Narrow Results

We study the exotic particles symmetry in the background of noncommutative two-dimensional phase-space leading to realize in physics the deformed version of C_λ-extended Heisenberg algebra and ω_∞ symmetry.

The focus of the paper is mainly on the existence of limit cycles of a planar system with third-degree polynomial functions. A previously developed perturbation technique for computing normal forms of differential equations is employed to calculate the focus values of the system near equilibrium points. Detailed studies have been provided for a number of cases with certain restrictions on system parameters, giving rise to a complete classification for the local dynamical behavior of the system. In particular, a sufficient condition is established for the existence of k small amplitude limit cycles in the neighborhood of a high degenerate critical point. The condition is then used to show that the system can have eight and ten small amplitude (local) limit cycles for a set of particular parameter values.

In this paper, we prove the existence of twelve small (local) limit cycles in a planar system with third-degree polynomial functions. The best result so far in literature for a cubic order planar system is eleven limit cycles. The system considered in this paper has a saddle point at the origin and two focus points which are symmetric about the origin. This system was studied by the authors and shown to exhibit ten small limit cycles: five around each of the focus points. It will be proved in this paper that the system can have twelve small limit cycles. The major tasks involved in the proof are to compute the focus values and solve coupled enormous large polynomial equations. A computationally efficient perturbation technique based on multiple scales is employed to calculate the focus values. Moreover, the focus values are perturbed to show that the system can exactly have twelve small limit cycles.

Equilibrium manifold is a manifold that consists of equilibrium points. Planar systems with one-dimensional equilibrium manifolds are considered in this paper. First, for such planar systems, a unified equation with the horizontal axis as the equilibrium curve is formulated. Then, according to the corresponding linearized systems, different cases are discussed: For the nondegenerate case, the simplest normal form of a system with simplified Bogdanov–Takens singularities is obtained; for the general first-order degenerative case, the simplest normal forms are completely characterized; finally, for the general higher-order degenerative case, deduction of the simplest normal form is illustrated.

Using analytic methods from the dynamical systems theory, some new nonlinear wave equations are investigated, which have exact explicit parametric representations of breaking loop-solutions under some fixed parameter conditions. It is shown that these parametric representations are associated with some families of open level-curves of traveling wave systems corresponding to such nonlinear wave equations, each of which lies in an area bounded by a singular straight line and the stable and the unstable manifolds of a saddle point of such a system.

Smooth planar vector fields containing two hyperbolic saddles may possess contours formed by heteroclinic connections between these saddles. We present an overview of known results on bifurcations of these contours. Additionally, two new explicit polynomial systems containing such contours are derived, which are studied using the bifurcation software matcont and are shown to exhibit the theoretically predicted phenomena, including series of heteroclinic connections.

The problem of Mittag–Leffler stabilization (MLS) is studied for a class of nonlinear non-integer order systems. The stabilizer is constructed by using the Lyapunov function and backstepping algorithm. The continuous controller is designed to ensure that the state of the nonlinear fractional-order closed-loop system converges to the equilibrium. Two simulation examples are given to illustrate the effectiveness of the method.