Nonautonomous saddle-node bifurcations: random and deterministic forcing

We study the effect of external forcing on the saddle-node bifurcation pattern of interval maps. By replacing fixed points of unperturbed maps by invariant graphs, we obtain direct analogues to the classical result both for random forcing by measure-preserving dynamical systems and for deterministic forcing by homeomorphisms of compact metric spaces. Additional assumptions like ergodicity or minimality of the forcing process then yield further information about the dynamics. The main difference to the unforced situation is that at the critical bifurcation parameter, two alternatives exist. In addition to the possibility of a unique neutral invariant graph, corresponding to a neutral fixed point, a pair of so-called pinched invariant graphs may occur. In quasiperiodically forced systems, these are often referred to as 'strange non-chaotic attractors'. The results on deterministic forcing can be considered as an extension of the work of Novo, Nunez, Obaya and Sanz on nonautonomous convex scalar differential equations. As a by-product, we also give a generalisation of a result by Sturman and Stark on the structure of minimal sets in forced systems.Comment: 17 pages, 5 figure

Early warning signals for non-smooth bifurcations in population dynamics

Die klassische Sattel-Knotten Verzweigung ist ein paradigmatisches Beispiel eine sogenannten Kritischen Verzweigung (critical transition). Sie wurde in einer Vielzahl von verschiedenen zusammenhängen herausgezogen, insbesondere in der Ökologie und der Klimawissenschaft um die Rolle von langsamen Ruckkehrraten (Recoveryrates) und erhöhten Autokorrelationen als Frühwarnsignale solcher Übergänge zu motivieren. In dieser Arbeit untersuchen wir den Einfluss von externen Störungen auf Sattel-Knotten Verzweigung und die entsprechenden Frühwarnsignale. Unsere Hauptbeispiele sind dabei zwei spezifische Parameterfamilien, die eine Sattelknoten- Bifurkation durchlaufen: die $\arctan$-Familie in diskreter Zeit und populationsdynamische Modelle mit Allee-Effekt in kontinuierlicher Zeit. In beiden Fällen untersuchen wir den Einfluss von quasiperiodischem Antrieb, beschränkten zufälligen Rauschen und einer Kombination aus beiden Antriebsprozessen, die wir als hybriden Antrieb bezeichnen. Wir zeigen, dass die Existenz dieser externen Faktoren zu sogenannten nicht-glatten Verzweigungen führen kann, die durch das Auftreten von seltsamen nicht-chaotischen Attraktoren (SNA) und Repellern charakterisiert sind. Dies hat einen signifikanten Einfluss auf das Verhalten der Lyapunov-Exponenten (und damit der Ruckkehrraten). Während die kritische Verlangsamung (critical slowing down) durch verschwindende Lyapunov-Exponenten am Verzweigungspunkt charakterisiert ist, bleiben die Lyapunov-Exponenten des Attraktors und des Repellers bei einer nicht-glatten Sattel-Knotten Verzweigung von Null verschieden und lassen eine sogenannte Lyapunov Lücke'' (Lyapuno gap) zwischen den beiden Exponenten. Untersuchen wir auch Lyapunov Exponenten in endliche zeit (Finite Time Lyapunov Exponenten, FTLE) und beide können als Frühwarnsignale für kritische Übergänge in unseren Modellsystemen angesehen werden

Strange attractors of forced one-dimensional systems: existence and geometry

Non-smooth saddle-node bifurcations of one-parameter families of quasiperiodically driven dynamical systems on the real line give rise to strange non-chaotic attractors. In this thesis, we provide a class of families which has non-empty interior in the C2-topology and whose elements undergo non-smooth saddle-node bifurcations. Within this class, we study the geometry of the corresponding strange attractors by computing different fractal dimensions. In particular, we show that the Hausdorff dimension differs from the box-counting dimension. We further obtain a description of the minimal set at the bifurcation as a maximal invariant set. Our results treat both the discrete and the continuous time case. A number of explicit examples emphasise the applicability of our findings

Periodic time dependent Hamiltonian systems and applications

[eng] A dynamical system is one that evolves with time. This definition is so diffuse that seems to be completely useless, however, gives a good insight of the vast range of applicability of this field of Mathematics has. It is hard to track back in the history of science to find the origins of this discipline. The works by Fibonacci, in the twelfth century, concerning the population growth rate of rabbits can be already considered to belong to the above mentioned field. Newton's legacy changed the prism through the humankind watched the universe and established the starting shot of several areas of knowledge including the study of difierential equations. Newton's second law relates the acceleration, the second derivative of the position of a body with the net force acting upon it. The formulation of the law of universal gravitation settled the many body problem, the fundamental question around the field of celestial mechanics has grown. Newton itself solved the two body problem, providing an analytical proof of Kepler's laws. In the subsequent years a number of authors, among of them Euler and Lagrange, exhausted Newton's powerful ideas but none of them was able to find a closed solution of the many body problem. By the end of the nineteenth century, Poincaré changed again the point of view: The French mathematician realized that the many body problem could not be solved in the sense his predecessors expected, however, many other fundamental questions could be addressed by studying the solutions of not quantitatively but by means of their geometrical and topological properties. The ideas that bloomed in Poincaré's mind are nowadays a source of inspiration for modern scientist facing problems located along all the spectrum of human knowledge. Poincaré understood that invariant structures organize the long term behaviour of the solutions of the system. Invariant objects are, therefore, the skeleton of the dynamics. These invariant structures and their linear normal behaviour are to be analyzed carefully and this shall lead to a good insight on global aspects of the phase space. For nonintegrable systems the task of studying invariant objects and their stability is, in general, a problem which is hard to be handled rigorously. Usually, the hypotheses needed to prove specific statements on the solutions of the systems reduce the applicability of the results. This is especially relevant in physical problems: Indeed, we cannot, for instance, choose the mass of Sun to be suficiently small. The advent of the computers changed the way to undertake studies of dynamical systems. The task of writing programs for solving, numerically, problems related to specific examples is, at the present time, as important as theoretical studies. This has two main consequences: On the first hand, more involved models can be chosen to study real problems and this allow us to understand better the relation between abstract concepts and physical phenomena. Secondly, even when facing fundamental questions on dynamics, the numerical studies give us data from which build our theoretical developments. Nowadays, a large number of commercial (or public) software packages helps scientist to study simple problems avoiding the tedious work to master numerical algorithms and programming languages. These programs are coded to work in the largest possible number of different situations, therefore, they do not have the eficiency that programs written specifically for a certain purpose have. Some of the computations presented in this dissertation cannot be performed by using commercial software or, at least, not in a reasonable amount of time. For this reason, a large part of the work presented here has to do with coding and debugging programs to perform numerical computations. These programs are written to be highly eficient and adapted to each problem. At the same time, the design is done so that specific blocks of the code can be used for other computations, that is, there exist a commitment between eficiency and reusability which is hard to achieve without having full control on the code. Under these guiding principles we undertake the study of applied dynamical systems according to the following stages: From a particular problem we get a simple model, then perform a number of numerical experiments that permits us to understand the invariant objects of the system, with that information, we can isolate the relevant phenomena and identify the key elements playing a role on it. Next, we try to find an even simpler model in which we can develop theoretical arguments and produce theorems that, with more effort, can be generalized or related to other problems which, in principle, seem to be difierent to the original one. Paraphrasing Carles Simó, from a physical problem we can take the lift to the abstract world, use theoretical arguments, come out with conclusions and, finally, lift down to the real world and apply these conclusions to specific problems (maybe not only the original one). This methodology has been developed in the last decades over the world when it turned out to outstand among the most powerful approaches to cope with problems in applied mathematics. The group of Dynamical Systems from Barcelona has been one of the bulwarks of this development from the late seventies to the present days. Following the guidelines presented in the previous section, we concern with several problems, mostly from the field of celestial mechanics but we also deal with a phenomenon coming from high energy physics. All these situations can be modeled by means of periodically time dependent Hamiltonian systems. To cope with those investigations, we develop software which can be used to perform computations in any periodically perturbed Hamiltonian system. We split the contents of this dissertation in two parts. The first one is devoted to general tolos to handle periodically time dependent Hamiltonians, even though we fill this first part with a number of illustrating examples, the goal is to keep the exposition in the abstract setting. Most of the contents of Part I deal with the development of software used to be applied in the second part. Some of the software has not been applied to the specific contents of Part II, this is left for future work. We also devote a whole chapter to some theoretical issues that, while are motivated by physical problems, they fall out of the category of periodic time dependent Hamiltonians. This splitting of contents has the intention of reecting, somehow, the basic methodological principles presented in the previous paragraph, keeping separated the abstract and the physical world but keeping in mind the lift

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio