# Image Entropy for the Identification of Chimera States of Spatiotemporal Divergence in Complex Coupled Maps of Matrices

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Notes and the Objective

#### 2.1. A Network of Coupled Maps

#### 2.2. A Network of Coupled Map of Matrices

## 3. Chimera States of Spatiotemporal Divergence in Regular NCMMs

#### 3.1. Spatiotemporal Divergence in a Regular NCMM

#### 3.2. Chimera States of Spatiotemporal Divergence in a Regular Feed-Forward NCMM

## 4. Chimera States of Spatiotemporal Divergence in a Complex NCMM

#### 4.1. Chimera States of Spatiotemporal Divergence in the Erdős-Rényi NCMM

#### 4.2. Chimera States of Spatiotemporal Divergence in the Small-World NCMM

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kuramoto, Y.; Battogtokh, D. Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst.
**2002**, 5, 380–385. [Google Scholar] - Zakharova, A.; Kapeller, M.; Schöll, E. Chimera Death: Symmetry Breaking in Dynamical Networks. Phys. Rev. Lett.
**2014**, 112, 154101. [Google Scholar] [CrossRef] [Green Version] - Panaggio, M.J.; Abrams, D.M. Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators. Nonlinearity
**2015**, 28, 67–87. [Google Scholar] [CrossRef] - Bukh, A.; Strelkova, G.; Anishchenko, V. Spiral wave patterns in a two-dimensional lattice of nonlocally coupled maps modeling neural activity. Chaos Soliton Fractals
**2019**, 120, 75–82. [Google Scholar] [CrossRef] - Martens, E.A.; Thutupalli, S.; Fourrière, A.; Hallatschek, O. Chimera states in mechanical oscillator networks. Proc. Natl. Acad. Sci. USA
**2013**, 110, 10563–10567. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hart, J.D.; Bansal, K.; Murphy, T.E.; Roy, R. Experimental observation of chimera and cluster states in a minimal globally coupled network. Chaos
**2016**, 26, 094801. [Google Scholar] [CrossRef] [PubMed] - Li, X.W.; Bi, R.; Sun, Y.X.; Zhang, S.; Song, Q.Q. Chimera states in Gaussian coupled map lattices. Front. Phys.
**2018**, 13, 130502. [Google Scholar] [CrossRef] - Xu, H.Y.; Wang, G.L.; Huang, L.; Lai, Y.C. Chaos in Dirac electron optics: Emergence of a relativistic quantum chimera. Phys. Rev. Lett.
**2018**, 120, 124101. [Google Scholar] [CrossRef] - Nkomo, S.; Tinsley, M.R.; Showalter, K. Chimera States in Populations of Nonlocally Coupled Chemical Oscillators. Phys. Rev. Lett.
**2013**, 110, 244102. [Google Scholar] [CrossRef] [Green Version] - Totz, J.F.; Rode, J.; Tinsley, M.R.; Showalter, K.; Engel, H. Spiral wave chimera states in large populations of coupled chemical oscillators. Nat. Phys.
**2018**, 14, 282–286. [Google Scholar] [CrossRef] - Majhi, S.; Bera, B.K.; Ghosh, D.; Perc, M. Chimera states in neuronal networks: A review. Phys. Life Rev.
**2018**. [Google Scholar] [CrossRef] - Hizanidis, J.; Kanas, V.G.; Bezerianos, A.; Bountis, T. Chimera states in networks of nonlocally coupled Hindmarsh-Rose neuron models. Int. J. Bifurc. Chaos
**2014**, 24, 1450030. [Google Scholar] [CrossRef] - Shepelev, I.A.; Bukh, A.V.; Strelkova, G.I.; Vadivasova, T.E. Chimera states in ensembles of bistable elements with regular and chaotic dynamics. Nonlinear Dyn.
**2017**, 90, 2317–2330. [Google Scholar] [CrossRef] - Malchow, A.K.; Omelchenko, I.; Schöll, E.; Hövel, P. Robustness of chimera states in nonlocally coupled networks of nonidentical logistic maps. Phys. Rev. E
**2018**, 98, 012217. [Google Scholar] [CrossRef] - Bogomolov, S.A.; Slepnev, A.V.; Strelkova, G.I.; Schöll, E.; Anishchenko, V.S. Mechanisms of appearance of amplitude and phase chimera states in ensembles of nonlocally coupled chaotic systems. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 43, 25–36. [Google Scholar] [CrossRef] - Bukh, A.; Rybalova, E.; Semenova, N.; Strelkova, G.; Anishchenko, V. New type of chimera and mutual synchronization of spatiotemporal structures in two coupled ensembles of nonlocally interacting chaotic maps. Chaos Interdiscipl. J. Nonlin. Sci.
**2017**, 27, 111102. [Google Scholar] [CrossRef] [PubMed] - Laing, C.R. Chimeras in networks with purely local coupling. Phys. Rev. E
**2015**, 92, 050904(R). [Google Scholar] [CrossRef] [PubMed] - Clerc, M.G.; Coulibaly, S.; Ferré, M.A.; García-Ñustes, M.A.; Rojas, R.G. Chimera-type states induced by local coupling. Phys. Rev. E
**2016**, 93, 052204. [Google Scholar] [CrossRef] [Green Version] - Kundu, S.; Majhi, S.; Bera, B.K.; Ghosh, D.; Lakshmanan, M. Chimera states in two-dimensional networks of locally coupled oscillators. Phys. Rev. E
**2018**, 97, 022201. [Google Scholar] [CrossRef] [Green Version] - Schmidt, L.; Krischer, K. Clustering as a Prerequisite for Chimera States in Globally Coupled Systems. Phys. Rev. Lett.
**2015**, 114, 034101. [Google Scholar] [CrossRef] - zur Bonsen, A.; Omelchenko, I.; Zakharova, A.; Schöll, E. Chimera states in networks of logistic maps with hierarchical connectivities. Eur. Phys. J. B
**2018**, 91, 65. [Google Scholar] [Green Version] - Omelchenko, I.; Provata, A.; Hizanidis, J.; Schöll, E.; Hövel, P. Robustness of chimera states for coupled FitzHugh-Nagumo oscillators. Phys. Rev. E
**2015**, 91, 022917. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lopes, M.A.; Goltsev, A.V. Distinct dynamical behavior in Erdős-Rényi networks, regular random networks, ring lattices, and all-to-all neuronal networks. Phys. Rev. E
**2019**, 99, 022303. [Google Scholar] [CrossRef] [PubMed] - Sawicki, J.; Omelchenko, I.; Zakharova, A.; Schöll, E. Chimera states in complex networks: Interplay of fractal topology and delay. Eur. Phys. J. Spec. Top.
**2017**, 226, 1883–1892. [Google Scholar] [CrossRef] - Zhu, Y.; Zheng, Z.; Yang, J. Chimera states on complex networks. Phys. Rev. E
**2014**, 89, 022914. [Google Scholar] [CrossRef] - Li, B.; Saad, D. Chimera-like states in structured heterogeneous networks. Chaos
**2017**, 27, 043109. [Google Scholar] [CrossRef] [Green Version] - Ghosh, S.; Zakharova, A.; Jalan, S. Non-identical multiplexing promotes chimera states. Chaos Soliton Fractals
**2018**, 106, 56–60. [Google Scholar] [CrossRef] [Green Version] - Hizanidis, J.; Kouvaris, N.E.; Zamora-López, G.; Díaz-Guilera, A.; Antonopoulos, C.G. Chimera-like States in Modular Neural Networks. Sci. Rep.
**2016**, 6, 19845. [Google Scholar] [CrossRef] [PubMed] - Makarov, V.V.; Kundu, S.; Kirsanov, D.V.; Frolov, N.S.; Maksimenko, V.A.; Ghosh, D.; Dana, S.K.; Hramov, A.E. Multiscale interaction promotes chimera states in complex networks. Commun. Nonlinear Sci.
**2019**, 71, 118–129. [Google Scholar] [CrossRef] - May, R.M. Simple mathematical models with very complicated dynamics. Nature
**1976**, 261, 459–467. [Google Scholar] [CrossRef] - Zhang, Y.Q.; He, Y.; Wang, X.Y. Spatiotemporal chaos in mixed linear-nonlinear two-dimensional coupled logistic map lattice. Phys. A
**2018**, 490, 148–160. [Google Scholar] [CrossRef] - Huang, T.; Zhang, H. Bifurcation, chaos and pattern formation in a space-and time-discrete predator-prey system. Chaos Soliton Fractals
**2016**, 91, 92–107. [Google Scholar] [CrossRef] - Fernandez, B. Selective chaos of travelling waves in feedforward chains of bistable maps. arXiv
**2018**, arXiv:1811.08310. [Google Scholar] - Guangqing, L.; Smidtaite, R.; Navickas, Z.; Ragulskis, M. The effect of explosive divergence in a coupled map lattice of matrices. Chaos Soliton Fractals
**2018**, 113, 308–313. [Google Scholar] - Navickas, Z.; Smidtaite, R.; Vainoras, A. The logistic map of matrices. Discret. Cont. Dyn. B
**2011**, 3, 927–944. [Google Scholar] - Miranda, G.H.B.; Machicao, J.; Bruno, O.M. Exploring spatio-temporal dynamics of cellular automata for pattern recognition in networks. Sci. Rep.
**2016**, 6, 37329. [Google Scholar] [CrossRef] - Zheng, Y.H.; Lu, Q.S. Spatiotemporal patterns and chaotic burst synchronization in a small-world neuronal network. Phys. A Stat. Mech. Its Appl.
**2008**, 387, 3719–3728. [Google Scholar] [CrossRef] - Sakyte, E.; Ragulskis, M. Self-calming of a random network of dendritic neurons. Neurocomputing
**2011**, 74, 3912–3920. [Google Scholar] [CrossRef] - Cross, M.C.; Hohenberg, P.C. Pattern formation outside of equilibrium. Rev. Mod. Phys.
**1993**, 65, 851. [Google Scholar] [CrossRef] - Goldman, M.S. Memory without feedback in a neural network. Neuron.
**2009**, 61, 621–634. [Google Scholar] [CrossRef] [PubMed] - Zankoc, C.; Fanelli, D.; Ginelli, F.; Livi, R. Desynchronization and pattern formation in a noisy feed-forward oscillator network. Phys. Rev. E
**2019**, 99, 012303. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Solé, R.V.; Valverde, S. Information theory of complex networks: On evolution and architectural constraints. In Complex Networks; Springer: Berlin/Heidelberg, Germany, 2004; pp. 189–207. [Google Scholar]
- Erdős, P.; Rényi, A. On Random Graphs. Publ. Math.
**1959**, 6, 290–297. [Google Scholar] - Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small-world’networks. Nature
**1998**, 393, 440. [Google Scholar] [CrossRef] - Shinoda, K.; Kaneko, K. Chaotic Griffiths Phase with Anomalous Lyapunov Spectra in Coupled Map Networks. Phys. Rev. Lett.
**2016**, 117, 254101. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The transient dynamics of a regular NCMM comprising 200 nodes ($a=3.699956$; $\epsilon =0.4$; ${\lambda}^{\left(0\right)}\left(i\right)$; $i=1,2,\dots ,200$ are randomly distributed in the interval $\left(\right)$) represented by the variation of ${\mu}^{\left(t\right)}\left(i\right)$. The network diverges at $r=0.02$ (part (

**a**)); generates complex patterns at $r=0.025$ (part (

**b**)); and calms down at $r=0.03$ (part (

**c**)). Numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are truncated to 5 for the clarity of presentation.

**Figure 2.**The evolution of ${\mu}^{\left(t\right)}\left(1\right)$, ${\mu}^{\left(t\right)}\left(50\right)$, ${\mu}^{\left(t\right)}\left(100\right)$, ${\mu}^{\left(t\right)}\left(150\right)$ and ${\mu}^{\left(t\right)}\left(200\right)$ in time interval $500\le t\le 1000$ at the set of system parameters corresponding to Figure 1b. The numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are cropped to 5 in part (

**a**) and are shown uncropped in part (

**b**).

**Figure 3.**Image entropy of patterns is calculated for regular network when parameter r is set to $0.05$. Network coupling parameter $\epsilon $ is set to $0.2$, $0.366$, $0.6$, $0.823$ and $0.9$ in parts (

**a**), (

**b**), (

**c**), (

**d**) and (

**e**). Image entropy is equal to $2.29$ and $0.785$ in parts (

**b**) and (

**d**) respectively.

**Figure 4.**The visualization of chimera states of spatiotemporal divergence for networks of different structure: a regular NCMM (part (

**a**), parameter plane $\epsilon -r$); a regular unidirectional NCMM (part (

**b**), parameter plane $\epsilon -r$); the Erdős-Rényi NCMM (part (

**c**), parameter plane $\epsilon -d$); the small-world NCMM (part (

**d**), parameter plane $\epsilon -r$). The colorbar denotes numerical values of the entropy computed for steady-state evolution of the networks.

**Figure 5.**The transient dynamics of a regular directional NCMM comprising 200 nodes ($a=3.699956$; $\epsilon =0.4$; ${\lambda}^{\left(0\right)}\left(i\right)$; $i=1,2,\dots ,200$ are randomly distributed in the interval $\left(\right)$) represented by the variation of ${\mu}^{\left(t\right)}\left(i\right)$. The network diverges at $r=0.025$ (part (

**a**)); generates complex fractal-type patterns at $r=0.035$ (part (

**b**)); and calms down at $r=0.045$ (part (

**c**)). Numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are truncated to 5 for the clarity of presentation.

**Figure 6.**The transient dynamics of the Erdős-Rényi NCMM comprising 200 nodes ($a=3.699956$; $\epsilon =0.4$; ${\lambda}^{\left(0\right)}\left(i\right)$; $i=1,2,\dots ,200$ are randomly distributed in the interval $\left(\right)$) represented by the variation of ${\mu}^{\left(t\right)}\left(i\right)$. The network diverges at $d=0.031$ (part (

**a**)); generates complex fractal-type patterns at $d=0.033$ (part (

**b**)); and calms down at $d=0.035$ (part (

**c**)). Numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are truncated to 5 for the clarity of presentation.

**Figure 7.**The transient dynamics of the small-world network NCMM comprising 200 nodes ($a=3.699956$; $\epsilon =0.4$; ${\lambda}^{\left(0\right)}\left(i\right)$; $i=1,2,\dots ,200$ are randomly distributed in the interval $\left(\right)$; $\beta =0.2$) represented by the variation of ${\mu}^{\left(t\right)}\left(i\right)$. The network diverges at $r=0.015$ (part (

**a**)); generates complex patterns of spatiotemporal divergence at $r=0.02$ (part (

**b**)); and calms down at $r=0.025$ (part (

**c**)). Numerical values of ${\mu}^{\left(t\right)}\left(i\right)$ are truncated to 5 for the clarity of presentation.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Smidtaite, R.; Lu, G.; Ragulskis, M.
Image Entropy for the Identification of Chimera States of Spatiotemporal Divergence in Complex Coupled Maps of Matrices. *Entropy* **2019**, *21*, 523.
https://doi.org/10.3390/e21050523

**AMA Style**

Smidtaite R, Lu G, Ragulskis M.
Image Entropy for the Identification of Chimera States of Spatiotemporal Divergence in Complex Coupled Maps of Matrices. *Entropy*. 2019; 21(5):523.
https://doi.org/10.3390/e21050523

**Chicago/Turabian Style**

Smidtaite, Rasa, Guangqing Lu, and Minvydas Ragulskis.
2019. "Image Entropy for the Identification of Chimera States of Spatiotemporal Divergence in Complex Coupled Maps of Matrices" *Entropy* 21, no. 5: 523.
https://doi.org/10.3390/e21050523