Bifurcation of limit cycles from a 4-dimensional center in 1:n resonance

We study the bifurcation of limit cycles from the periodic orbits of a linear differential system in R⁴ in resonance 1:n perturbed inside a class of piecewise linear differential systems, which appear in a natural way in control theory. Our main result shows that at most 1 limit cycle can bifurcate using expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving this result we use the averaging theory in a form where the differentiability of the system is not needed.     

Phase portraits for quadratic homogeneous polynomial vector fields on $$ \mathbb{S}^2 $$

Let X be a homogeneous polynomial vector field of degree 2 on $ \mathbb{S}^2 $ \mathbb{S}^2 . We show that if X has at least a non-hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on $ \mathbb{S}^2 $ \mathbb{S}^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16 ^th Hilbert’s problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on $ \mathbb{S}^2 $ \mathbb{S}^2 of degree n.     

Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities

In this paper we classify the centers, the cyclicity of its Hopf bifurcation and their isochronicity for the polynomial differential systems in R² of arbitrary degree d?3 odd that in complex notation z=x+iy can be written as

     

A second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter

We deal with nonlinear T-periodic differential systems depending on a small parameter. The unperturbed system has an invariant manifold of periodic solutions. We provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation. In the end two applications are done. The key tool for proving the main result is the Lyapunov-Schmidt reduction method applied to the T-Poincaré-Andronov mapping.     

Sliding Mode Control in a Bioreactor Model

This article deals with an example of nonlinear control systems and the interlacing between a biochemical system, the mathematical model and the constrains derived from the discrete implementation of a continuous control policy. The theory is developed on a simplified model of a bioreactor to be regulated, and the sliding mode control is presented as a robust control technique. The biological interpretation of the results derived from the mathematical model is pointed out, especially of those more closely involved with the implementation, as is the case of sample period, which seems to be very enough with respect to the minimum time needed for sample analysis.