Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

Homoclinic and heteroclinic solutions to a hepatitis C evolution model

Homoclinic and heteroclinic solutions to a standard hepatitis C virus (HCV) evolution model described by T. C. Reluga, H. Dahari and A. S. Perelson, (SIAM J. Appl. Math., 69 (2009), pp. 999–1023) are considered in this paper. Inverse balancing and generalized differential techniques enable derivation of necessary and sufficient existence conditions for homoclinic/heteroclinic solutions in the considered system. It is shown that homoclinic/heteroclinic solutions do appear when the considered system describes biologically significant evolution. Furthermore, it is demonstrated that the hepatitis C virus evolution model is structurally stable in the topological sense and does maintain homoclinic/heteroclinic solutions as diffusive coupling coefficients tend to zero. Computational experiments are used to illustrate the dynamics of such solutions in the hepatitis C evolution model.

Chebyshev Distance

Summary In [21], Marco Riccardi formalized that ℝN-basis n is a basis (in the algebraic sense defined in [26]) of ${\cal E}_T^n $ and in [20] he has formalized that ${\cal E}_T^n $ is second-countable, we build (in the topological sense defined in [23]) a denumerable base of ${\cal E}_T^n $ . Then we introduce the n-dimensional intervals (interval in n-dimensional Euclidean space, pavé (borné) de ℝn [16], semi-intervalle (borné) de ℝn [22]). We conclude with the definition of Chebyshev distance [11].

Automatized Search for Complex Symbolic Dynamics with Applications in the Analysis of a Simple Memristor Circuit

An automatized method to search for complex symbolic dynamics is proposed. The method can be used to show that a given dynamical system is chaotic in the topological sense. Application of this method in the analysis of a third-order memristor circuit is presented. Several examples of symbolic dynamics are constructed. Positive lower bounds for the topological entropy of an associated return map are found showing that the system is chaotic in the topological sense.

REVISITING PARRONDO'S PARADOX FOR THE LOGISTIC FAMILY

The aim of this paper is to investigate the existence of Parrondo's paradox for the logistic family fa(x) = ax(1 - x), x ∈ [0, 1], when the parameter value a ranges over the interval [1, 4]. We find that a paradox of type "order + order = chaos" arises for both physically observable and topological chaos, while a "chaos + chaos = order" paradox can be only detected for the case of physically observable chaos. In addition, we raise the question of whether the paradox "chaos + chaos = order" can appear in the topological sense or whether, as our computations seem to show, it is impossible for the logistic family.

Wild Milnor attractors accumulated by lower-dimensional dynamics

We present new examples of open sets of diffeomorphisms such that generic diffeomorphisms in those sets have no dynamically indecomposable attractors in the topological sense and have infinitely many chain-recurrence classes. We show that all other classes except one are contained in periodic surfaces. This study allows us to obtain the existence of Milnor attractors as well as study ergodic properties of the diffeomorphisms in those open sets by using ideas and results from Bonatti and Viana [SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly contracting. Israel J. Math.115 (2000), 157–193] and Buzzi and Fisher [Entropic stability beyond partial hyperbolicity. Preprint, 2011, arXiv:1103:2707].

A GENERAL APPROXIMATION ALGORITHM FOR PLANAR MAPS WITH APPLICATIONS

Given an unknown target planar map, we present an algorithm for constructing an approximation of the unknown target based on information gathered from linear probes of the target. Our algorithm is a general purpose reconstruction algorithm that can be applied in many settings. Our algorithm is particularly suited for the setting where computing the intersection of a line with an unknown target is much simpler than computing the unknown target itself. The algorithm maintains a triangulation from which the approximation of the unknown target can be extracted. We evaluate the quality of the approximation with respect to the target both in the topological sense and the metric sense. The correctness of the algorithm and the evaluation of its time complexity are also presented. Finally, we present some experimental results. For example, since generalized Voronoi diagrams are planar maps, our algorithm presents a simpler alternative method for constructing approximations of generalized Voronoi diagrams, which are notoriously difficult to compute.

Intermediate values and inverse functions on non-Archimedean fields

Continuity or even differentiability of a function on a closed interval of a non-Archimedean field are not sufficient for the function to assume all the intermediate values, a maximum, a minimum, or a unique primitive function on the interval. These problems are due to the total disconnectedness of the field in the order topology. In this paper, we show that differentiability (in the topological sense), together with some additional mild conditions, is indeed sufficient to guarantee that the function assumes all intermediate values and has a differentiable inverse function.