Plenary speakers

Abstract: A map is said to exhibit trivial dynamics when all omega limit sets are subsets of fixed points. We review some trivial dynamics in some ecological maps by way of topological methods, monotone systems theory and Lyapunov functions. Then we look specifically at the case where the map is planar and non-injective, such as for the Ricker map, and obtain interesting results on stable and unstable sets that can be viewed as a version of the competitive exclusion principle that is well-known in ecology.

Abstract: Our main focus is on the introduction and analysis of discrete-time population models that exhibit multiple attractors. Most widely studied single-species density-dependent models without structure - such as the Beverton-Holt equation, the Leslie-Gower competition model, and the Ricker competition model-are generally incapable of supporting multiple attractors. However, as demonstrated in [1], even the Leslie-Gower model can admit multiple boundary attractors when interior equilibria are absent. In this talk, we extend our investigation to include a wide variety of structured and unstructured models - single-species and multi-species alike-that support multistability. 

When multiple attractors occur, the long-term fate of a population becomes strongly dependent on initial conditions. This implies that one cannot predict all possible outcomes based on a single trajectory or data set. The phenomenon of deterministic uncertainty is exacerbated in the presence of environmental noise, which can induce transitions between basins of attraction. When coexisting attractors include cycles or chaotic sets, and the system exhibits sensitive dependence on initial conditions, the basins of attraction may become intricately intermingled, further complicating prediction [4]. 

A particularly well-known mechanism leading to multiple attractors is the Allee effect, where the per capita growth rate increases at low population densities due to biological advantages of aggregation, such as enhanced mating success, group foraging, or protection from predators [2, 5, 6]. A strong Allee effect occurs when the per capita growth rate fails to exceed one until the population crosses a critical threshold, resulting in two attractors: extinction and a survival state. These effects arise from positive density dependence in components of fitness (e.g., birth rate or survival rate), in contrast to the negative density dependence found in classical models like Beverton-Holt or Ricker. 

We examine both unstructured and structured models exhibiting multistability through various mechanisms, including strong Allee effects, double backward bifurcations, and hysteresis loops in bifurcation diagrams [6]. Examples include semelparous Leslie models where juvenile survival increases with density, yielding three attractors: extinction, a stable positive equilibrium, and a stable 2-cycle. We also study two-patch dispersal models that display bistability in the absence of Allee effects, as well as multi-species competition models, predator-prey models, age-structured models, and epidemic systems. These models illustrate how biologically motivated nonlinearities and structure can yield rich dynamical outcomes. Our treatment follows and expands the approaches of [1] and [3], combining bifurcation analysis with biological interpretation.

References

[1] J. C. Alexander, J. A. Yorke, Z. You, and I. Kan, ”Riddled basins,” International Journal of Bifurcation and Chaos, 2(4):795-813, 1992.

[2] F. Courchamp, L. Berec, and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008.

[3] J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, 1998.

[4] S. Elaydi and R. Sacker, Population models with Allee effects: A new model, Journal of Biological Dynamics, 4(4):397-408, 2010.

[5] S. Elaydi and J. M. Cushing, Discrete Mathematical Models in Population Biology: Ecological, Epidemic, and Evolutionary Dynamics, Springer, 2025.

[6] S. Elaydi, E. Kwessi, G. Livadiotis, Hierarchical competition models with the Allee effect III: multispecies, Journal of Biological Dynamics, 12(1): 271-287, 2018.

Abstract: Time-delay systems are omnipresent in control applications where transmission, actuation, or sensing delays play a critical role. Understanding their behavior through the lens of positive invariance opens the door to robustness and constraint satisfaction guarantees in control design. This talk bridges classical delay-independent invariance concepts with emerging delay-dependent formulations.

Starting from the foundational notion of D-invariance, we explore the set-theoretic characterization of invariant sets for linear discrete-time delay systems. Particular attention is given to how D-invariance offers a delay-independent perspective with fixed prescribed complexity, yet lacking full characterizability.

Building on this, we introduce recent developments, which incorporate delay-dependent invariance notions. These offer refined tools by adapting the invariant set definition to account for specific delay values and their variability. This shift enables improved conservativeness in controller synthesis and stability analysis, especially when dealing with time-varying or uncertain delays.

The talk will illustrate the theoretical underpinnings, necessary and sufficient conditions, and set factorization approaches, along with practical algorithms and examples. We will emphasize how the combination of geometric, algebraic, and iterative tools leads to a more nuanced understanding of system invariance in the presence of delays.

Abstract: In a smooth world, curves, surfaces, and more generally, manifolds intersect each other transversely, if at all; tangencies are rare events that cannot be observed in experiments. In dynamical systems theory, therefore, it was long assumed that tangencies between invariant manifolds occur at isolated points when a parameter is varied, and the transition from tame to chaotic dynamics is mediated by a single tangential event. Recent theoretical work by Bonatti, Diaz, and others has shown that the boundary between tame and chaotic dynamics is, actually, more like a thick grey world that challenges our geometric intuition: tangencies may occur robustly, which is called wild chaos. This type of dynamics requires at least three dimensions for discrete-time systems, or four for a system of ordinary differential equations. This higher dimensionality has been an impediment to our understanding of how steady states, periodic solutions, and their invariant manifolds organise wild chaotic dynamics.

In this talk, I will discuss how manifolds can have persistent tangential intersections as a parameter is varied. The starting point will be classical chaos in the planar H$\Acute{e}$non map, a simple polynomial dynamical system. I will then consider a three-dimensional extension and explain its counter-intuitive properties: this map has one-dimensional invariant manifolds that cannot be avoided by other smooth curves. Hence, these one-dimensional manifolds behave as though they are two dimensional. With a careful combination of dynamical systems theory and advanced computational methods, I will show what wild chaos and robust tangencies look like, how they arise, and why this matters for applications.

Abstract: The present lecture devotes to the review of recently obtained numerical results concerning the impact of noise on the dynamics and spatio-temporal patterns in networks of discrete-time nonlinear systems. As individual elements three well-known chaotic maps are chosen, the Hénon map, the Lozi map, and the Hénon-Lozi map. The ring network of nonlocally coupled Hénon maps is characterized by the appearance of chimera states which accompany the transition from complete synchronization to spatio-temporal chaos with decreasing the coupling strength between nodes [4]. The chimera state corresponds to the coexistence of clusters with coherent and incoherent dynamics of elements of a network [2]. The ring network of nonlocally coupled Lozi maps demonstrates another scenario of the transition, which is related to the appearance of solitary states [3]. Solitary states denote network regimes when solitary nodes coexist with coherent or synchronized groups of elements [8]. The Hénon-Lozi map represents a synthesis of the Hénon and Lozi maps and combines their dynamic properties [1]. Thanking this feature, the ring network of coupled Hénon-Lozi maps can exhibit both chimera states and solitary states.

It is shown that chimeras and solitary states demonstrate different response and robustness toward external noise perturbations. The dynamics of three aforementioned networks of coupled maps are systematically studied in the presence of noise, including Lévy noise, when the coupling parameters of the networks and the noise characteristics are varied. It is shown that additive and multiplicative noise can increase the probability of observing chimera states in ring networks of of nonlocally coupled Hénon maps and Hénon-Lozi maps [5, 6, 7]. This means that there is an optimum noise level at which the interval of the coupling strength within which chimeras are observed with a high or even maximum probability is the widest. This effect constitutes the constructive role of noise in analogy with stochastic and coherence resonance and is referred to as chimera resonance [5].

It is revealed that even weak noise can sustain the solitary state regime in the range of weak nonlocal coupling in the networks of nonlocally coupled Lozi maps and Hénon-Lozi maps. However, stronger noise causes the solitary nodes to disappear for any values of the coupling strength [7].

Thus, the obtained results clearly indicate that external noise can serve as an effective tool for controlling the spatio-temporal dynamics and structures of complex networks.

[1] M. Aziz-Alaoui, C. Robert, C. Grebogi, Dynamics of a H´enon-Lozi-type map. Chaos, Solitons and Fractals 12(12), 2323 (2001).

[2] Y. Kuramoto and D. Battogtokh, Nonlinear Phenom. Complex Syst. 5, 380 (2002).

[3] Y. Maistrenko, B. Penkovsky, and M. Rosenblum, Phys. Rev. E 89, 060901 (2014).

[4] I. Omelchenko, B. Riemenschneider, P. H¨ovel, Y. Maistrenko, E. Sch¨oll, Phys. Rev. E 85(2), 026212 (2012).

[5] E. Rybalova, V. Nechaev, E. Sch¨oll, and G. Strelkova, Chaos 33, 093138 (2023).

[6] E. Rybalova, E. Sch¨oll, and G. Strelkova, J. Diff. Equ. Appl. 29(9-12), 909 (2023).

[7] E.V. Rybalova, V.M. Averyanov, R. Lozi, G.I. Strelkova, Chaos, Solitons and Fractals 184, 115051 (2024).

[8] N. Semenova, A. Zakharova, E. Sch¨oll, V. Anishchenko, Europhys. Lett. 112(4), 40002 (2015).

Abstract:  This talks suggests discrete fractional calculus for numerical discretization of fractional differential equations. First, the physical meaning and background of fractional calculus are revisited. The traditional numerical methods face challenges in initial value problems. Then, a fractional difference equation approach is employed to address the initial value non-smoothness’s problem. The convergence result is provided. Finally, applications in data-driven fractional differential equations and machine learning are considered.

Abstract:  In this presentation, I will review our results [1, 2], which demonstrate that a high-dimensional map in the form of a deep neural network (DNN) can be modelled using a single delay-differential equation with multiple delays. This single-neuron DNN comprises only a single nonlinearity and appropriately adjusted modulations of the feedback signals. Network states emerge over time as a temporal unfolding of the neuron’s dynamics. By adjusting the feedback modulation within the loops, we adapt the network’s connection weights. These connection weights are determined via a back-propagation algorithm, in which both the delay-induced and local network connections must be taken into account. Our approach can fully represent standard DNN and extends the DNN concept toward dynamical systems implementations. This new method is called Folded-in-Time DNN (Fit-DNN), and has been tested in a set of benchmark tasks.

References

[1] Florian Stelzer, Andr´e R¨ohm, Raul Vicente, Ingo Fischer, Serhiy Yanchuk, Deep neural networks using a single neuron: folded-in-time architecture using feedback-modulated delay loops. Nat Commun 12, 5164 (2021).

[2] Stelzer, Yanchuk, Emulating complex networks with a single delay differential equation. Eur. Phys. J. Spec. Top. 230, 2865–2874 (2021).

Abstract: 

Abstract: Usually, the complexity of dynamics is understood as a result of iterated nonlinearity. In this talk we show that some “tumors”--bad terms or bad points can be erased by iteration. We introduce results on degree-preserving and repair of continuity and smoothness under iteration.

Abstract: