used Brody model. The model considered variables such as local temperature, age, sex, origin, and pregnancy status to determine the optimal combination of parameters for sheep growth in our dataset. The results have shown that moving the old sheep from the steppe to the valley resulted in an increase of an average of 1 kg in weight, but their offspring had an asymptotic weight of 65 kg, 17 kg more than their mothers. The optimum temperature for the growth rate was 15.7+/-0.56 C and 8.7+/-6.3C for the asymptotic weight.

Nonparametric estimation of nonautonomous uncertain differential equations with application to temperature models

In recent years, there has been a great development in parameter estimation methods for uncertain differential equations (UDEs). However, the observations we can obtain in real life are limited, in which case the form of function in a UDE is unknown. When dealing with such UDEs, we may use observational data to make nonparametric estimates. There are many nonautonomous systems in real life, and nonautonomous UDEs can simulate some uncertain nonautonomous dynamical systems well. In this paper, a nonparametric estimation method based on the nonautonomous UDEs of the binary Legendre polynomial is proposed. Then, three numerical examples are given to verify the reliability of nonparametric estimation. As an application, a real data example of global average monthly temperatures is used to illustrate the effectiveness of our method.