Problem Of Finding Periodic Solutions Research Articles

Multiple symmetric periodic solutions of differential systems with distributed delay

In this paper, we study the existence of periodic solutions to the following delay differential system:{x1′(t)=∫−10f1(x2(t+θ))dθ,x2′(t)=∫−10f2(x1(t+θ))dθ. By transforming the problem of searching for periodic solutions of the above system to the problem of finding periodic solutions of an associated ordinary differential system with boundary value conditions and using pseudoindex theory, some sufficient conditions are obtained to guarantee the existence of multiple W-symmetric 2-periodic solutions. We also present three specific examples to illustrate our results.

Periodic solutions to a class of distributed delay differential equations via variational methods

Abstract In this article, we study the existence of periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.

FORCED DIFFUSION EQUATION WITH PIECEWISE CONTINUOUS TIME DELAY

In this article, we study the boundary value problem (BVP) for forced diffusion equation with piecewise constant arguments. We give explicit formula for solving BVP. The problem of finding periodic solutions of some differential equations with piecewise constant arguments (DEPCA) is reduced to solving a system of algebraic equations. Using the method of finding periodic solutions of DEPCA, the solutions of BVP are given in several examples which are periodic in time.

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.

Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments

Abstract In this paper, we describe a method to solve the problem of finding periodic solutions for second-order neutral delay-differential equations with piecewise constant arguments of the form x″(t) + px″(t − 1) = qx([t]) + f(t), where [⋅] denotes the greatest integer function, p and q are nonzero real or complex constants, and f(t) is complex valued periodic function. The method reduces the problem to a system of algebraic equations. We give explicit formula for the solutions of the equation. We also give counter examples to some previous findings concerning uniqueness of solution.

Periodic solutions of Euler–Lagrange equations in an anisotropic Orlicz–Sobolev space setting

We consider the problem of finding periodic solutions of certain Euler–Lagrange equations which include, among others, equations involving the p-Laplace operator and, more generally, the pp, qq-Laplace operator. We employ the direct method of the calculus of variations in the framework of anisotropic Orlicz–Sobolev spaces. These spaces appear to be useful in formulating a unified theory of existence of solutions for such a problem.

Some existence results on periodic solutions of Euler–Lagrange equations in an Orlicz–Sobolev space setting

In this paper we consider the problem of finding periodic solutions of certain Euler–Lagrange equations. We employ the direct method of the calculus of variations, i.e. we obtain solutions minimizing certain functional I. We give conditions which ensure that I is finitely defined and differentiable on certain subsets of Orlicz–Sobolev spaces W1LΦ associated to an N-function Φ. We show that, in some sense, it is necessary for the coercivity that the complementary function of Φ satisfy the Δ2-condition. We conclude by discussing conditions for the existence of minima of I.

Some Notes on the Grassmann Manifolds and Nonlinear System

The differential geometry as a new tool has been introduced to research the control system, especially the nonlinear system. In this paper, by considering how to construct a manifold from a quotient space, we investigate the structure of Grassmann manifold concretely. This is beneficial to study the problem of finding periodic solutions of the matrix Riccati equations of control theory and the two point boundary problem.

Time switched differential equations and the Euler polynomials

We discuss differential equations depending non-smoothly on the integration time of the form $$\begin{aligned} y^{(n)}=\mathop {\text{ sgn}}\nolimits (\sigma (t)) + F(t) \end{aligned}$$where \(n\in \mathbb N , n>0\), and \(F, \sigma \) are piecewise-\({\fancyscript{C}}^{\infty }\) periodic functions. The main results deal with the existence of periodic solutions of such equations as well as their computation by explicit formulas. No infinite series appear, and it is indeed established that these periodic solutions are explicitly computable by means of finitely many Euler polynomials. We also introduce a wider class of piecewise-\({\fancyscript{C}}^{\infty }\) equations where the problem of finding periodic solutions is finitely solvable as well.

Uncertainty Quantification and Bifurcation Analysis of an Airfoil with Multiple Nonlinearities

In order to calculate the limit cycle oscillations and bifurcations of nonlinear aeroelastic system, the problem of finding periodic solutions with maximum vibration amplitude is transformed into a nonlinear optimization problem. An algebraic system of equations obtained by the harmonic balance method and the stability condition derived from the Floquet theory are used to construct the general nonlinear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the proposed approach is validated, and the effects of structural parameter uncertainty on the limit cycle oscillations and bifurcations of an airfoil with multiple nonlinearities are studied. Numerical examples show that the coexistence of multiple nonlinearities may lead to low amplitude limit cycle oscillation.

BIFURCATIONS IN A BOUNDARY-VALUE PROBLEM OF A NONLINEAR MODEL FOR STRESS-INDUCED PHASE TRANSITIONS

In this paper, some methodology in nonlinear dynamics is used to study a boundary-value problem of a nonlinear model arisen in phase transitions in a slender cylinder composed of a compressible hyperelastic material. We transform the original system of boundary-value problem to an initial-value (dynamical) problem of finding periodic solutions of coupled nonlinear autonomous oscillators in a four-dimensional space. Hopf-like bifurcation analysis of the periodic solutions of the system is studied. Both analytical and numerical solutions are obtained by using a nonlinear transformation formulation. The accuracy of analytical solutions is investigated by comparing with the numerical solutions. The engineering stress-strain curve is plotted and compared with that from the normal form equation, which is a simplification of the original system.

Finding the Periodic Solution of Differential Equation via Solving Optimization Problem

In this paper, we propose a new method to find the periodic solutions of differential equations. The key technique is to convert the problem of finding periodic solutions of differential equations into an optimization problem. Then by solving the corresponding optimization problem, we can find the periodic solutions of differential equations. Finally, some numerical results are presented to illustrate the utility of the technique.

Closed characteristics of fourth-order twist systems via braids

For a large class of second order Lagrangian dynamics, one may reformulate the problem of finding periodic solutions as a problem in solving second-order recurrence relations satisfying a twist condition. We project periodic solutions of such discretized Lagrangian systems onto the space of closed braids and apply topological techniques. Under this reformulation, one obtains a gradient flow on the space of braided piecewise linear immersions of circles. We derive existence results for closed braided solutions using Morse–Conley theory on the space of singular braid diagrams.

Finding periodic solutions of ordinary differential equations via homotopy method

This paper deals with the problem of finding periodic solutions of ordinary differential equations. In our approach, by means of the homotopy method, the equation considered is linked to a simpler equation by introducing a parameter. As a result, the problem of finding periodic solutions is transformed into that of finding suitable solutions of the Cauchy problem; following the path of solutions of the Cauchy problems starting from the Cauchy problem of a simpler equation, one can obtain the desired periodic solutions. In general, the solutions of the Cauchy problem of the simpler equation are easily determined, relative to periodic solutions of the equation considered. Consequently, an efficient and global method of finding periodic solutions is established.

Finding periodic solutions of ordinary differential equations via homotopy method

This paper deals with the problem of finding periodic solutions of ordinary differential equations. In our approach, by means of the homotopy method, the equation considered is linked to a simplier equation by introducing a parameter. As a result, the problem of finding periodic solutions is transformed into that of finding suitable solutions of the Cauchy problem. Following the path of solutions of the Cauchy problems starting from the Cauchy problem of a simpler equation, one can obtain the desired periodic solutions. In general, the solutions of the Cauchy problem of the simpler equation are easily determined, relative to periodic solutions of the equation considered. Consequently, an efficient and global method of finding periodic solutions is established.

On the existence of multiple solutions for semilinear equations with monotone nonlinearities crossing a finite number of eigenvalues

LET 0 be a bounded domain in R” (n 2 1) with smooth boundary. In this paper we study in L,(Q) nonlinear equations of the form Lu = N(u), (1.1) where L is a densely defined closed self-adjoint linear operator with closed range and N is a Nemytskii operator generated by a Caratheodory function (w, s) + g(w, s) from Q x R to lR with g( *, 0) = 0. We are interested in the existence of multiple solutions of (1.1) in the case where N interacts with a finite number of eigenvalues of finite multiplicity of L. It is essential in our study that the kernel of L may be infinite dimensional and that L has a pure point spectrum which may be unbounded both from above and below. Hence the results obtained can be applied to the problem of finding periodic solutions for semilinear wave equations of the form

Periodic bounce solutions of dynamical conservative systems and their minimal period

SEVERAL important studies have been made on the problem of finding periodic solutions of a conservative dynamical system. An important part of such a study is concerned with the case in which the minimal period of the trajectory is given (see e.g. [l, 3, 6, 9, 14)). In this paper we consider the problem of existence of periodic trajectories with prescribed minimal period, when the material point moves in a convex billiard in I?” under the action of a conservative field and it bounces in a completely elastic way against the boundary, hitting it orthogonally. If Q is a bounded open convex subset of I?” (not necessarily regular), T > 0 and V is a real Cz-function in a neighbourhood of 0, we consider the nonconstant periodic orbits q : R” -+ Q (which are called “principal bounce trajectories” @.b.t.)) having period 2T such that:

Lie and morse theory for periodic orbits of vector fields and matrix riccati equations, I: General lie-theoretic methods

The problem of finding periodic solutions of the matrix Riccati equations of linear control theory is interpreted geometrically as a problem of finding periodic orbits of certain one-parameter transformation groups on Grassmann manifolds. For certain control problems the vector fields which generate these groups can be written as a sum of two commuting vector fields, one a gradient vector field, the other a Killing vector field, i.e., an infinitesimal isometry of a metric on the Grassman manifold. For such vector fields, the methods of Morse theory can be adapted to study the periodic orbits. The topological data that is needed to count periodic orbits, i.e., the Poincare polynomial of certain submanifolds of the Grassmann manifold, can be derived from results proved by A. Borel.

A numerical method for multipoint boundary value problems with application to a restricted three body problem

A type of parallel shooting method is proposed for the solution of nonlinear multipoint boundary value problems. It extends the usual quasilinearization method and a previous shooting method developed for such problems, and reduces to usual multiple shooting techniques for two point boundary value problems. The effectiveness of the method for stiff problems is illustrated by an application to the problem of finding periodic solutions of a restricted three body problem with given Jacobian constant and unknown period.

Periodic solutions of a class of functional differential equations

where x(t) is a real n-vector, n > 1, and x, is an element of the Banach Space C = C([ -7, 01, R”), z b 0 fixed, defined through the relation x@) = x(t + @, -z G 8 < 0. The norm of an element cpofCis)cpJ, = _,qlt o ljcp(B)/ where II .I) is a norm in R”. .-. We assume that 9: C -+ R” is linear and continuous, g is periodic in t, ‘higher order’ in the second variable and g(t, 0,O) = 0. Our objective is to derive computable sufficient conditions for the existence of periodic solutions of (0.1) with E # 0 and 1.~1 small, which are ‘close’ to the 0 solution of (0.1) with E = 0. The treatment is done for the resonance case. The nonresonant case was considered by Halanay [l]. The motivation for looking at this problem comes from an earlier paper of Perello [2], in which he reduces the problem of finding periodic solutions of (0.1) to that of solving a finite dimensional nonlinear equation, known as the determining equation or bifurcation equation. However, since the determining equation is not given explicitly, Perello’s method is very difficult to apply at least without some further analysis. Our approach is to carry out a finite dimensional Lyapunov Schmidt reduction and to apply techniques of topological degree to the finite dimensional problem. This approach has been used by Cronin [3] and [4] for the case of o.d.e. In Section 1 we summarize parts of the theory of linear systems of f.d.e. due to Hale [5], which are relevant to our study. Refer to Hale [5] and [6] for all the results stated in this section. In Section 2 we present an abstract formulation of the bifurcation problem as a nonlinear equation in a Banach Space. In Section 3 we apply the techniques of Section 2 to equation (0.1). In Section 4 we illustrate the applicability of the method with an example.