Periodic Solutions Of Differential Equations Research Articles

Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Delay

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the (μ1,μ2)-exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the (μ1,μ2)-exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research.

Periodic solutions of differential equations with periodic constraints

We study the harmonic solutions of periodically perturbed differential equations subjected to a possibly time‐dependent constraint. We obtain a degree theoretic condition ensuring a “global branching” result for the nontrivial periodic solutions. The argument stems from a combination of techniques from the theory of topological degree and differential‐algebraic equations. As an application, the set of harmonic solutions of periodically perturbed implicit differential equations is investigated.

On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales

Abstract This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.

On Periodic Solutions of Differential Equations with Continuous Right-Hand Sides on Lie Groups

On Periodic Solutions of Differential Equations with Continuous Right-Hand Sides on Lie Groups

The Brouwer fixed point theorem and periodic solutions of differential equations

The Brouwer fixed point theorem is a key ingredient in the proof that a periodic differential equation has a periodic solution in a set that satisfies a suitable tangency condition on its boundary. The main goal of this note is to show that both results are in fact equivalent.

Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator

We obtain the existence of multiple pairs of periodic solutions for difference equations of type −Δ(Δu(n − 1)1 − |Δu(n − 1)|2)=λg(u(n))(n∈Z), where g:R→R is a continuous odd function with anticoercive primitive, and λ>0 is a real parameter. The approach is variational and relies on the critical point theory for convex, lower semicontinuous perturbations of C1-functionals.

An Operator Approach to Periodic Solutions of Differential Equations on Lie Groups

An Operator Approach to Periodic Solutions of Differential Equations on Lie Groups

WEYL ALMOST PERIODIC FUNCTIONS ON TIME SCALES AND WEYL ALMOST PERIODIC SOLUTIONS OF DYNAMIC EQUATIONS WITH DELAYS

<abstract> Due to the incompleteness of the space composed of Weyl almost periodic functions, there are few results on the existence of Weyl almost periodic solutions of differential equations. In addition, as a discrete analogs of differential equations, there is almost no result of the existence of Weyl almost periodic solutions of difference equations. Because dynamic equations on time scales can unify the study of differential equations and difference equations. Therefore, in this paper, we first propose a concept of Weyl almost periodic functions on time scales. Then, taking a Clifford-valued neural network with time-varying delays on time scales as an example of dynamic equations on time scales, we study the existence and stability of Weyl almost periodic solutions of this neural network on time scales. Even when the system we consider degenerates into a real-valued system, our results are new. A numerical example is given to illustrate the feasibility of our results. </abstract>

Noel Glynne Lloyd, 1946–2019

Noel Lloyd was born in Llanelli, Carmarthenshire, on 26 December 1946. He was educated first at Ysgol Gymraeg Dewi Sant (St David's Welsh School) and then at Llanelli Grammar School for Boys, where his father was his mathematics teacher. He subsequently attended Queens' College, Cambridge, where he obtained a First Class in all Parts of the Mathematical Tripos. Noel proceeded to work for his PhD under the supervision of Sir Peter Swinnerton-Dyer; his thesis, which he defended in 1972, was entitled ‘Periodic solutions of differential equations’. After two years at St John's College, Cambridge, Noel was appointed to a Lectureship in Pure Mathematics at the University of Wales, Aberystwyth. He remained at Aberystwyth for the remainder of his working life where he was appointed Head of Department, Dean of the Faculty of Science, Pro-Vice-Chancellor and Vice-Chancellor of the University and also served on numerous academic, scientific and governmental committees well into his retirement; these included the Commission on Devolution in Wales. Noel's research centred on systems of polynomial differential equations and their connection to the famous outstanding Hilbert's Sixteenth Problem, with his research group being early users of Computer Algebra to make progress with complex calculations. In his free time he was a talented organist and active within the Presbyterian Church of Wales where he pursued his deep religious beliefs and acted as a prominent member of the church's administrative leadership. He died at the age of 72 on 7 June 2019.

PERIODIC SOLUTIONS OF DIFFERENTIAL EQUATION WITH PIECEWISE CONSTANT ARGUMENT

In this paper, differential equations with piecewise constant arguments are studied. The problem of finding the periodic solution of differential equations with piecewise constant arguments (DEPCA) is brought to the solution of the system of algebraic equations. The periodic solution over time using the periodic solution of differential equations with piecewise constant arguments is given.

Almost Periodic Solutions of Differential Equations

We construct the Favard–Amerio theory for almost periodic differential equations in Banach spaces without using the ℋ-classes of these equations. For linear equations, we present the first examples of almost periodic operators that have no analogs in the classical Favard–Amerio theory.

Vibration Stability of the Impulse System of the Electric Drive Control

Purpose of reseach is Study of vibration stability of the impulse system of direct current electric drive in order to ensure operating modes with specified dynamic characteristics. Methods. The stability analysis of periodic solutions of differential equations with discontinuous right-hand side is reduced to the problem of studying local stability of fixed map points. Results. The analysis of stability is carried out depending on the supply voltage of the electric drive and the gain of the correcting link in the feedback circuit. It is revealed that the boundary of the stability region on the plane of the variable parameters has a pronounced extremum in the form of a maximum at the bifurcation point of codimension two, also called the resonance point 1: 2. On one side of this point, the stability region is bounded by the NeimarkSacker bifurcation line, and on the other, by the period-doubling bifurcation line. This means that with a change in the parameters, the radius of the stability region first increases, reaching a maximum at the resonance point 1: 2, and then decreases. This important conclusion can be used in optimization calculations. Conclusion. The analysis of the vibration stability of the impulse system of direct current electric drive, the behavior of which is described by differential equations of the discontinuous right-hand side, is carried out. The problem of finding periodic solutions to differential equations is reduced to the problem of finding fixed points of the map. The fixed points of the map satisfy a system of nonlinear equations, which was solved numerically by the NewtonRaphson method. The stability of periodic solutions of differential equations corresponds to the stability of fixed points of the corresponding map. The studies were carried out with variation of the gain in the feedback circuit and the supply voltage. It is revealed that the loss of a fixed point occurs through the supercritical Neimark-Sacker bifurcation, when the complex-conjugate pair of multipliers leaves the unit circle when the parameters change. However, with an increase in the supply voltage, the Neimark-Saker bifurcation boundary passes into the perioddoubling bifurcation boundary at the 1: 2 resonance point.

Almost periodic and periodic solutions of differential equations driven by the fractional Brownian motion with statistical application

ABSTRACT We show that the unique solution to a semilinear stochastic differential equation with almost periodic coefficients driven by a fractional Brownian motion is almost periodic in a sense related to random dynamical systems. This type of almost periodicity allows for the construction of a consistent estimator of the drift parameter in the almost periodic and periodic cases.

Measure pseudo almost periodic solutions for differential equations with reflection

ABSTRACT In this research paper, we give some new results for the existence and uniqueness of doubly measure pseudo almost periodic (Or -pap) solutions for some differential equations with reflection. We will use the Banach fixed-point theorem and some properties of pseudo almost periodic functions with measurements and we will discuss both linear and nonlinear cases. Finally, we finish with some applications to illustrate our results.

On the Poincaré-Adronov-Melnikov method for the existence of grazing impact periodic solutions of differential equations

The persistence of periodic, grazing and impact solutions is studied for periodically perturbed ordinary differential equations with impacts. An approach of the Poincaré-Adronov-Melnikov method is applied. It is based on introducing and studying an appropriate impact Poincaré mapping. Examples are presented to illustrate theoretical results.

Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments

AbstractWe study scalar advanced and delayed differential equations with piecewise constant generalized arguments, in short DEPCAG of mixed type, that is, the arguments are general step functions. It is shown that the argument deviation generates, under certain conditions, oscillations of the solutions, which is an impossible phenomenon for the corresponding equation without the argument deviations. Criteria for existence of periodic solutions of such equations are discussed. New criteria extend and improve related results reported in the literature. The efficiency of our criteria is illustrated via several numerical examples and simulations.

Периодические решения дифференциальных уравнений

Периодические решения дифференциальных уравнений

Existence and Uniqueness Fourth-order Differential Equations in Banach Space Having Fourier Type

The aim of this work is to study the existence of a periodic solutions of differential equations $\frac{d^{4} }{ dt^{4} }x(t) = Ax(t) + f(t)$. Our approach is based on the M-boundedness of linear operators, Fourier type, $B^{s}_{p, q}$-multipliers and Besov spaces.

Method of Guiding Functions for Existence Problems for Periodic Solutions of Differential Equations

We provide a review and systematic explanation of various generalizations of the guiding function method. The current state of the said method and its applications to various kinds of problems for nonlinear periodic systems described by differential and functional differential equations are considered.

On optimal periodic solution of differential equations

this paper dedicated to the construction of solution of a three time scale periodic singular perturbed non-linear quadratic optimal control problem by using the direct method. The algorithm of the method is the direct substitution of the postulated asymptotic expansion of the solution of the problem and then by the conditions of the problem we constructed of a series of problem and find terms of the asymptotic. We find the solution by using the Hamilton's function and maximum principle