**Plenary Speakers**

**Viktor 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

**Title:** The Beverton–Holt Equation

**Abstract:** In this talk, we will present the Beverton–Holt equation as used in fisheries and other population models, in many different scenarios (discrete case, continuous case, time scales case, quantum case, periodic case, with and without harvesting etc.).

M. Bohner, J. Mesquita, and S. Streipert, Periodicity on isolated time scales. Math. Nachr., 295(2):259--280, 2022.

M. Bohner, J. Mesquita, and S. Streipert. The Beverton–Holt model on isolated time scales. Math. Biosci. Eng., 19(11):11693--11716, 2022.

M. Bohner, F. M. Dannan, and S. Streipert. A nonautonomous Beverton–Holt equation of higher order. J. Math. Anal. Appl., 457(1):114--133, 2018.

M. Bohner and S. H. Streipert. The second Cushing–Henson conjecture for the Beverton-Holt q-difference equation. Opuscula Math., 37(6):795--819, 2017.

M. Bohner and S. Streipert. Optimal harvesting policy for the Beverton–Holt model. Math. Biosci. Eng., 13(4):673--695, 2016.

M. Bohner and S. Streipert. Optimal harvesting policy for the Beverton–Holt quantum difference model. Math. Morav., 20(2):39--57, 2016.

M. Bohner and S. Streipert. The Beverton–Holt equation with periodic growth rate. Int. J. Math. Comput., 26(4):1--10, 2015.

M. Bohner and S. H. Streipert. The Beverton–Holt q-difference equation with periodic growth rate. In Difference equations, discrete dynamical systems and applications, volume 150 of Springer Proc. Math. Stat., pages 3--14. Springer, Cham, 2015.

M. Bohner and R. Chieochan. The Beverton–Holt q-difference equation. J. Biol. Dyn., 7(1):86--95, 2013.

M. Bohner, S. Stevic, and H. Warth. The Beverton–Holt difference equation. In Discrete dynamics and difference equations, pages 189--193. World Sci. Publ., Hackensack, NJ, 2010.

M. Bohner and H. Warth. The Beverton–Holt dynamic equation. Appl. Anal., 86(8):1007--1015, 2007.

**Elena Braverman**

** **

University of Calgary

**Title:** From continuous delayed differential systems to discrete models: can stochastic perturbations improve control outcome for difference equations?

**Abstract:** Differential and difference equations provide an adequate description of physical and biological phenomena. Significant interest to discrete models is stimulated by complicated types of behaviour exhibited even by simple maps. Differential equations have less sophisticated dynamics unless delay is included. Delay differential equations combine features of ordinary differentiial and difference equations, including chaotic oscillations in equations, not system. Chaos, though predicted in discrete maps, is not as easily observed in nature as cyclic dynamics, and stochastic perturbations are considered as a regulating force. Noise is an integral part of our world, it can be intrinsic, due to internal system processes, as well as extrinsic, due to the influence of the environment. To exclude undesired behaviour of discrete maps, control can be introduced, which can also involve a stochastic component. Introduction of noise into discrete modeling, as well as into controls, is quite a natural part of a model design. While sometimes noise destroys stability of a system, we concentrate on the opposite situation, when deterministic control does not stabillize but stochastic one with the same expected value does. For difference equations and systems, we investigate the influence of stochastic perturbations on population survival, chaos control, eventual cyclic behavior, and on local and global stability. We distinguish between the cases when noise can and when it cannot improve stability. The main purpose is to highlight an active role of stochastic perturbations in stabilization of controlled difference models.

**Guanrong (Ron) Chen**

** **

City University of Hong Kong

**Title:** Chaos in Finite-Dimensional Linear Systems with Weak Topology

**Abstract: **In this talk, we discuss a case of Li-Yorke chaos from a linear system of differential equations in a finite-dimensional Euclidean space with a weak topology, where a solution flow of the system is proved to be Li-Yorke chaotic under certain conditions. This is in sharp contradiction to the well-known fact that linear differential equations cannot be chaotic in a finite-dimensional space with a strong topology. We will also show that there is a sequential version of Li-Yorke chaos generated by iterating a bounded linear map on a finite-dimensional space with a weak topology under some conditions.

**Jim Cushing**

** **

Department of Mathematics, Interdisciplinary Program in Applied Mathematics, University of Arizona, Tucson, USA.

**Title:** Discrete-time Darwinian Dynamics

**Abstract:** Difference equations have a long history of use in defining discrete-time models for the dynamics of biological populations. The vast majority of difference equations that have been used for this purpose are time autonomous and, therefore, assume that model coefficients remain forever constant in time. This assumption is, of course, biologically unrealistic because the coefficients represent biological parameters and mechanisms that, for any number of reasons, typically change over time, especially in natural settings

(but also in even controlled settings). One particularly notable reason for this is related to the most basic principle in biology, namely Darwinian evolution by natural selection. While evolution is often viewed as a slow process (in generation time units), it has been observed in recent decades that it can proceed at a remarkably fast pace (even within a few generations), in both natural and laboratory circumstances. This emphasizes that it is of interest to include evolution population models, especially those whose asymptotic dynamics are investigated.

In this lecture I will review one methodology for including evolution in any discrete-time population model of interest, namely the methodology of Darwinian dynamics (sometimes called evolutionary game theory). I will describe a few general theorems about the equilibrium dynamics of such Darwinian models (motivated by the extinction versus survival struggle). I will briefly discuss trait (strategy) driven-evolution and invasion-driven evolution with the associated concept of an evolutionarily stable strategy (ESS). Darwinian versions of the classic discrete logistic (Beverton-Holt) equation will be used as simple illustrative examples.

**Peter Kloeden**

** **

University Tubingen

**Title:** Spatial discretisation of dynamical systems

**Abstract:** Many insights into complex behaviour of nonlinear dynamical systems are derived from computer studies, so it is important to understand just what effects arise from simulation of continuous processes on the discrete phase space of a finite state machine. This is not the same as the accuracy of representing differential equations by finite difference schemes, but concerns the consequences of discretizing the state space for system dynamics whose ``true" values lie in a continuum.

Such spatial discretizations can produce artifacts such as spurious fixed points and limit cycles that do not exist in the true system evolution in a continuous space. A basic problem here is that any trajectory of the discretized system is eventually periodic, even if. the original system is chaoztic.

What then is the relationship between a chaotic dynamical system and the behaviour of systems based on spatial discretizations?

It will be shown here that the appropriate characterizations to investigate and compare are invariant measures. There are two essentially equivalent approaches depending on how the discretized function is constructed: a random choice of closest values on the discretized space or a set-valued choice of all such points.

P. Diamond and P. E. Kloeden, Spatial discretization of mappings, J. Computers Math. Applns., 26, (1993), 85--94.

P. Diamond, P.E. Kloeden and A. Pokrovskii, Cycles of spatial discretizations of shadowing dynamical systems, Mathematische Nachrichten, 171, (1995), 95--110.

P. Diamond, P.E. Kloeden and A. Pokrovskii, Interval stochastic matrices, a combinatorial lemma, and the computation of invariant measures, J. Dynamics & Diff. Eqns., 7, (1995), 341--364.

P. Imkeller and P.E. Kloeden, On the computation of invariant measures in random dynamical systems, Stochastics & Dynamics, 3, (2003), 247--265.

**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)**∈**R^{2}.

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)**∈** R^{2} (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)**∈**R^{2} there exists n_{x∈N} (that depends on **x**) such that F^{n}_{x=(x,y)}_{∈Γ}.

From the dynamical viewpoint, and among other results, for each b**∈**R we characterize when F|_{Γ} has positive or zero topological entropy.

**Theorem:** Set a=-1. For each b**∈**R consider the map F restricted to the corresponding invariant graph Γ. Then

(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.

The results of this talk have been obtained in collaboration with Anna Cima, Armengol Gasull and Francesc Manosas, and can be found in Cima *et al* (2024).

B. Aiewcharoen, R. Boonklurb and N. Konglawan. Global and local behavior of the system of piecewise linear difference equations x_{n+1} = |x_{n}|-y_{n}-b and y_{n+1} = x_{n}-|y_{n}|+1 where b≤4, Mathematics, 9(12) (2021), 1390-27pp.

I. Bula and A. Sile. About a system of piecewise linear difference equations with many periodic solutions. Talk at ICDEA21, https://pmf.unsa.ba/wp-content/uploads/2021/07/Abstracts-Book-ICDEA-2021.pdf.

A. Cima, A. Gasull, V. Manosa and F. Manosas. Invariant graphs and dynamics of a family of piecewise continuous maps. Preprint, (2024).

R. L. Devaney. A piecewise linear model for the zones of instability of an area-preserving map. Phys. D, 10(3) (1984), 387--393.

E. A. Grove and G. Ladas. Periodicities in nonlinear difference equations, Advances in Discrete Mathematics and Applications, 4, Chapman & Hall/CRC, Boca Raton, FL, 2005.

R. Lozi. Un attracteur étrange (?) du type attracteur de Henon. J. Phys. Colloques, 39 (1978), C5--9--C5--10.

W. Tikjha, Y. Lenbury, E. Lapierre. Equilibriums and periodic solutions of related systems of piecewise linear difference equations. Int. J. Math. Comput. Sim., (2013), 323--335.

W. Tikjha, E. Lapierre and T. Sitthiwirattham, The stable equilibrium of a system of piecewise linear difference equations. Adv. Difference Equ., (2017), Paper No. 67, 10pp.

**Maia Martcheva**

** **

University of Florida

**Title:** Hybrid Discrete-Continuous Epidemic Models

**Abstract:** Discrete models are used for species with non-overlapping generations while continuous ODE models are best suited for species with continuous births and deaths. When species with discrete and continuous generations interact the most adequate mathematical model is a hybrid discrete-continuous ODE model. In this talk I would introduce such a novel class of models and the analytical techniques for studying of such models. The techniques are used to compute the reproduction number of the hybrid system. It is shown that the disease-free equilibrium is locally asymptotically stable if the reproduction number is less than one and unstable, if the reproduction number is grater than one. Numerical methods for computation with such models are introduced and used to simulate the system.

**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.

**Title:** Discontinuous maps and their applications: Exploring the bifurcation structures

Abstract: Nonsmooth dynamical systems, in particular, piecewise smooth maps, are currently actively studied by many researchers from various theoretical and applied fields. Quite often, the main focus is on the study of possible attractors, their basins, as well as the bifurcations that these attractors may undergo. In the parameter space, the existence regions of various attractors can form bifurcation structures that significantly differ from those observed in the smooth case, mainly due to border collision bifurcations occurring when an attractor collides with a border separating different definition domains of the system. The most studied are bifurcation structures associated with 1D piecewise smooth maps, continuous and discontinuous, defined on two partitions (see, for example, Avrutin et al (2019) where bifurcation structures related to attracting cycles and chaotic attractors of 1D piecewise monotone maps are described in detail). Among 2D nonsmooth maps, it is worth mentioning the continuous piecewise linear map known as 2D border collision normal form, which for several decades continues to attract the attention of researchers. Many important results have been obtained, but a complete description of the bifurcation structure of the parameter space of this map is still missing. This talk aims to recall some known results on various bifurcation structures in discontinuous maps and to present new results related to 2D discontinuous piecewise linear maps defined on three partitions. The study of such maps is interesting from a theoretical point of view, and is additionally motivated by their application, for example, in modeling the dynamics of financial markets with heterogeneous agents (see, e.g. Sushko and Tramontana (2024)).

V. Avrutin, L.Gardini, I. Sushko, F.Tramontana. Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps: Invariant Sets and Bifurcation structures. World Scientific Series on Nonlinear Science Series A: Vol. 95, World Scientific, 2019.

I. Sushko, F. Tramontana. Regular and Chaotic dynamics in a 2D discontinuous financial market model with heterogeneous traders. Mathematics and Computers in Simulation, 2024.

**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.