Periodic solutions of periodically perturbed planar autonomous systems: A topological approach

Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x_0 of least period T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In the case when T_0/T_1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation e>0 sufficiently small, the existence of klT_0-periodic solutions x_e of the perturbed system which converge to the trajectory x_1 of the limit cycle as e->0. Moreover, we state conditions under which T=klT_0 is the least period of the periodic solutions x_e. We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T_0/T_1 is an irrational number we show the nonexistence, whenever T>0 and e>0, of T-periodic solutions x_e of the perturbed system converging to x_1. The employed methods are based on the topological degree theory

On the Cycle Spaces Associated to Orbits of Semi-simple Lie Groups

Let G be a semi-simple Lie group and Q a parabilic subgroup of its complexification G^\mathbb C, then Z:=G^\mathbb C/Q is a compact complex homogeneous manifold. Moreover, G as well as K^\mathbb C, the complexification of the maximal compact subgroup of G, acts naturally on Z with finitely many orbits. For any G-orbit, there exist a K^\mathbb C-orbit so that their intersection is non-empty and compact. This duality relation with consideration of cycle intersection at the boundary of a G-orbit lead to the definition of the cycle space associated to any G-orbit. Methods involving Schubert varieties, transversal Schubert slices together with geometric properties of a certain complementary incidence hypersurface and results about the open orbits yield a complete characterisation of the cycle space associated to an arbitrary G-orbit. In particular, it is shown that all the cycle spaces except in a few Hermitian cases are equivalent to the domain $\Omega_{AG}$. In the exceptional Hermitian cases, the cycle spaces are equivalent to the associated bounded domain.Comment: 33 page manuscript, submitte

Visualizing Chaos

Chaos is typically visualized on an infinite 2D plane. By using stereographic projection, my colleague Preston Hardy and I utilized a third dimension to plot basin maps of iterative root finding methods on a subset of the complex plane onto a sphere. These spheres are then shaded in accordance to the speed in which the particular initial point converges, creating images that can be used to visualize all basins of attraction on the complex plane on a finite 3D surface. The resulting images are used to explore efficiency of root finding methods as well as evaluating the choice of addition or subtraction n the denominator of the Hansen-Patrick root finding method. There are many theories suggesting the sign choice for positive alpha values; however, in the case of a negative alpha value, these theories do not hold. Using programs based off of those developed by Andrew Nicklawsky and Dr. Robert Hesse, we developed rules to dictate this choice between addition and subtraction in order to maximize the speed of convergence for negative and imaginary alpha values