Alex Solynin (USA)
Tuesday (05/08): 09:50AM -10:15AM
We will discuss a broad variety of problems on the heat distribution, ranging from an everyday question on where to place heater to feel yourself most comfortable in your favorite chair to the question on what should be the shape of a heating patch attached to the surface of toroidal solid, aka Tokamak, to provide an optimal temperature at the marked position in this solid? In a 3D setting, the total heat flux from the surface of a solid equals the Newtonian capacity of this solid and in the planar case it is related to the logarithmic capacity. Thus, some problems on the Newtonian capacity of configurations consisting of n balls in space and on the logarithmic capacity of n disks in the plane will be discussed. This study was initiated by M.L. Glasser and S.G. Davison in 1978, who considered the so-called “Sleeping armadillos problem”, that is the problem on the distribution of heat in systems of n warm-blooded creatures when they bundle together. We will identify configurations which minimize the Newtonian capacity or logarithmic capacity under certain geometrical restrictions. Then, we will prove that the linear string of balls maximizes the Newtonian capacity among all strings consisting of n equal balls and that the circular necklace maximizes the logarithmic capacity over the set of all necklaces consisting of n equal disks. Several open questions on the capacities of constellations of balls in space and disks in the plane also will be also discussed.
André L P Livorati (BRA)
Tuesday (05/08): 04:45PM – 05:10PM
We investigate some statistical and transport properties of the relativistic standard map. Through the Hamiltonian of a wave packet under an electric potential, we are able to obtain a relativistic version of the standard map, where there are two control parameters that rules the dynamics, $K$ which is the classical intensity parameter and $\gamma$, that controls the relativity. The phase space is mixed, and has a confined local chaos for $\gamma$ near the unity,
which approaches the integrability. When $\gamma$ is diminished, i.e., in the semi classical regime, the diffusion in the action variable start to act. However, the phase space looses its axial symmetry and a invariant curve appears to limit the diffusion as $\gamma$ gets smaller. Considering this, we investigate the diffusion in the action variable as function of the number of iterations. The root mean square action grows for a small number of iterations and bend towards a saturation regime for long times. Scaling properties were set up for this behavior as function of $\gamma$, and a perfect collapse for the curves
were obtained indicating a scaling invariance. In addition, we investigated the transport properties concerning the evolution of the survival probability of initial conditions, where a escape region were set up near the saturation region of the root mean square action curves. The decay rate of the survival probability are mainly exponential, and power law tails. As we range the value of $\gamma$, the escape rates become slower and also obey a scaling in their decay.
Arturo C. Marti (URU)
Wednesday (06/08): 09:50AM – 10:15AM
The Mackey–Glass system is a paradigmatic example of a delayed dynamical model whose complexity stems from features such as its infinite-dimensional phase space and pronounced multistability, including the coexistence of numerous periodic and chaotic attractors. Predicting the system’s long-term behavior is particularly challenging, as initial conditions must be specified as functions over a finite time interval. To address this, we extend the concept of basin entropy, a measure of predictability in multistable systems, to infinite-dimensional delayed systems. By employing a stochastic sampling strategy in high-dimensional spaces and complementing it with the analysis of basin fractions, we gain insight into the intricate organization and intermingling of basins of attraction. Furthermore, we demonstrate that basin entropy can serve as a diagnostic tool: it captures qualitative changes in basin structure near bifurcations, such as Hopf bifurcations, though it may be insensitive to others, like pitchfork bifurcations that do not affect basin geometry. Our findings show that basin entropy is a powerful tool for quantifying predictability and understanding bifurcation-related transitions in time-delayed systems. These results not only offer a framework for probing the global dynamics of infinite-dimensional systems but also point toward new directions. These results offer a framework for probing the global dynamics of infinite-dimensional systems. Current work is actively exploring transient dynamics, extensions to noisy or externally driven systems, and potential applications to experimental settings.
References
[1] Tarigo, J. P., Stari, C., Masoller, C., & Martí, A. C. (2024). Basin entropy as an indicator of a bifurcation in a time-delayed system. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(5).
[2] Tarigo, J. P., Stari, C., & Martí, A. C. (2024). Basin of attraction organization in infinite-dimensional delayed systems: A stochastic basin entropy approach. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(12).
[3] Tarigo, J. P., Stari, C., Cabeza, C., & Marti, A. C. (2022). Characterizing multistability regions in the parameter space of the Mackey–Glass delayed system. The European Physical Journal Special Topics, 231(3), 273-281.
Carla M. A. Pinto (POR)
Friday (08/08): 09:25AM – 09:50AM
We present a novel mathematical model describing the co-dynamics of dengue and COVID-19 using a nonlinear system of differential equations. The basic reproduction number is derived and analyzed for outbreak prediction and sensitivity. A stochastic Itô formulation extends the deterministic model, supported by numerical simulations. Model validation is performed using Colombian epidemiological data via global and staged fitting. Key challenges include parameter unidentifiability and data scarcity. Certain parameters were fixed to enable optimization, warranting cautious interpretation of results.
Dimitri Volchenkov (USA)
Thursday (07/08): 09:50AM -10:15AM
This invited talk presents a unified mathematical framework for modeling belief formation and propagation in complex social networks, synthesizing two recent contributions. In the first work, we develop a nonlinear reaction–diffusion theory of belief dynamics in hierarchical systems, inspired by contrasting apostolic archetypes. The model integrates classical FKPP propagation with a novel degenerating diffusion (DD) mechanism that accounts for saturation-induced inhibition. We analytically characterize the spatiotemporal structure of belief fronts under varying diffusivity and obstinacy, demonstrating the emergence of quasi-synchronous activation in network interiors when traditional diffusion fails to reach peripheral agents. These findings challenge canonical assumptions about centralized influence and suggest that robust consensus can emerge from mutual reinforcement even in the absence of strong top-down signals. In the second contribution, we introduce a probabilistic theory of social conformity and belief adoption grounded in the matrix logistic differential equation. This formulation models belief as a dynamically evolving probability, incorporating both structural influence and individual resistance. By deriving explicit bounds for mean learning time (MLT) under correlated and uncorrelated influence regimes, we demonstrate how network topology and initial conditions shape the speed and reach of consensus. The theory yields closed-form results on saturation thresholds, autopoietic amplification of weak signals, and logistic-optimal centrality in tree-like and fully connected graphs. Taken together, these models offer a comprehensive and analytically tractable approach to understanding belief dynamics across a spectrum of social structures. By separating fast local consensus from slow global diffusion, and distinguishing between pulled and pushed belief propagation mechanisms, we derive robust constraints on the effectiveness of propaganda and identify critical conditions under which peripheral agents may resist or accelerate system-wide adoption. Implications include structural limits on misinformation, emergent coherence without central authority, and new diagnostic tools for assessing influence strategies in political, organizational, and algorithmic contexts.
Everton S Medeiros (BRA)
Tuesday (05/08): 05:10PM – 05:35PM
Utilizing a paradigmatic model for the motion of interacting self-propelled particles, we demonstrate that local accelerations at the level of individual particles can drive transitions between different collective dynamics, leading to a control process. We find that the ability to trigger such transitions is hierarchically distributed among the particles and can form distinctive spatial patterns within the collective. Chaotic dynamics occur during the transitions, which can be attributed to fractal basin boundaries mediating the control process. The particle hierarchies described in this presentation offer decentralized capabilities for controlling artificial swarms.
Gonzalo Marcelo Ramírez-Avila (BOL)
Wednesday (06/08): 11:35AM -12:00PM
Taking into consideration six dynamical systems (three continuous and three discrete), and combining Lyapunov exponents and periodicity indicators, we describe in the parameter space pseudofractal regions, quasiperiodicity, chirality, and a variety of structures characterizing regular dynamical behavior.
Even though our findings appear as simple curiosities, they open the possibility to go in-depth in the fundamental aspects of dynamical systems theory, and also, in several situations, can lead to potential practical applications in technological and biomedical systems.
Haris Skokos (ZAF)
Thursday (07/08): 05:35PM – 06:00PM
We numerically study the dynamics of initially localized excitations in a one-dimensional stub lattice model in the presence of disorder and nonlinearity. The model’s piecewise frequency spectrum is comprised by a near flat band and two non-flat spectra separated by distinct gaps when the disorder strength is below a threshold value. We theoretically predict and numerically observe three different dynamical regimes induced by chaos, namely the weak and strong chaos spreading regimes, and the self-trapping regime. Our numerical simulations show subdiffusive spreading for relatively large disorder strengths for both the weak and strong chaos regimes, which are characterized by specific exponents in the power law increase of the wave packets’ second moment evolution in time. The system’s chaoticity is quantified through numerical computations of the finite time maximum Lyapunov exponent, which is diminishing to zero following power law decays. Our findings show that the presence of frequency gaps does not have any significant effect on the wave packet spreading in the weak chaos regime, while they remain rather inconclusive for the strong chaos case, indicating the need for further investigations.
Hilda Cerdeira (BRA)
Monday (04/08): 11:35AM – 12:00PM
Multi-agent systems are ubiquitously found in nature in the form of schools of fish, honey bees, locust swarms etc. The emergence of coordinated movements without any central controller remains of interest. Systems of oscillators called Swarmalators, whose phase and spatial dynamics are coupled, have been used to describe the dynamics of some living systems. Their collective behavior presents simultaneous aggregation in space and synchronization in phase which in turn leads
in some cases to explosive synchronization in a finite population as a function of the coupling parameter between the phases of the internal dynamics. This phenomenon is described using the order parameter and the Hamiltonian formalism. We study the synchronization transition of the internal phases of the particles, which can be of the first or second order. Introducing delay in these systems gives rise to several new phases which will be presented.
José Roberto C. Piqueira (BRA)
Friday (08/08): 09:00AM – 09:25AM
Clock distribution systems are used in many applications requiring accurate time basis: integrated circuits, computer networks, satellite
communications, and Global Position System (GPS). Trying to automatize the design of clock distribution systems, this work presents a general
formulation considering the possible topologies and parameters, building a computational tool allowing to design and evaluate the performance of any case to be studied. The developed tool enables to simulate networks setting the number of nodes, internal node parameters, topology, perturbations, and signal propagation delays. The simulation results include response times, synchronization quality, and stability for all types of arrangements: mutually connected, chains, rings, stars, and mixed architectures
Marcus Aloizio Martinez de Aguiar (BRA)
Thursday (07/08): 11:35AM – 12:00AM
The Kuramoto model was recently extended to higher dimensions by reinterpreting the oscillators as particles moving on the surface of unit spheres in a D-dimensional space. Each particle is then represented by a D-dimensional unit vector; for D= 2 the particles move on the unit circle and the vectors can be described by a single phase, recovering the original Kuramoto model. This multidimensional description can be further extended by promoting the coupling constant between the particles to a matrix K that acts on the unit vectors. The coupling matrix changes the direction of the vectors and can be interpreted as a generalized frustration. In this talk I will discuss the effects of coupling matrices in D=2 and higher dimensions. I will show that, for identical particles, the system converges either to a stationary synchronized state, given by one of the real eigenvectors of K, or to an effective two-dimensional rotation, defined by one of the complex eigenvectors of K. The stability of these states depends on the set eigenvalues and eigenvectors of the coupling matrix, which controls the asymptotic behavior of the system, and therefore, can be used to manipulate these states.
Marcus Werner Beims (BRA)
Friday (08/08): 09:50AM – 10:15AM
Mark Edelman (USA)
Thursday (07/08): 04:45PM – 05:10PM
Fractional systems, which are systems with power-law-like memory, do not have periodic solutions/points except the fixed points. But they do have asymptotically periodic solutions/points. Study of discrete fractional systems (maps) may be separated into two significant parts. The first part is the study of finite time evolution. This evolution is characterized by slow, as a power law, convergence to asymptotically periodic points, strong dependance on the initial conditions, and cascade of bifurcations type trajectories (CBTT), on which cascades of bifurcations and inverse cascades of bifurcations occur on single trajectories. The second part is the study of the asymptotic behavior of fractional systems. This requires calculations of the asymptotically periodic and bifurcation points, conditions of their asymptotic stability, and the study of asymptotic fractional chaotic attractors. While the finite time evolution is very complicated and different from the evolution of regular (with no memory) systems, the asymptotic behavior seems to be quite similar to the behavior of regular systems, and the fractional Feigenbaum number d is equal to the regular Feigenbaum d. The recently derived algebraic equations defining periodic and bifurcations points and conditions of stability of fixed points in fractional maps depend on coefficients which are slowly converging series. In this presentation, we derive the analytic expressions for these coefficients and therefore complete the analytic formulation of the problem of periodic and bifurcation points calculations for fractional difference maps. This derivation turns out to be too simple to be published, but, nevertheless, the result makes it possible for those who investigate fractional difference maps to calculate asymptotically periodic and bifurcation pints and to draw asymptotic bifurcation diagrams.
Miguel A. F. Sanjuán (ESP)
Monday (04/08): 05:35PM – 06:00PM
We present a novel two-player game in a chaotic dynamical system where players have opposing objectives regarding the system’s behavior. The game is analyzed using a methodology from the field of chaos control known as partial control. Our aim is to introduce the utility of this methodology in the scope of game theory. These algorithms enable players to devise winning strategies even when they lack complete information about their opponent’s actions. To illustrate the approach, we apply it to a chaotic system, the logistic map. In this scenario, one player aims to maintain the system’s trajectory within a transient chaotic region, while the opposing player seeks to expel the trajectory from this region. The methodology identifies the set of initial conditions that guarantee victory for each player, referred to as the winning sets, along with the corresponding strategies required to achieve their respective objectives. This is joint work with Gaspar Alfaro and Rubén Capeáns from URJC, Madrid, Spain.
Norma Valencio (BRA)
Tuesday (05/08): 11:35AM – 12:00PM
In Brazil, direct links are often made between the occurrence of severe or extreme events of different natures and the immediate action of local authorities to declare an emergency, without duly considering the political and economic factors involved in interpreting this type of crisis. The detailing of the last one’s characteristics and dynamics should not be neglected if the institutional and social purpose is to overcome the naïve narratives and policies about resilience. However, once this cognitive barrier persists in both technical and governmental discourses and practices across the three levels of power (national, state, and municipal), it reveals a conscious and structural resistance, thereby avoiding too much scrutiny of the sensitive points of “normalised exceptionalities”. The question is: What is there to hide regarding the political and economic specificities of this recurring crisis? Using techniques from information theory, initially developed to tackle neuroscience problems, we can infer the complex structure of political and economic alliances and dynamics involved in disaster decreeing over the last twenty-two years (2003-2024). Firstly, we created and tested clusters with different combinations of economic strata (three variables: GDP, tax revenue, and Gini Index) and political spectra (two variables: party alliances and ideological orientation) to compose neurons. By adopting bio-inspired models from the brain as analogues for key municipal political and economic factors, we identified specific patterns in recurrent disasters through simulations. The results show that, during the mentioned period, there was a relative indistinction between the power architectures of public administration with different ideological and political party backgrounds, particularly in municipalities that adopted this type of “state of exception” as a new style of crisis governance. However, we have observed that economic features drive emergency declarations. This demonstrates that the appropriate approach to disasters in Brazil should focus on the power pacts that lead to public management. The simulations of economic-political networks under recurrent disasters reveal that these systems easily become flawed, which may help to trigger future disasters. Hence, the concern about the development model underlying such pacts should be at the root of the disaster risk reduction debate.
Sabrina Camargo (ARG)
Thursday (07/08): 05:10PM – 05:35PM
Scale invariance is a ubiquitous observation in the dynamics of large distributed complex systems. The computation of its scaling exponents, which provide clues on its origin, is often hampered by the limited available sampling data, making an appropriate mathematical description a challenge. This work investigates the behavior of correlation functions in fractal systems under conditions of severe subsampling. Analytical and numerical results reveal a striking robustness: the correlation functions continue to capture the expected scaling exponents despite substantial data reduction. This behavior is demonstrated numerically for the random 2D Cantor set and the Sierpinski gasket, both consistent with exact analytical predictions. Similar robustness is observed in 1D time series both synthetic and experimental, as well as in high resolution images of a neuronal structure. Overall, these findings are broadly relevant for the structural characterization of biological systems under realistic sampling constraints.
Suani Tavares Rubim de Pinho (BRA)
Monday (04/08): 09:50AM – 10:15PM
The study of the dynamics of infectious diseases, from the mathematical modelling point of view, is not a new subject in the literature, although it has gained wide visibility during COVID-19 pandemics. Since its first steps in XVIII century, based on population dynamics concepts, the nonlinear compartment models, such as the famous SIR (Susceptible – Infected – Recovered) model, proposed by Kermack and Mc-Kendrick at the beginning of twentieth century, lead to the development of Mathematical Epidemiology. From there on, the nonlinearity presented in the population-based and individual-based models makes evident complex scenarios observed on actual epidemic and endemic processes resulted from the transmission of infectious diseases. In this talk, I intend to highlight the nonlinear effect of data-based models through analytical results and computational simulations of some models of directly communicable diseases, such as COVID-19 [1] and Tuberculosis [2], and vector-borne transmitted diseases such as Dengue and Zika. I also emphasize the advantage of using the reproduction number, a suitable measure based on both data and models, to investigate complex scenarios such as co-circulation of pathogens [3,4] or the flux of cases between cities considering both the propagation dynamics of disease and the human movement [5].
References
[1] Oliveira, JF; Jorge, DCP; Veiga, RV; Rodrigues, MS; Torquato, MF; da Silva, NB; Fiaccone, RL; Cardim, LL; Pereira, FAC; de Castro, CP; Paiva, ASS; Amad, AAS; Lima, EABF; Souza, DS; Pinho, STR; Ramos, PIP; Andrade, RFS; Rede CoVida working group. Mathematical modeling of COVID-19 in 14.8 million individuals in Bahia, Brazil. Nature Commun. 12 (2021) 333.
[2] Pinho, STR.; Rodrigues, P; Andrade, RFS; Serra, H; Lopes, JS; Gomes, MGM. Impact of tuberculosis treatment length and adherence under different transmission intensities. Theor. Pop. Biol. 104 (2015) 68-77.
[3] Hirata, FMR; Jorge,, DCP; Pereira, FAC; Skalinski, LM; Cruz-Pacheco, G; M.L.M. Esteva, MLM; and Pinho, STR; Cocirculation of Dengue and Zika viruses: A modelling approach applied to epidemics data, Chaos Solit. Fractals 173 (2023) 113599.
[4] de Araujo, RGS; Jorge, DCP; Dorn, RC; Cruz-Pacheco, ´G; Esteva, MLM and Pinho, STR; Applying a multi-strain dengue model to epidemics data, Math. Biosc. 360 (2023) 109013.
[5] Jorge, DCP.; Oliveira, JF.; Miranda, JGV; Andrade, RFS; Pinho, STR; Estimating the effective reproduction number for heterogeneous models using incidence data. R. Soc. Open Sci. 9 (2022) 220005.
Ulrike Feudel (GER)
Tuesday (05/08): 05:35PM – 06:00PM
Permanent and transient chaotic dynamics has been found in many applications like mechanical oscillators, laser physics, neuroscience, ecology and coupled systems of different kind to name only a few. Recently, the focus has shifted to transient chaos on chaotic saddles as a phenomenon which provides new opportunities for complex dynamics. We show how unstable, and hence, transient chaotic dynamics can lead to extraordinary long chaotic transients or even permanent chaos in coupled oscillatory systems or in non-autonomous dynamical systems. We discuss the role of chaotic saddles in complex networks where a single perturbation in one node can lead to desynchronization of the whole network, which can either be destructive or constructive depending on the system under consideration. Additionally, we demonstrate how coupling can stabilize transient chaotic
motion to prevent the extinction of species in coupled ecological systems. Finally, we discuss the role of transient chaos for the control of swarms of oscillators transitioning from translational motion to rotational motion.