Synchronization and chimera state in a mechanical system
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ORIGINAL PAPER
Synchronization and chimera state in a mechanical system Phablo R. Carvalho · Marcelo A. Savi
Received: 7 September 2019 / Accepted: 16 December 2019 © Springer Nature B.V. 2020
Abstract Synchronization phenomenon appears in several natural systems being associated with physical, chemical and biological processes. In brief, synchronization may be understood as a harmonization of two or more system behaviors following some patterns. This paper deals with synchronization analysis of a mechanical pendulum-chart system composed by a hierarchical network of three pendula coupled to each other through their own chart that receives continuous supply of energy via a base excitation. Dynamical patterns are classified and investigated in order to understand the conditions to each one of them. Asynchronous behaviors are analyzed including the chimera state defined as a symmetry break of the behavior of identical oscillator network. Numerical simulations indicate that patterns do not have a direct relation with periodicity. An energetic approach is proposed in order to define a measure of synchronized states, providing new insights about the origin of chimera state and its relationship with natural frequencies.
P. R. Carvalho · M. A. Savi (B) Center for Nonlinear Mechanics, COPPE – Department of Mechanical Engineering, Universidade Federal do Rio de Janeiro, 21.941.972, P.O. Box 68.503, Rio de Janeiro, RJ, Brazil e-mail: [email protected] P. R. Carvalho e-mail: [email protected]
Keywords Nonlinear dynamics · Chaos · Synchronization · Chimera states · Mechanical system · Mechanical oscillators
1 Introduction Rhythms are an essential characteristic of natural systems being associated with several patterns. Synchronization is one of these phenomena, being defined as a combination of two or more system behaviors following some pattern. Huygens [1] was the first scientist that discussed the synchronization phenomenon in 1665 investigating clocks based on pendulum oscillations. Although the analysis was essentially the pendulum synchronization, the phenomenon was called “sympathy” between the clocks [2]. Basically, the analysis was developed considering a system of two pendulum clocks hanging on the same base and the idea was to exploit the clock oscillation on the same frequency, either in-phase or anti-phase. Results of Huygens [1] was recreated by Kapitaniak et al. [3] and Bennet et al. [4]. Nature presents several examples of synchronous behavior as the synchronization of fireflies on the southeast of Asia [5], applauses [6] or chirps of crickets [7]. Thereby, the synchronization is a collective behavior of animals and physical system as the Huygens’ clocks [1]. The first mathematical investigation of synchronization was provided by Wiener [8,9] that developed a
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model for the neural system based on Fourier integrals, obtaining a frequency distribution similar to brain α-rhythm. Winfree [10] reformulated the problem in terms of a large population of coup
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