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Plenary speakersPlenary SpeakersViktor Avrutin
Institute for Systems Theory and Automatic Control, University of Stuttgart, Germany Title: Chaotic Attractors and their Bifurcations in 1D and 2D maps Abstract: Chaotic attractors can be considered from several different perspectives. They may be appreciated for their aesthetic beauty, or analyzed for the geometric properties, distinguishing between attractors with fractal and non-fractal dimensions. Keeping all these aspects in mind, in this talk, we consider chaotic attractors from the bifurcation theory point of view. Here, the central questions are how chaotic attractors appear and which transformations they may eventually undergo. There is a long tradition behind the first question, frequently referred to as "routes to chaos". As for the second question, the complete picture is still missing. It is known that many of such transformations are related to some homoclinic bifurcations, leading, for example, to a sudden expansion of a chaotic attractor (so-called interior crises, also known as expansion bifurcations). In addition to that, in piecewise smooth systems, robust chaotic attractors may undergo border collision bifurcations. In general, these transformations of chaotic attractors are not related to homoclinic bifurcations and can be explained by mainly geometric reasoning. The effects caused by border collision bifurcations of chaotic attractors are diverse. In particular, such a bifurcation may cause an arbitrary number of additional bands of the attractor to occur, as well as an arbitrary number of additional holes inside the existing bands. Under specific conditions, more effects are possible, including a sudden expansion of the attractor or a sudden jump in its fractal dimension (a transition from a non-fractal to a fractal chaotic attractor). This presentation will provide an overview of bifurcation leading to transformations of chaotic attractors, starting with classical scenarios related to homoclinic bifurcations and concluding with current findings on border collision bifurcations of chaotic attractors. V. Avrutin, A. Panchuk and I. Sushko, Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities, Proc. R. Soc. A, 477, 20210432, 2021. V. Avrutin, A. Panchuk and I. Sushko, Can a border collision bifurcation of a chaotic attractor lead to its expansion?, Proc. R. Soc. A, 479, 20230260, 2023. V. Avrutin and I. Sushko, Border collision bifurcations of chaotic attractors in two-dimensional discontinuous maps, forthcomnig.
Soumitro Banerjee
Indian Institute of Science Education & Research, Kolkata, India Title: Bifurcations of tori Abstract: It is known that invariant closed curves in maps, representing tori in continuous-time phase space, undergo various bifurcations. This includes two types of torus doubling, the generation of a third frequency, and the merger and disappearance of stable and unstable invariant closed curves. In the past, a few approaches have been proposed to analyse and predict the outcome of such bifurcations. We review these approaches and show that some of these may fail under certain conditions. We propose an alternative approach that applies to both resonant and ergodic tori.
Martin Bohner
Missouri S&T, USA
Elena Braverman
University of Calgary
Guanrong (Ron) Chen
City University of Hong Kong
Jim Cushing
Department of Mathematics, Interdisciplinary Program in Applied Mathematics, University of Arizona, Tucson, USA.
Peter Kloeden
University Tubingen Agnieszka Malinowska
Faculty of Computer Science, Bialystok University of Technology, Poland Title: Consensus in Fractional-Order Systems and Systems Defined on Time Scales. Abstract: Opinion dynamics is a field of study that encompasses mathematical models describing interactions among groups of entities known as agents, including humans, schools of fish, or swarms of robots. This talk focuses on a fundamental phenomenon in such systems, namely, the emergence of agreement or consensus. Consensus occurs when a group of agents agree on shared values or opinions, such as position, velocity, or price. By incorporating fractional difference operators instead of classical ones, memory effects are introduced into the system dynamics. It’s worth noting that individuals, animals, and electronic systems possess memory, which influences their behaviors. Models of opinion formation are particularly relevant when analyzed on discrete-time domains, where agent meetings and information exchanges occur intermittently. Leveraging time scales theory allows for a flexible exploration of various sampling or time step configurations in these processes. Results obtained highlight the significance of both memory and the duration of time gaps between information exchanges among agents in shaping opinion dynamics and achieving consensus. In situations where consensus is not reached, polarization or chaos may ensue. In such cases, one strategy to steer all agents toward consensus involves introducing a (virtual) leader and implementing control mechanisms within the system. This concept draws parallels from real-world phenomena, such as the relationship between a sheepdog and sheep or the influence of mass media on societal opinions. Controlling the system through a leader finds practical applications, such as crowd evacuation during emergencies or designing reference trajectories for guiding groups of robots. Control strategies may vary, depending on factors such as the coupling strength between agents or the desired level of external intervention. These strategies will also be presented. Through rigorous research and analysis, this talk offers insights into control strategies tailored for achieving consensus in fractional-order systems and systems operating on time scales.
Victor Manosa
Departament de Matematiques and Institut de Matematiques Universitat Politecnica de Catalunya Title: Invariant graphs and dynamics in a family of piecewise linear maps Abstract: Invariant sets are very important objects in the study of dynamical systems. Its significance is clear in the case of systems possessing a first integral (an invariant), but this is also the case of highly dissipative systems as the one that we will consider in the talk: the family of piecewise linear maps of the form: F(x,y) = (|x| - y + a, x - |y| + b), where (a,b)∈R2. Years ago, Grove and Ladas (2005) introduced the family of maps G(x,y)=(|x| + α y + βx + γ |y| + δ) with α,β,γ ∈R and δ∈{-1,0,1}, with the aim of generating a broader framework for studying generalized Lozi-type maps. These kind of maps were, in turn, introduced in the late 70ies by R. Lozi (1978). A particular case, known as the Gingerbread map, was studied in the 80ies by R. Devaney (1984). The family of maps F intersects the general family of Grove and Ladas. In the last years, some works have appeared analyzing different particular cases of the Grove-Ladas family, see for example Aiewcharoen et al (2021), Tikjha et al (2013), Tikjha et al (2017), just to cite some works that include subcases of the family F. Essentially, these works characterize cases in which all orbits converge to fixed points, periodic orbits, or are eventually-periodic. In fact, our motivation for the study of F was I. Bula's talk at the Sarajevo's 26th ICDEA, Bula and Sile (2021). In that talk it was expressed the conjecture that for a,b<0 all orbits are eventually periodic. As we will see, the global dynamics of the family F is substantially richer and the invariant sets of the map play an important role. First, we prove that for a≥0 all the orbits are eventually-periodic and moreover the set of periodic orbits has finite cardinality: Theorem A: If a≥0 then F has trivial dynamics. Moreover the set of periodic points of F has finite cardinality. The interesting situation occurs when a<0. The interesting situation occurs when a<0. For this case, we prove that for all b∈ R there exists a compact invariant graph γ, such that the orbit of all (x,y)∈ R2 (except perhaps a fixed points or some periodic orbits that are well characterized) converges to Γ in a finite number of iterates. Theorem B: Set a=-1. For all b∈R there exists a compact graph Γ which is invariant under the map F such that for all x=(x,y)∈R2 there exists nx∈N (that depends on x) such that Fnx=(x,y)∈Γ. (a) For b∈(-∞,-112/137]∪[-1/36,603/874]∪{1}∪[8,∞) the map F|Γ has zero entropy. (b) For b∈[-13/16,-1/36)∪[563/816,1)∪(1,8) the map F|Γ has positive entropy. (c) For b∈[-112/137,-13/16]∪[603/874,563/816] is non-decreasing in b, and there is a value of b in each of the above intervals, so that the entropy is zero in one of its subintervals and positive in the other. B. Aiewcharoen, R. Boonklurb and N. Konglawan. Global and local behavior of the system of piecewise linear difference equations xn+1 = |xn|-yn-b and yn+1 = xn-|yn|+1 where b≤4, Mathematics, 9(12) (2021), 1390-27pp.
Maia Martcheva
University of Florida
Adina Luminita Sasu
Faculty of Mathematics and Computer Science, West University of Timisoara and Academy of Romanian Scientists, Bucharest, Romania Title: Admissibility and Asymptotic Behaviors of Dynamical Systems: from Global Methods to Ergodic Theory Approaches Abstract: The admissibility methods represent some of the most important class of tools in exploring the asymptotic behaviors of dynamical systems. Their impact is due to both their effectiveness in describing various phenomena of uniform or nonuniform nature (such as stability, expansivity, dichotomy, trichotomy) through adequate solvability conditions imposed to certain input-output systems as well as due to their spectacular applications in control theory. In this lecture, we will present new admissibility methods for detecting the nonuniform/uniform asymptotic properties of discrete dynamical systems, that employ several representative classes of sequence spaces (invariant under translations or not), beginning with global conditions that take place on all the points of the parameter space and arriving to local-type conditions that are satisfied only on some subsets of positive measure. Starting from the techniques and results developed in [1-4], we present complete characterizations for both stability and expansiveness of discrete variational systems from three perspectives, as follows: in each case we discuss a global method and two local approaches (that rely on ergodic theory arguments) and we also highlight several new applications. In the first part, we provide input-output conditions for nonuniform and uniform exponential stability by employing input and output spaces from three main classes of sequence spaces. In the second part, we introduce several notions of exact admissibility to explore the nonuniform and uniform exponential expansiveness and in this aim we use three specific classes of sequence spaces. For both stability and expansiveness, besides giving existence criteria, we discuss the hypotheses and the minimal requirements on the underlying input or output spaces via relevant examples. After that, we present applications to the study of stability and expansiveness of continuous-time dynamics described by skew-product semiflows, by using nonuniform admissibility conditions with pairs of function spaces. Finally, inspired by some methods introduced in [5] we point out open problems, new directions and we discuss several future aims. [1] D. Dragicevic, A. L. Sasu, B. Sasu, On the asymptotic behavior of discrete dynamical systems - An ergodic theory approach, J. Differential Equations 268 (2020), 4786-4829. [2] D. Dragicevic, A. L. Sasu, B. Sasu, On stability of discrete dynamical systems: from global methods to ergodic theory approaches, J. Dynam. Differential Equations 34 (2022), 1107-1137. [3] D. Dragicevic, A. L. Sasu, B. Sasu, L. Singh, Nonuniform input-output criteria for exponential expansiveness of discrete dynamical systems and applications, J. Math. Anal. Appl. 515 (2022), 1-37. [4] D. Dragicevic, A. L. Sasu, B. Sasu, Input-output criteria for stability and expansiveness of dynamical systems, Appl. Math. Comput. 414 (2022), 1-22. [5] A. L. Sasu, B. Sasu, Input-output criteria for the trichotomic behaviors of discrete dynamical systems, J. Differential Equations 351 (2023), 277-323.
Iryna Sushko
Università Cattolica del Sacro Cuore, Milan, Italy. The National Science Foundation Amina Eladdadi
The National Science Foundation, Alexandria, VA, USA The U.S National Science Foundation (NSF) Program Director: A brief presentation of funding opportunities related to the NSF Division of Mathematical Sciences (DMS) Programs and international collaborations, followed by a Q&A with NSF Program Director.
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