Some contributions to the analysis of piecewise linear systems.

This thesis consists of two parts, with contributions to the analysis of dynamical systems in continuous time and in discrete time, respectively. In the first part, we study several models of memristor oscillators of dimension three and four, providing for the first time rigorous mathematical results regarding the rich dynamics of such memristor oscillators, both in the case of piecewise linear models and polynomial models. Thus, for some families of discontinuous 3D piecewise linear memristor oscillators, we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible so to justify the periodic behavior exhibited by such three dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. By using the first-order Melnikov theory, we derive the bifurcation set for a three-parametric family of Bogdanov-Takens systems with symmetry and deformation. As an applications of these results, we study a family of 3D memristor oscillators where the characteristic function of the memristor is a cubic polynomial. In this family we also show the existence of an infinity number of invariant manifolds. Also, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role of invariant manifolds in these models. In a similar way than for the 3D case, we study some discontinuous 4D piecewise linear memristor oscillators, and we show that the dynamics in each stratum is topologically equivalent to a continuous 3D piecewise linear dynamical system. Some previous results on bifurcations in such reduced systems, allow us to detect rigorously for the first time a multiple focus-center-cycle bifurcation in a three-parameter space, leading to the appearance of a topological sphere in the original model, completely foliated by stable periodic orbits. In the second part of this thesis, we show that the two-dimensional stroboscopic map defined by a second order system with a relay based control and a linear switching surface is topologically equivalent to a canonical form for discontinuous piecewise linear systems. Studying the main properties of the stroboscopic map defined by such a canonical form, the orbits of period two are completely characterized. At last, we give a conjecture about the occurrence of the big bang bifurcation in the previous map

A locally active discrete memristor model and its application in a hyperchaotic map

© 2022 Springer Nature Switzerland AG. Part of Springer Nature. This is the accepted manuscript version of an article which has been published in final form at https://doi.org/10.1007/s11071-021-07132-5The continuous memristor is a popular topic of research in recent years, however, there is rare discussion about the discrete memristor model, especially the locally active discrete memristor model. This paper proposes a locally active discrete memristor model for the first time and proves the three fingerprints characteristics of this model according to the definition of generalized memristor. A novel hyperchaotic map is constructed by coupling the discrete memristor with a two-dimensional generalized square map. The dynamical behaviors are analyzed with attractor phase diagram, bifurcation diagram, Lyapunov exponent spectrum, and dynamic behavior distribution diagram. Numerical simulation analysis shows that there is significant improvement in the hyperchaotic area, the quasi-periodic area and the chaotic complexity of the two-dimensional map when applying the locally active discrete memristor. In addition, antimonotonicity and transient chaos behaviors of system are reported. In particular, the coexisting attractors can be observed in this discrete memristive system, resulting from the different initial values of the memristor. Results of theoretical analysis are well verified with hardware experimental measurements. This paper lays a great foundation for future analysis and engineering application of the discrete memristor and relevant the study of other hyperchaotic maps.Peer reviewedFinal Accepted Versio

Improving chaos-based pseudo-random generators in finite-precision arithmetic

One of the widely-used ways in chaos-based cryptography to generate pseudo-random sequences is to use the least significant bits or digits of finite-precision numbers defined by the chaotic orbits. In this study, we show that the results obtained using such an approach are very prone to rounding errors and discretization effects. Thus, it appears that the generated sequences are close to random even when parameters correspond to non-chaotic oscillations. In this study, we confirm that the actual source of pseudo-random properties of bits in a binary representation of numbers can not be chaos, but computer simulation. We propose a technique for determining the maximum number of bits that can be used as the output of a pseudo-random sequence generator including chaos-based algorithms. The considered approach involves evaluating the difference of the binary representation of two points obtained by different numerical methods of the same order of accuracy. Experimental results show that such estimation can significantly increase the performance of the existing chaos-based generators. The obtained results can be used to reconsider and improve chaos-based cryptographic algorithms

Entropy in Image Analysis II

Image analysis is a fundamental task for any application where extracting information from images is required. The analysis requires highly sophisticated numerical and analytical methods, particularly for those applications in medicine, security, and other fields where the results of the processing consist of data of vital importance. This fact is evident from all the articles composing the Special Issue "Entropy in Image Analysis II", in which the authors used widely tested methods to verify their results. In the process of reading the present volume, the reader will appreciate the richness of their methods and applications, in particular for medical imaging and image security, and a remarkable cross-fertilization among the proposed research areas