Intermittent chaotic chimeras for coupled rotators

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Intermittent chaotic chimeras for coupled rotators

Simona Olmi, Erik A. Martens, Shashi Thutupalli, and Alessandro Torcini

Phys. Rev. E 92, 030901(R) – Published 9 September 2015

Abstract

Two symmetrically coupled populations of $N$ oscillators with inertia $m$ display chaotic solutions with broken symmetry similar to experimental observations with mechanical pendulums. In particular, we report evidence of intermittent chaotic chimeras, where one population is synchronized and the other jumps erratically between laminar and turbulent phases. These states have finite lifetimes diverging as a power law with $N$ and $m$ . Lyapunov analyses reveal chaotic properties in quantitative agreement with theoretical predictions for globally coupled dissipative systems.

Received 6 May 2015
Revised 14 August 2015

DOI:https://doi.org/10.1103/PhysRevE.92.030901

Authors & Affiliations

Simona Olmi^1,2,*, Erik A. Martens^3,4,5,†, Shashi Thutupalli^6,7,‡, and Alessandro Torcini^1,2,8,9,§

¹Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, I-50019 Sesto Fiorentino, Florence, Italy
²INFN Sezione Firenze, Via Sansone 1, I-50019 Sesto Fiorentino, Florence, Italy
³Department of Biomedical Sciences, University of Copenhagen, Blegdamsvej 3, 2200 Copenhagen, Denmark
⁴Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
⁵Group of Biophysics and Evolutionary Dynamics, Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Lower Saxony, Germany
⁶Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
⁷Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
⁸Aix-Marseille Université, Inserm, INMED UMR 901 and Institut de Neurosciences des Systèmes UMR 1106, 13000 Marseille, France
⁹Aix-Marseille Université, Université de Toulon, CNRS, CPT, UMR 7332, 13288 Marseille, France

^*simona.olmi@fi.isc.cnr.it
^†erik.martens@sund.ku.dk
^‡shashi@princeton.edu
^§alessandro.torcini@cnr.it

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Vol. 92, Iss. 3 — September 2015

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Images

Figure 1
Order parameters $R^{(1)}$ (solid black curve) and $R^{(2)}$ (dashed red curve) for the two coupled populations versus time: (a)–(c) experimental measurements and (d)–(f) numerical simulations of the model (1) with $m = 10$ . Initial conditions are (a), (b), (d), and (e) BSCs and (c) and (f) UCs and $N = 15$ . The experiments are carried out with (a) and (b) $f = 160$ beats per minute (bpm) and $l = 17 cm$ and (c) $f = 184$ bpm and $l = 25$ cm and experimental time is measured in seconds.
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Figure 2
Experimental setup: sketch of the mechanical system studied in [10], composed of two swings (A and B) coupled with a spring mechanism, each of which is loaded with $N = 15$ metronomes.
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Figure 3
(a) Average maximal LE $〈 λ_{1} 〉$ vs $m$ for $N = 100$ : black circles (red squares) refer to BSCs (UCs). (b) Order parameter $R$ (black solid line) and finite time LE $Λ$ (red dashed line) for the chaotic population vs time $t$ for $N = 200$ and $m = 10$ . The corresponding $P (Λ)$ (solid red line) is shown in (c) with the Gaussian fit (black dashed line). The inset shows the probability $p_{0}$ versus $N$ . The values $〈 λ_{1} 〉$ are obtained by following each realization for a time span $t = 50 000$ and by averaging over 100 different initial conditions.
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Figure 4
(a) Plot of $Λ^{(*)}$ (symbols) versus $1 / ln (N)$ for $50 \leq N \leq 1600$ and the corresponding fit; the dot-dashed black curve with the shaded area denotes $λ_{M} (\infty)$ with its error bar. (b) Lyapunov spectrum for $N = 100$ ; the dashed red line is $λ^{(s)}$ . (c) Positive part of the Lyapunov spectra for various sizes, the dashed (green) line is $λ^{(c)}$ . (d) Average lifetimes $〈 τ 〉$ of the ICC vs $N$ for inertia $m = 8$ (red squares) and $m = 10$ (black triangles). The inset shows $〈 τ 〉$ vs $m$ for $N = 30$ for ICCs. The blue dashed line in the main panel (inset) refers to a power law with exponent 1.60 (2.50). (a)–(c) The values of $〈 τ 〉$ are averaged over 200–3000 different realizations of BSCs and $m = 10$ .
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Figure 5
(a) Order parameters $R^{(1)}$ (black solid line) and $R^{(2)}$ (red dashed line) vs time, (b) imaginary vs real part of the complex fields $ρ^{(1)}$ and $ρ^{(2)}$ , (c) Lyapunov spectra obtained for three different realizations of UCs, and (d) averaged components ${\bar{ξ}}_{i}$ (red diamonds) together with the corresponding averaged frequencies of the oscillators ${\bar{ω}}_{i} \equiv \frac{\bar{d θ_{i}}}{d t}$ (black circles) for both populations. All the data refer to C2P states and to $m = 9$ ; the system sizes are (c) $N = 200$ and (a), (b), and (d) $N = 50$ . The time averages have been performed at time $T ≃ 5 \times 10^{4}$ .
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