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The problem of stabilization of unstable periodic orbits of discrete nonlinear systems is considered in the article. A new generalization of the delayed feedback, which solves the stabilization problem, is proposed. The feedback is…
We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in $\mathbb{R}$. We prove that, after an appropriate translation, each of them is necessarily an even function which…
A general nonautonomous Mackey-Glass equation for the regulation of the hematopoiesis with several non-constant delays is studied. Using topological degree methods, we prove the existence and multiplicity of positive periodic solutions.
The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our…
We describe an analytical method for computing the orbital parameters of a planet from the periodogram of a radial velocity signal. The method is very efficient and provides a good approximation of the orbital parameters. The accuracy is…
This paper is concerned with the derivative nonlinear Schr\"{o}dinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small…
Comet-type periodic orbits of the circular restricted three-body problem (CR3BP) are periodic solutions that are generated from very large retrograde and direct circular Keplerian motions around the common center of mass of the primaries.…
Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x_0 of least period T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In the…
In the present paper, we establish the existence and orbital instability results of cnoidal periodic waves for the quintic Klein-Gordon and nonlinear Schr\"odinger equations. The spectral analysis for the corresponding linearized operator…
In this paper, we give a proof of the existence of stationary dark soliton solutions or heteroclinic orbits of nonlinear equations of Schr\"odinger type with periodic inhomogeneous nonlinearity. The result is illustrated with examples of…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…
We study the Schr\"{o}dinger equation: \begin{eqnarray} - \Delta u+V(x)u+f(x,u)=0,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{eqnarray} where $V$ is periodic and $f$ is periodic in the $x$-variables, $0$ is in a gap of the spectrum of…
A variational principle for determining unstable periodic orbits of flows as well as unstable spatio-temporally periodic solutions of extended systems is proposed and implemented. An initial loop approximating a periodic solution is evolved…
We extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and…
We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that,…
Traditional analytical theories of celestial mechanics are not well-adapted when dealing with highly elliptical orbits. On the one hand, analytical solutions are quite generally expanded into power series of the eccentricity and so limited…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
This paper concerns the existence of multiple rotating quasi-periodic solutions for second order Hamiltonian systems with sub-quadratic potential. Such solutions have the form $x(t+T)=Qx(t)$ for some orthogonal matrix $Q$. To deal with such…
This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with…
If one has to attain high accuracy over long timescales during the numerical computation of the N-body problem, the method called Lie-integration is one of the most effective algorithms. In this paper we present a set of recurrence…