Bifurcations of a Bouncing Ball Dynamical System

A deceptively simple difference system of a bouncing ball is investigated. Applying the center manifold theorem and the normal form analysis, we first give the parameter conditions for the flip bifurcation. Then we discuss the 1:2 resonance. Because the nondegeneracy conditions are not satisfied, we use the approximation of mappings by a flow and change this difference system into an ordinary differential system with dimension two. With the aid of the Melnikov method, the homoclinic bifurcation and the Poincaré bifurcation are analyzed, which imply the existence of the invariant circles for the difference system. Furthermore, we compute the normal forms to provide the parameter conditions for the Chenciner bifurcation and also obtain the stability of the fixed point. Finally, some numerical simulations are presented to verify the obtained results.

The Reflection Map and Infinitesimal Deformations of Sphere Mappings

The reflection map introduced by D’Angelo is applied to deduce simpler descriptions of nondegeneracy conditions for sphere maps and to the study of infinitesimal deformations of sphere maps. It is shown that the dimension of the space of infinitesimal deformations of a nondegenerate sphere map is bounded from above by the explicitly computed dimension of the space of infinitesimal deformations of the homogeneous sphere map. Moreover a characterization of the homogeneous sphere map in terms of infinitesimal deformations is provided.

Some properties of ultrafilters of widely understood measurable spaces

Filters and ultrafilters (maximal filters) on the π-system with “zero” and “unit” are considered (here, π-system is a nonempty family of sets closed with respect to finite intersections); so, our π-system contains including and empty sets. Characteristic properties of ultrafilters obtained from special representations for bases of two typical topologies connected with construction of Wallman extension and Stone compactums are investigated. The topology of Wallman type on the ultrafilters set of arbitrary π-system with “zero” and “unit” is defined. In addition, initial set is transformed in a compact T1-space with points in the form of ultrafilters of above-mentioned π-system. Under equipment of the resulting ultrafilter set with two topologies (by sense, Stone and Wallman topologies), bitopological space with comparable topologies is obtained; for this space, the degeneracy (in the sense of coincidence for above-mentioned topologies) and nondegeneracy conditions are indicated. The initial set is immersed in above-mentioned bitopological space as everywhere dense subset. Resulting construction is oriented on application in extensions of abstract attainability problems with constraints of asymptotic character (we keep in mind the possible application of ultrafilters as generalized elements).

Codimension-3 Flip Bifurcation of a Class of Difference Equations

In this paper, we consider a one-dimensional difference equation with three parameters, the derivative of which at a fixed point has an eigenvalue [Formula: see text] as the parameters are all zero. In the case that both nondegeneracy conditions of the flip bifurcation and the generalized flip bifurcation are not satisfied, by computing normal form, we give the nondegeneracy condition and transversality condition of the codimension-3 flip bifurcation. Moreover, by discussing the number of positive zeros of a cubic function in a neighborhood of the origin, we show the bifurcation scenario and give the parameter conditions, respectively, that the normal form of the equation possesses three 2-cycles, two 2-cycles, only one 2-cycle or none.

Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman’s method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).

EXISTENCE OF STRONG TRACES FOR GENERALIZED SOLUTIONS OF MULTIDIMENSIONAL SCALAR CONSERVATION LAWS

In the half-space t > 0 a multidimensional scalar conservation law with only continuous flux vector is considered. For the wide class of functions including generalized entropy sub- and super-solutions to this equation, we prove existence of the strong trace on the initial hyperspace t = 0. No nondegeneracy conditions on the flux are required.