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Characterizing multistability regions in the parameter space of the Mackey–Glass delayed system

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Proposed to study the dynamics of physiological systems in which the evolution depends on the state in a previous time, the Mackey–Glass model exhibits a rich variety of behaviors including periodic or chaotic solutions in vast regions of the parameter space. This model can be represented by a dynamical system with a single variable obeying a delayed differential equation. Since it is infinite dimensional requires to specify a real function in a finite interval as an initial condition. Here, the dynamics of the Mackey–Glass model is investigated numerically using a scheme previously validated with experimental results. First, we explore the parameter space and describe regions in which solutions of different periodic or chaotic behaviors exist. Next, we show that the system presents regions of multistability, i.e. the coexistence of different solutions for the same parameter values but for different initial conditions. We remark the coexistence of periodic solutions with the same period but consisting of several maximums with the same amplitudes but in different orders. We characterize the multistability regions by introducing families of representative initial condition functions and evaluating the abundance of the coexisting solutions. These findings contribute to describe the complexity of this system and explore the possibility of possible applications such as to store or to code digital information.

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The authors would like to thank the Uruguayan institutions Programa de Desarrollo de las Ciencias Básicas (MEC-Udelar, Uruguay) and Comisión Sectorial de Investigación Científica (Udelar, Uruguay) for the grant Física Nolineal (ID 722) Programa Grupos I+D. The numerical experiments presented here were performed at the ClusterUY (site:

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Correspondence to Arturo C. Marti.

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Tarigo, J.P., Stari, C., Cabeza, C. et al. Characterizing multistability regions in the parameter space of the Mackey–Glass delayed system. Eur. Phys. J. Spec. Top. 231, 273–281 (2022).

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