Practical initialization of homoclinic orbits from a Bogdanov-Takens point

In a recent paper [IJBC, 24(04):1450057, 2014], we improved the theoretical base for the initialization of homoclinic orbits. However, practical application of this method is not very robust without the consideration of some numerical issues. We deal with these issues and provide examples from a robust implementation of the initialization procedure in the software package MatCont [ACM Trans. Math. Software, 29(2):141–164, 2003]

Numerical periodic normalization for codim 2 bifurcations of limit cycles : computational formulas, numerical implementation, and examples

Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time

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

Faculty of Sciences

A comprehensive study of fuzzy rough sets and their application in data reductio