Numerical bifurcation analysis of homoclinic orbits embedded in one-dimensional manifolds of maps

We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing one-dimensional stable and unstable manifolds. The methods are implemented in MatContM, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits, and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows us to detect all known codimension 1 and 2 bifurcations in three-dimensional (3D) maps, including tangencies and generalized tangencies. MatContM provides a graphical user interface, enabling interactive control for all computations. As the prime new feature, we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing us to compute all homoclinic orbits to saddle points in 3D maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis, and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features

Капіталізація як результат управління грошовими активами публічного акціонерного товариства

Досліджено вплив використання грошових активів публічного акціонерного товариства на фінансові результати господарювання та визначення рівня його капіталізації. Ключові слова: капіталізація, публічне акціонерне товариство, грошові активи, флоут.Исследовано воздействие использования денежных активов публичного акционерного общества на финансовые результаты хозяйствования и определение уровня его капитализации. Ключевые слова: капитализация, публичное акционерное общество, денежные активы, флоут.The effect of the use of monetary assets of public company on financial results and determination of the level of its capitalization are analysed. Keywords: capitalization, public company, monetary assets, float

Using MatContM in the study of a nonlinear map in economics

MatContM is a MATLAB interactive toolbox for the numerical study of iterated smooth maps, their Lyapunov exponents, fixed points, and periodic, honioclinic and heteroclinic orbits as well as their stable and unstable invariant iwniifolds. The bifurcation;1.t aysis is based on continuation methods, tracing out solution iiianifolds of various types of objects while some of the parameters of the map vary. In particular, MatContM computes codimension 1 bifurcation curves of cycles and supports the computation of the normal form coefficients of their codimension two bifurcations, and allows branch switching from codimension 2 points to secondary curves. MatContM builds on an earlier command-line MATLAB package Cl_MatContM but provides new coinputational routines and functionalities, as well as a graphical user interface, enabling internctive, control of all computations, data handling and archi Ong. We, apply MatContM in our study of the monopoly model of T. Puu with cubic price and quadratic marginal cost functions. Using MatContM, we analyze the fixed points and their stability and we compute branches of solutions of period 5, 10, 13 17. The chaotic and periodic behavior of the monopoly model is further analyzed by computing the largest Lyapunov exponents