Organization and identification of solutions in the time-delayed Mackey-Glass model

Abstract

Copyright 2015 AIP Publishing. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP PublishingMultistability in the long term dynamics of the Mackey-Glass (MG) delayed model is analyzed by using an electronic circuit capable of controlling the initial conditions. The system's phase-space is explored by varying the parameter values of two families of initial functions. The evolution equation of the electronic circuit is derived and it is shown that, in the continuous limit, it exactly corresponds to the MG model. In practice, when using a finite set of capacitors, an excellent agreement between the experimental observations and the numerical simulations is manifested. As the delay is increased, different periodic or aperiodic solutions appear. We observe abundant periodic solutions that have the same period but a different alternation of peaks of dissimilar amplitudes and propose a novel symbolic method to classify these solutions.Peer ReviewedPostprint (published version

Resonant Doppler effect in systems with variable delay

“…The so-called slowly oscillating periodic solutions, which are characterized by slow oscillations within a period of the order of the delay, and the connected transition layer phenomena where intensively investigated for constant delay [54,61,62] and for state-dependent delay [34]. For constant delay, the multistability of these solutions was also investigated [63][64][65]. The analysis in [39] goes beyond those periodic solutions and establishes a theory of the dynamics of a slowly oscillating chaotic solution, which is called laminar chaos and is found only in systems with dissipative delay.…”

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

“…It can be shown that this numerical method based on the exact discretization is a reliable representation of the experimental electronic system and a good approximation of Eq. ( 2) for sufficiently large values of N , as well as being more efficient than standard numerical algorithms [14,15].…”

“…A remarkable characteristic of this approach is that the temporal integration is exact, thus, experimental and numerical simulations agree very well with each other enabling the study of large regions in the parameter space. In a subsequent investigation [15], this approach was used to explore the parameter space described in terms of the dimensionless delay and production rate. This study unmasked the existence of periodic and chaotic solutions intermingled in vast regions of the parameter space.…”

Bifurcation of Periodic Solutions of the Mackey– Glass Equation

“…Уравнение (1) обладает богатой динамикой. Оно изучалось в ряде работ ( [2] - [7]), в которых анализировались различные свойства решений и на основе чис-ленного интегрирования показано существование разнообразных периодических ре-шений, а также сложных, в том числе хаотических колебаний. В [5] рассмотрены некоторые обобщения уравнения (1).…”

“…В [5] рассмотрены некоторые обобщения уравнения (1). В работе [7] приведены результаты модели-рования динамики уравнения (1) посредством электронного устройства. Отмечено существование хаотических колебаний, изучаются их спектральные свойства.…”