Observatorio de I+D+i UPM

Memorias de investigación
Book chapters:
From Chaos to Order in a Ring of Coupled Oscillators with Frequency Mismatch
Research Areas
  • Physics chemical and mathematical,
  • Engineering
Synchronization is commonly understood as a dynamical process to reach a collective state of coupled oscillatory systems. Generally, synchronization requires some relation between the functionals of different processes due to their interaction [1, 2]. As a result of synchronization, coupled systems adjust their individual frequencies in a certain ratio. Numerous examples of synchronization can be found in almost all fields of science and in nature, from mechanics and electronics to physics, chemistry, and biology. Therefore, the study of synchronization of coupled oscillators is of extreme importance for understanding the dynamical processes underlying this phenomenon. The mechanisms behind the cooperative synchronous behavior are different for each system and determined by its properties. The knowledge of these mechanisms will allow the understanding of the principles of self-organization of the matter. The notion of synchronization was first used to describe the cooperative behavior of periodic systems, such as the Huygens?s clocks, but it was later extended to chaotic systems [3] able to adjust their individual behaviors from uncorrelated oscillations to a completely identical motion as the coupling strength increases. The ever-present demand for secure communication was one of the primary motivations for studying the synchronization of chaotic systems. This demand, especially in military applications, led Louis Pecora and Thomas Carroll [4] to develop a method for synchronizing two chaotic systems that continue to be a reason for innovation in this field nowadays [5, 6]. They revealed the necessary conditions needed for synchronization and indicated that secure communication using a chaotic career was possible. If a receiver synchronizes with a transmitter, a message can be extracted from the mask. Although their simple method for encrypting a signal in chaos is easily defeated by cryptanalysis, such as synchronization attacks [7], synchronous chaotic oscillators continue to stimulate many researchers in developing new increasingly sophisticated methods to improve communication security [8]. Synchronization of simple systems composed by two identical chaotic oscillators has been extensively studied (see, e.g., [1, 9] and references therein) and is now relatively well understood. Many types of chaos synchronization have been identified and characterized using typical synchronization measures, such as synchronization error, cross-correlation, similarity function, and phase difference. Among them, it is worth mentioning complete synchronization [4], phase synchronization [10], anti-phase synchronization [11], lag synchronization [12], anticipating synchronization [13], and generalized synchronization [14].A route to synchronization depends on both types of coupling and the coupling configuration. In the simplest case of two identical chaotic oscillators, the most common synchronization scenario from asynchronous motion to complete synchronization, as the coupling strength is increased, is as follows: imperfect phase synchronization?perfect phase synchronization?complete synchronization [1]. A different scenario occurs in the presence of a mismatch between natural frequencies of coupled oscillators. When the natural frequency of the slave oscillator is smaller than the natural frequency of the master oscillator, the oscillators synchronize with lag; in the opposite case, the slave oscillator anticipates the master oscillator dynamics [15, 16].
Book Edition
Book Publishing
Springer, Cham
Nonlinear Systems and Complexity
Book title
Regularity and Stochasticity of Nonlinear Dynamical Systems
From page
To page
  • Autor: Alexander Pisarchik (UPM)
  • Autor: Mariano Alberto Garcia Vellisca (UPM)
Research Group, Departaments and Institutes related
  • Creador: Grupo de Investigación: Tecnologías para Ciencias de la Salud
S2i 2020 Observatorio de investigación @ UPM con la colaboración del Consejo Social UPM
Cofinanciación del MINECO en el marco del Programa INNCIDE 2011 (OTR-2011-0236)
Cofinanciación del MINECO en el marco del Programa INNPACTO (IPT-020000-2010-22)