Isochronicity of a class of piecewise continuous oscillators

Motivated by a classical pendulum clock model suggested by Andrade in 1920, we study the equation and prove that for a nonlinear analytic the origin is never an isochronous focus or an isochronous center.

     

Two inverse problems for analytic potential systems

In this paper, we solve a basic problem about the existence of an analytic potential with a prescribed period function. As an application, it is shown how to extend to the whole phase plane an arbitrary potential defined on a semiplane in order to get isochronicity.     

Note on the Markus-Yamabe conjecture for gradient dynamical systems

Let be a C¹ vector field which has a singular point O and its linearization is asymptotically stable at every point of Rn. We say that the vector field v satisfies the Markus-Yamabe conjecture if the critical point O is a global attractor of the dynamical system . In this note we prove that if v is a gradient vector field, i.e. v=∇f (f∈C²), then the basin of attraction of the critical point O is the whole Rn, thus implying the Markus-Yamabe conjecture for this class of vector fields. An analogous result for discrete dynamical systems of the form xm+1=∇f(xm) is proved.     

New periodic recurrences with applications

We develop two methods for constructing several new and explicit m-periodic difference equations. Then we apply our results to two different problems. Firstly we show that two simple natural conditions appearing in the literature are not necessary conditions for the global periodicity of the difference equations. Secondly we present the first explicit non-linear analytic potential differential system having a global isochronous center.     

Simple examples of planar involutions with non-global Montgomery–Bochner linearizations

We give two planar polynomial involutions, one preserving and the other one reversing orientation, for which the Montgomery–Bochner linearization is not a global diffeomorphism.