ICDEA2023

Galina Filipuk

University of Warsaw, Poland

Special functions: from orthogonal polynomials to discrete Painleve equations

Abstract

Classical orthogonal polynomials are well-known and are used in many areas of modern mathematics and mathematical physics. What happens if one goes to the semi-classical world? Remarkably, it turns out that in many cases coefficients of the so-called three-term recurrence relation for semi-classical orthogonal polynomials satisfy discrete Painleve equation, which are certain nonlinear discrete dynamical systems. Several examples, appearing in joint papers with Anton Dzhamay, Xing Li, Alexander Stokes, and Da-jun Zhang, will be discussed.

David Simpson

Massey University, New Zealand

Piecewise-linear maps: where do they come from and what do they do?

Abstract

Piecewise-linear maps (generalising the Lozi family) are highly nonlinear yet relatively amenable to analysis. For this reason they provide a useful medium for us to explore nonlinear dynamics at a detailed level, but they also arise as approximations to bifurcations of limit cycles in piecewise-smooth systems of ODEs. In this talk I will overview some of the ways in which the dynamics of piecewise-linear maps differs from those of smooth maps: homoclinic corners instead of homoclinic tangencies, periodicity regions with sausage-string structures, and low-dimensional robust chaos, with results obtained by constructing trapping regions, identifying invariant expanding cones, and using renormalisation.

Bo Zheng

Guangzhou University, Guangzhou, China

Wolbachia -release Dynamics in Mosquito Populations via Discrete Models

Abstract

A promising strategy to eliminate diseases such as dengue or Zika is to suppress or replace the population of the virus transmission vector Aedes mosquitoes by releasing Wolbachia-infected mosquitoes. It has stimulated a great interest in mathematics to model the mosquito population dynamics with Wolbachia interference. In this talk, we first present a discrete model of population suppression driven by the implementation of Wolbachia-release in two islands of the mega-city of Guangzhou, China, which enabled near elimination of Aedes albopictus, the sole dengue vector in Guangzhou. Our model (Nature 2019) accurately predicted target population dynamics in the semi-field cage experiments, and supported the notion that 5:1 ratio of Wolbachia-infected to wild-type males is sufficient for effective population suppression and/or elimination. The second model is for population replacement that characterizes the Wolbachia spread dynamics in mosquito populations, where the environmental change is random. The model consists of a first-order rational difference equation with variable coefficients. To give a preliminary glimpse of its dynamical complexity, we consider an ideal case that environmental conditions change periodically. In this simplified case, by qualitatively analyzing the discrete Poincar \( \acute{e} \) map, we find that the introduction threshold of Wolbachia-infected mosquitoes that must be surpassed for Wolbachia establishment becomes an unstable periodic solution, and the infection will ultimately reach a stable periodic solution. We finally present some mathematical questions that arose from the general model, whose extensive studies may enrich the field of discrete dynamical systems. The talk is based on the joint studies with Prof. Jianshe Yu's research team at Guangzhou University, and Profs. Moxun Tang and Zhiyong Xi at Michigan State University.

Lyudmila Efremova

Department of Differential Equations, Mathematical and Numerical Analysis, Nizhny Novgorod State University

Department of Fundamental Mathematics, Moscow Institute of Physics and Technology Russia

Skew Products and Geometrically Integrable Maps: Current State, Problems and Prospects

Abstract

We describe the historical background and give a survey, including the latest results on skew products and geometrically integrable maps on \(n\)-dimensional \(n \geq 2\) cells, cylinders and tori.
We also formulate some problems, the solution of which, in our opinion, can significantly expand the class of discrete dynamical systems under consideration.

Miguel Sanjuan

King Juan Carlos University, Spain.

Partial Control and Beyond: Controlling Chaotic Transients with the Safety Function

Abstract

A new control algorithm based on the partial control method has been developed. The general situation we are considering is an orbit starting in a certain phase space region Q having a chaotic transient behavior affected by a bounded noise, so that the orbit will definitely escape from Q in an unpredictable number of iterations. Thus, the goal of the algorithm is to control in a predictable manner when to escape. While partial control has been used as a way to avoid escapes, here we want to adapt it to force the escape in a controlled manner. We have introduced new tools such as escape functions and escape sets that once computed makes the control of the orbit straightforward. The partial control method aims to avoid the escape of orbits from a phase space region Q where the transient chaotic dynamics takes place. The technique is based on finding a special subset of Q called the safe set. The chaotic orbit can be sustained in the safe set with a minimum amount of control. We have developed a control strategy to gradually lead any chaotic orbit in Q to the safe set by using the safety function. With the technique proposed here, the safe set can be converted into a global attractor of Q. In addition, we deal with the Hénon and the Lozi maps for a choice of parameters where they show transient chaos, and we compute their safety functions showing the strong dependence of the safety function with the strength of the bounded noise affecting the maps, drastically impacting the controlled orbits.

Michal Misiurewicz

Indiana University-Purdue University Indianapolis, USA

Dynamics of congestion games played by a diverse population

Abstract

Traditional evolutionary game theory is a powerful tool for analyzing statistics of a large population participating in a game. However, the behavior of the individual agents are based on simple memoryless dynamics and this collective behavior is typically represented by a single distribution encoding the frequency of the different actions played deterministically by all the infinitesimal agents. Here we study a more general model where there is a large population of agents of different types. The state of the system is captured not by a single discrete distribution but involves more complex measures describing all possible heterogeneous learning states of the population of agents. We apply this advanced learning model in congestion games, which are well known to admit an essentially unique equilibrium solution. We show that our learning dynamics can exhibit convergence to numerous asymmetric equilibrium states as well as transitions to chaos. Remarkably, even in the chaotic regime, precise predictions can be made about the system performance, as the time-average costs of all actions are shown to be equal to each other, and in fact agree with their values at equilibrium.

Tomasz Downarowicz

Wrocław University of Science and Technology, Poland

Normality versus determinism - Rauzy's approach

Abstract

A number \(x\in\mathbb R\) is normal in base \(b\) if, for any \(k\in\mathbb N\), every block of digits of length \(k\) appears in the \(b\)-ary expansion of \(x\) with density \(b^{-k}\). Equivalently, \(x\) is generic under the map \(x\mapsto bx\!\!\mod 1\) for the Lebesgue measure on the unit interval. Normality can be also described in terms of complexity: the expansion has the highest possible rate of exponential growth. On the other hand, a number \(y\) is deterministic (in base \(b\)) if all shift-invariant measures quasi-generated by the \(b\)-ary expansion of \(y\) under the shift (equivalently, all measures on the interval quasi-generated by \(x\) under the map \(y\mapsto by\!\!\mod 1\)) have entropy zero. This in turn translates to a condition of subexponential growth of complexity. Rauzy characterizes deterministic numbers as exactly those which preserve normality by addition, i.e., \(y\) is deterministic iff \(x+y\) is normal for every normal \(x\). In the talk I will present a strikingly simple proof of this fact, using only elementary ergodic theory (joinings, factors, entropy), and briefly present several possible generalizations resulting from this proof. I will also discuss how subtle and sensitive to even a slight change of the context this phenomenon is. This research is contained in our joint paper which will be posted on Arxiv and submitted soon.

Laura Gardini

University of Urbino Carlo Bo, Italy

Border collision bifurcations in a PWL stock market model described by a 2D discontinuous map

Abstract

This talk summarises the results of some works [1,2,3] in collaboration with D. Radi , N. Schmitt, I. Sushko , F. Westerhoff in which we consider a 2D piecewise-linear discontinuous map driving the dynamics of a simple stock market model. We describe the characteristic properties and bifurcations first in the map defined in two partitions, and then the novelties related to three partitions. We show that a 2-cycle can only be a saddle, and its first homoclinic bifurcation determines, in the parameter space, the boundary of the region associated with bounded trajectories, where the attracting set has the same properties of a Belykh chaotic attractor. There are families of attracting cycles whose periodicity regions are issuing, in the parameter plane, from the center bifurcation line of the fixed point. For these families, we consider the simplest cases of rotation number \(1/n\), \(n\geq 3\), to detect in explicit form the bifurcation boundaries of the corresponding periodicity regions. We show that in the case of three partitions, these regions form two overlapping period-adding structures that issue from the center bifurcation line. That is, each point of this line, associated with a rational rotation number, is an issue point for two different periodicity regions related to attracting cycles with the same rotation number but with different symbolic sequences. Since these regions overlap with each other and with the domain of a locally stable fixed point, we describe the corresponding basins of attraction. We hope that our results contribute to the understanding of the excessively volatile boom-bust nature of stock markets, as well as to the development of the bifurcation theory for discontinuous maps.

Kenneth James Palmer

National Taiwan University, Taiwan

Exponential dichotomies in noninvertible linear difference equations

Abstract

This is a report on two papers written jointly with Flaviano Battelli and Matteo Franca. The theory of exponential dichotomies for invertible difference equations case is similar, although not identical, to that for differential equations. For noninvertible difference equations though, there are substantial differences. It should be mentioned here that all our work is in finite dimensions and we are dealing with uniform exponential dichotomies.

One difference with the invertible case is that in the definition of dichotomy in the noninvertible case, the condition that the coefficient matrix be invertible on the unstable subspace, must be added. One other difference is that results for dichotomies on \([a,\infty)\) do not yield results for \((-\infty,a]\) simply by reversing time.

As in the invertible case, the stable (resp. unstable) subspace for systems on \([a,\infty)\) (resp. \((-\infty,a]\)) is unique, and a system has an exponential dichotomy on \((-\infty,\infty)\) if and only if it has exponential dichotomies on both \([a,\infty)\) and \((-\infty,a]\) with projections of the same rank and has no nontrivial bounded solution.

In the invertible case, when \(a < b\), an exponential dichotomy on \([b,\infty]\) (resp. \((-\infty,a]\)) can always be extended to \([a,\infty]\) (resp. \((-\infty,b]\)). In the noninvertible case, this does not always hold. X.-B. Lin gave necessary and sufficient conditions that the extension could be carried out. In the first of our papers we also determined when the extension could be carried without altering the projection at \(b\) (resp. \(a\)).

The roughness theorem for exponential dichotomies in the invertible case extends to the noninvertible case without any problem. In our first paper we showed that the roughness theorem also holds in the noninvertible case when the perturbations are multiplicative.

Our second paper was concerned with the relation between the dichotomy properties of a block triangular system and its associated block diagonal system. When the latter has an exponential dichotomy, it follows from the roughness theorem that the block triangular system has a dichotomy also, provided the off diagonal elements in the coefficient matrix are bounded. In the invertible case on an interval \([a,\infty)\) or \([a,\infty)\), if the block triangular system has a dichotomy, then the diagonal system has a dichotomy also. However this does not hold for \((-\infty, \infty)\) or for noninvertible systems. Nevertheless if certain conditions are added, then the diagonal system does have a dichotomy and it is also shown that these conditions are needed, since it turns out that if both the triangular and diagonal systems have dichotomies, then these extra conditions must hold.