Topological entropy and periods of self–maps on compact manifolds

Let (M; f) be a discrete dynamical system induced by a self{map f defined on a smooth compact connected n{dimensional manifold M. We provide sufficient conditions in terms of the Lefschetz zeta function in order that: (1) f has positive topological entropy when f is C1, and (2) f has infinitely many periodic points when f is C1 and f(M) ⊆ Int(M). Moreover, for the particular manifolds Sn, Sn x Sm, CPn and HPn we improve the previous sufficient conditions.The first author of this work was partially supported by MINECO grant number MTM2014-51891-P and Fundación Séneca de la Región de Murcia grant number 19219/PI/14. The second author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant number 2014SGR-568, and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338

Topological entropy of continuous self–maps on a graph

Let G be a graph and f be a continuous self–map on G. We provide sufficient conditions based on the Lefschetz zeta function in order that f has positive topological entropy. Moreover, for the particular graphs: p–flower graph, n-lips graph and (p+r1L1+:::+rsLs)–graph we are able to go further and state more precise conditions for having positive topological entropy.The second author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grants MTM-2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911

Detecting simple dynamics in Cournot-like models

AbstractWe introduced the so-called Cournot-like models, i.e. n-dimensional discrete dynamical systems which constitute the mathematical environment for modeling some economic and biological processes. The main aim of this work is to present a characterization of the dynamical simplicity for these types of systems through the property “to have zero topological entropy”. Cournot-like systems generalize the well-known economic situation of competition in a duopolistic market introduces by Cournot in 1838

Periods of homeomorphisms on closed surfaces

Publicació amb motiu de la International Conference on Difference Equations and Applications (July 22-27, 2012, Barcelona, Spain) amb el títol Difference Equations, Discrete Dynamical Systems and ApplicationsThe goal of this paper is to show what information on the set of periodic points of a homeomorphism on a closed surface can be obtained using the action of this homeomorphism on the homological groups of the closed surface

Solving a novel designed second order nonlinear Lane-Emden delay differential model using the heuristic techniques

The aim of the present study is to present a new model based on the nonlinear singular second order delay differential equation of Lane–Emden type and numerically solved by using the heuristic technique. Four different examples are presented based on the designed model and numerically solved by using artificial neural networks optimized by the global search, local search methods and their hybrid combinations, respectively, named as genetic algorithm (GA), sequential quadratic programming (SQP) and GA-SQP. The numerical results of the designed model are compared for the proposed heuristic technique with the exact/explicit results that demonstrate the performance and correctness. Moreover, statistical investigations/assessments are presented for the accuracy and performance of the designed model implemented with heuristic methodology.This paper has been partially supported by Ministerio de Ciencia, Innovación y Universidades, Spain grant number PGC2018-0971-B-100 and Fundación Séneca de la Región de Murcia, Spain grant number 20783/PI/18

Topological entropy of continuous self-maps on closed surfaces

Altres ajuts: MCIU grant number PGC2018-097198-B-I00 and Fundación Séneca de la Región de Murcia grant number 20783/PI/18The objective of this work is to present sufficient conditions for having positive topological entropy for continuous self-maps defined on a closed surface by using the action of this map on the homological groups of the closed surface

Skew-product maps with base having closed set of periodic points

In [Proc. ECIT-89, World Scientific, (1991), 177–183], A. N. Sharkovski˘ı and S.F. Kolyada stated the problem of characterization skew-product maps having zero topological entropy. It is known that, even under some additional assumptions, this aim has not been reached. In [J. Math. Anal. Appl., 287, (2003), 516–521], J. L. G. Guirao and J. Chudziak partially solved the problem in the class of skew-product maps with base map having closed set of periodic points. The present paper has two aims for this class of maps, on one hand to improve that solution showing the equivalence between the property “to have zero topological entropy” and the fact “not to be Li-Yorke chaotic in the union of the ω-limit sets of recurrent points”. On other hand, we show that the properties “to have closed set of periodic points” and “all nonwandering points are periodic” are not mutually equivalent properties, for doing this we disprove a result from Efremova of 1990

The global sliding mode tracking control for a class of variable order fractional differential systems

In this paper, a novel variable order fractional control approach is proposed for tracking control of both of variable order fractional and constant order fractional order system with uncertain and external disturbance terms. In term of the global sliding mode control theory and terminal sliding mode control method, the system states are guaranteed to stay on the switching surface from the initial time, and then converge to the origin by the designed controllers which are continuous input signals. Such controllers avoid the undesirable chattering and remove the effects of uncertain and external disturbance completely. Finally, the comparison between the variable order fraction controller and the constant order fractional controller is given by numerical simulation, furthermore, numerical results on the effects of the tracking control are provided.This paper has been supported by National Natural Science Foundation of China (No.12002194; No.12072178; No.11732005), Natural Science Foundation of Shandong Province (No.ZR2020QA037; No.ZR2020MA054), Ministerio de Ciencia, Innovación y Universidades (No. PGC2018-097198-B-I00) and Fundación Séneca de la Región de Murcia (No.20783/PI/18)

New trends on nonlinear dynamics and its applications

This paper is the Preface of a special issue devoted to Nonlinear Dynamics and Complexity generated by the contributions of participants at the conference NDC 2015, La Manga, Spain (http://ndc.lhscientificpublishing.com) organized by the Dynamical System Research Group of Región of Murcia (http://www.um.es/sistdinamicos/)

A novel design of fractional Mayer wavelet neural networks with application to the nonlinear singular fractional Lane-Emden systems

In this study, a novel stochastic computational frameworks based on fractional Meyer wavelet artificial neural network (FMW-ANN) is designed for nonlinear-singular fractional Lane-Emden (NS-FLE) differential equation. The modeling strength of FMW-ANN is used to transformed the differential NS-FLE system to difference equations and approximate theory is implemented in mean squared error sense to develop a merit function for NS-FLE differential equations. Meta-heuristic strength of hybrid computing by exploiting global search efficacy of genetic algorithms (GA) supported with local refinements with efficient active-set (AS) algorithm is used for optimization of design variables FMW-ANN., i.e., FMW-ANN-GASA. The proposed FMW-ANN-GASA methodology is implemented on NS-FLM for six different scenarios in order to exam the accuracy, convergence, stability and robustness. The proposed numerical results of FMW-ANN-GASA are compared with exact solutions to verify the correctness, viability and efficacy. The statistical observations further validate the worth of FMW-ANN-GASA for the solution of singular nonlinear fractional order systems.This paper is partially supported by Ministerio de Ciencia, Innovación y Universidades grant number PGC2018-097198-BI00 and Fundación Séneca de la Región de Murcia grant number 20783/PI/18