Stabilization of polynomial systems in ℝ3 via homogeneous feedback

In this paper, we study the stabilization problem of a class of polynomial systems of odd degree in dimension three. The constructed stabilizing feedback is homogeneous and guarantee the homogeneity of the closed loop system.mynotered In the end of the paper, we show the efficiency of such a study in the local stabilization of nonlinear systems affine in control.

Limit Cycle Bifurcations in a Class of Piecewise Smooth Polynomial Systems

In this paper, we consider the bifurcation problem of limit cycles for a class of piecewise smooth cubic systems separated by the straight line [Formula: see text]. Using the first order Melnikov function, we prove that at least [Formula: see text] limit cycles can bifurcate from an isochronous cubic center at the origin under perturbations of piecewise polynomials of degree [Formula: see text]. Further, the maximum number of limit cycles bifurcating from the center of the unperturbed system is at least [Formula: see text] if the origin is the unique singular point under perturbations.

H(curl2)-conforming quadrilateral spectral element method for quad-curl problems

In this paper, we propose an [Formula: see text]-conforming quadrilateral spectral element method to solve quad-curl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. [Formula: see text]-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is worth noting that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our [Formula: see text]-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.

On Behavior of the Periodic Orbits of a Hamiltonian System of Bifurcation of Limit Cycles

In light of the previous recent studies by Jaume Llibre et al. that dealt with the finite cycles of generalized differential Kukles polynomial systems using the first- and second-order mean theorem such as (Nonlinear Anal., 74, 1261–1271, 2011) and (J. Dyn. Control Syst., vol. 21, 189–192, 2015), in this work, we provide upper bounds for the maximum number of limit cycles bifurcating from the periodic orbits of Hamiltonian system using the averaging theory of first order.