Search

This research considers discontinuous dynamical systems, which have related vector fields that shift over a discontinuity surface. These systems appear in a variety of applications, including ecology, medicine, neuroscience, and nonsmooth mechanics. The purpose of this paper is to develop a perturbation technique that measures the effect of a nonsmooth perturbation on the period doubling bifurcation of an unperturbed system. The unperturbed system is assumed to be close to a period doubling orbit, such that when the bifurcation parameter varies, the response changes from a period one to a period two limit cycle. The generalized determination of the Poincaré map associated with perturbed systems subjected to nonsmooth transitions is derived. The main techniques used in the proof of the results are normal forms and Melnikov functions, which are defined in two zones. Various examples are presented to show that non-smoothness is responsible for period doubling. To illustrate the interesting period doubling phenomenon that emerges from an existing flat periodic orbit via the non-smooth perturbation, a simple and novel discontinuous system is provided. An additional example is provided to show the emergence of a perturbed period doubling orbit near an unperturbed one. [ABSTRACT FROM AUTHOR]

In this article, we study the intersection of Rauzy fractals associated with the k-bonacci substitution and its reversed substitution. Applying the balanced pair algorithm to these two substitutions, we characterise all minimal balanced pairs, and we obtain a general formula for the associated intersection substitution. This substitution is defined over letters. [ABSTRACT FROM AUTHOR]

In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions. The symmetry considered is reflection through the origin. Given an unimodular irreducible Pisot substitution, we consider the intersection of its Rauzy fractal with the Rauzy fractal of the reverse substitution. This set is symmetric and it is obtained by the balanced pair algorithm associated with both substitutions. [ABSTRACT FROM PUBLISHER]

In this article we introduce the balanced pair algorithm associated with 2 unimodular Pisot substitutions having the same incidence matrix. We are interested in beta-substitution related to the polynomial x³ -- ax² -- bx --1 for a ≥ b ≥ 1. Applying the balanced pair algorithm to these substitutions, we obtain a general formula for the associated intersection substitution. [ABSTRACT FROM AUTHOR]