Periodic Solutions of a Class of Duffing Differential Equations

In this work we study the existence of new periodic solutions for the well knwon class of Duffing differential equation of the form $x^{\prime\prime}+ c x^{\prime}+ a(t) x +b(t) x^3 = h(t)$, where $c$ is a real parameter, $a(t)$, $b(t)$ and $h(t)$ are continuous $T$--periodic functions. Our results are proved using three different results on the averaging theory of first order.     

Periodic Solutions of the Duffing Differential Equation Revisited via the Averaging Theory

We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations $\ddot y+ a \sin y= b \sin t$ and $\ddot y+a y-c y^3=b\sin t$, where $a$, $b$ and $c$ are real parameters.     

The solution of the second part of the 16th Hilbert problem for nine families of discontinuous piecewise differential systems

We provide the maximum number of limit cycles of some classes of discontinuous piecewise differential systems formed by two differential systems separated by a straight line, when these differential systems are linear centers or three families of cubic isochronous centers, giving rise to ten different classes of discontinuous piecewise differential systems. These maximum number of limit cycles vary from 0, 1, 2, 3, 5, 7 and 12 depending on the chosen class. For nine of these classes, we prove that the corresponding maximum number of limit cycles are reached. In particular, we have solved the extension of the second part of the 16th Hilbert problem to these classes of discontinuous piecewise differential systems. The main tool used for proving these results is based on the first integrals of the systems which form the discontinuous piecewise differential systems.

     

A new efficient short-step projective interior point method for linear programming

In this paper, we are interested in the performance of Karmarkar’s projective algorithm for linear programming. We propose a new displacement step to accelerate and improve the convergence of this algorithm. This purpose is confirmed by numerical experimentations showing the efficiency and the robustness of the obtained algorithm over Schrijver’s one for small problem dimensions.     

Asymptotic Study of a Boundary Value Problem Governed by the Elasticity Operator with Nonlinear Term

In this paper, a nonlinear boundary value problem in a three dimensional thin domain with Tresca's friction law is considered. The small change of variable z = x₃/ε transforms the initial problem posed in the domain Ω^εinto a new problem posed on a fixed domain Ω independent of the parameter ε. As a main result, we obtain some estimates independent of the small parameter. The passage to the limit on ε, permits to prove the results concerning the limit of the weak problem and its uniqueness.