Predictibilidad en sistemas con retardo temporal: Cuencas de atracción, multiestabilidad y dinámica de atractores en espacios de dimensión infinita

Making long-term predictions about natural ocurring phenomena is one of the main objectives of science. For this purpose, dynamics arises as a means to model events occurring in various disciplines and to determine their long-term behavior. However, most models of natural systems do not admit closed-form analytical solutions, which makes it necessary to use other techniques to study their evolution. This is aggravated by the problem that many of these systems are highly dependent on their initial state, and small deviations in these conditions lead to completely different long-term dynamics. More specifically, time-delay systems, whose evolution depends not only on their current state but also on its past states, are essential for modeling phenomena in various areas of science, such as biology and telecommunications, where information transmission times are comparable to processing times. These systems are particularly sensitive to uncertainties in the initial conditions due to being infinitedimensional. In this work, we explore the Mackey-Glass system, a paradigmatic example of a time-delay system that models the production of cells and their release into the bloodstream. We show that it presents a great diversity of solutions, including equilibrium, periodic, and aperiodic or chaotic solutions as system parameters vary. We also observe that, for some parameter values, the system exhibits multistability, meaning that more than one stable solution can coexist. To quantify the impact of multistability on its predictability, we employ various techniques, ranging from counting unique solutions and computing the volume of initial conditions that evolve toward each attractor to calculating basin entropy, adapted to time-delay systems. We also compute the first two Lyapunov exponents of the system to quantify the predictability of the aperiodic solutions and find that multiple such solutions coexist. Additionally, we present a method to reduce the dimensionality of any time-delay system to apply all these techniques. In this work, we propose methods to understand the phase spaces of t time-delay vii systems and to quantify their impact on predictability. Keywords: Delayed Systems, Attractors, Chaos, Multistability, Basin Entropy.