Small amplitude chimeras for coupled clocks
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ORIGINAL PAPER
Small amplitude chimeras for coupled clocks Dawid Dudkowski · Patrycja Jaros · Krzysztof Czołczynski ´ · Tomasz Kapitaniak
Received: 17 July 2020 / Accepted: 28 September 2020 © The Author(s) 2020
Abstract We report the arise of small amplitude chimera states in three coupled pendulum clocks suspended on an oscillating base. Two types of chimeras are identified and described by the character of the behaviour of particular units (which can be both regular or irregular). The regions of the appearance of the dynamical patterns are determined and the scenarios of their coexistence with typical synchronization states are discussed. We investigate the chimeras’ basins of attraction, showing that the arise of complex dynamics is not straightforward and highly depends on the system’s parameters and the initial conditions. The latter is confirmed by the probability analysis, exhibiting the rare character of the observed attractors. The scenarios of bifurcations between the chimeric patterns are studied and supported using the energy balance method, which allows to describe the changes of the energy flows between particular nodes of the system. The results presented in this paper confirm the ones D. Dudkowski (B)· P. Jaros · K. Czołczy´nski · T. Kapitaniak Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90–924 Lodz, Poland e-mail: [email protected] P. Jaros e-mail: [email protected] K. Czołczy´nski e-mail: [email protected] T. Kapitaniak e-mail: [email protected]
obtained for the previous models, extending the analysis with an additional degree of freedom. Keywords Chimera states · Pendulum clocks · Rare attractors · Energy balance method
1 Introduction The spontaneous coexistence of coherence and incoherence in networks of coupled systems [1] is one of the most surprising phenomena observed in the nonlinear dynamics discipline. The so-called chimera state [2], named after a hybrid creature from the Greek mythology, combines both regular and irregular types of patterns, showing that even though the coupling scheme applied to the system is homogeneous, the oscillators can split into many groups of different behaviours, uncovering unexpected scenarios of partial synchronization and chaos. Chimeras have been observed in various dynamical models, just to mention heterogeneous networks [3,4], chemical oscillators [5], mechanical systems [6,7] or the delayed ones [8,9]. Patterns of coexistence are reported in neuronal problems [10–14] (including such classical systems as the FitzHugh-Nagumo [12] or the Hindmarsh-Rose [13] ones) and can be related to the complex brain dynamics [14]. In [15], the authors present a solvable model exhibiting chimeric behaviours, while in [16], the spectral properties of chimeras are discussed. Coexisting states arise natu-
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rally on the road from complete synchronization to pure spatial chaos, which has been studied in [17,18] but can be also related to the transient dynamics occurring in th
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