American Institute of Mathematical Sciences

Advanced
July 2018, 23(5): 1931-1944. doi: 10.3934/dcdsb.2018189

Explosive synchronization in mono and multilayer networks

Inmaculada Leyva 1,2,, , Irene Sendiña-Nadal 1,2, and  Stefano Boccaletti 3,

1. 

Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
2. 

Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Madrid, Spain
3. 

CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy

* Corresponding author: I. Leyva

Received  April 2017 Revised  August 2017 Published  May 2018

Fund Project: Work partly supported by the Spanish Ministry of Economy under project FIS2013-41057-P and by GARECOM, Group of Research Excelence URJC-Banco de Santander. Authors acknowledge the computational resources and assistance provided by CRESCO, the supercomputing center of ENEA in Portici, Italy

Explosive synchronization, an abrupt transition to a collective coherent state, has been the focus of an extensive research since its first observation in scale-free networks with degree-frequency correlations. In this work, we report several scenarios where a first-order transition to synchronization occurs driven by the presence of a dependence between dynamics and network structure. Therefore, different mechanisms are shown to be able to prevent the formation of a giant synchronization cluster for sufficient large values of the coupling constant in both mono and multilayer networks. Using the Kuramoto model as a reference, we show how for an arbitrary network topology and frequency distribution, a very general weighting procedure acting on the weight of the links delays the synchronization transition forming independent synchronization clusters which suddenly merge above a critical threshold of the coupling constant. A completely different scenario in adaptive and multilayer networks is introduced which gives rise to the emergence of an explosive synchronization when a feedback between the dynamics and structure is operating by means of dependence links weighted through the order parameter.

Keywords: Complex networks, synchronization, chaos, phase transitions, coherence.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Inmaculada Leyva, Irene Sendiña-Nadal, Stefano Boccaletti. Explosive synchronization in mono and multilayer networks. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1931-1944. doi: 10.3934/dcdsb.2018189
References:
[1]

D. Achlioptas, R. M. D'Souza and J. Spencer, Explosive percolation in random networks, Science, 323 (2009), 1453–1455, URL http://www.sciencemag.org/content/323/5920/1453. abstract. doi: 10.1126/science.1167782.

[2]

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93–153, URL http://www.sciencedirect.com/science/ article/pii/S0370157308003384. doi: 10.1016/j.physrep.2008.09.002.

[3]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509–512, URL http://science.sciencemag.org/content/286/5439/509./ doi: 10.1126/science.286.5439.509.

[4]

S. Boccaletti, J. Almendral, S. Guan, I. Leyva, Z. Liu, I. Sendiña-Nadal, Z. Wang and Y. Zou, Explosive transitions in complex networks' structure and dynamics: Percolation and synchronization, Physics Reports, 660 (2016), 1–94, URL https://www.sciencedirect.com/ journal/physics-reports/vol/660. doi: 10.1016/j.physrep.2016.10.004.

[5]

S. Boccaletti, G. Bianconi, R. Criado, C. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang and M. Zanin, The structure and dynamics of multilayer networks, Phys. Rep., 544 (2014), 1–122, URL http://linkinghub.elsevier.com/retrieve/pii/S0370157314002105. doi: 10.1016/j.physrep.2014.07.001.

[6]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175–308, URL http://www.sciencedirect.com/science/article/pii/S037015730500462X. doi: 10.1016/j.physrep.2005.10.009.

[7]

M. M. Danziger, O. I. Moskalenko, S. A. Kurkin, X. Zhang, S. Havlin and S. Boccaletti, Explosive synchronization coexists with classical synchronization in the Kuramoto model Chaos, 26 (2016), 065307, 6 pp, URL http://scitation.aip.org/content/aip/journal/chaos/26/6/10.1063/1.4953345. doi: 10.1063/1.4953345.

[8]

P. Erdős and A. Rényi, On random graphs. i, Publicationes Mathematicae Debrecen, 6 (1959), 290-297.

[9]

J. Gómez-Gardeñes, S. Gómez, A. Arenas and Y. Moreno, Explosive synchronization transitions in scale-free networks, Phys. Rev. Lett., 106 (2011), 1–4, URL http://link.aps.org/doi/10.1103/PhysRevLett.106.128701
[10]

M. Kim, G. A. Mashour, S. Blain-Moraes, G. Vanini, V. Tarnal, E. Janke, A. G. Hudetz and U. Lee, Functional and topological conditions for explosive synchronization develop in human brain networks with the onset of anesthetic-induced unconsciousness, Front. Comput. Neurosci., 10 (2016), p1, URL https://www.ncbi.nlm.nih.gov/pubmed/26834616.

10.3389/fncom.2016.00001
[11]

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, 1984. doi: 10.1007/978-3-642-69689-3.

[12]

I. Leyva, A. Navas, I. Sendiña-Nadal, J. A. Almendral, J. M. Buldú, M. Zanin, D. Papo and S. Boccaletti, Explosive transitions to synchronization in networks of phase oscillators, Sci. Rep., 3 (2013), 1281, URL http://www.nature.com/doifinder/10.1038/srep01281. doi: 10.1038/srep01281.

[13]

I. Leyva, I. Sendiña-Nadal, J. A. Almendral, A. Navas, S. Olmi and S. Boccaletti, Explosive synchronization in weighted complex networks, Phys. Rev. E, 88 (2013), 042808, URL http://link.aps.org/doi/10.1103/PhysRevE.88.042808. doi: 10.1103/PhysRevE.88.042808.

[14]

I. Leyva, R. Sevilla-Escoboza, J. M. Buldú, I. Sendiña-Nadal, J. Gómez-Gardeñes, A. Arenas, Y. Moreno, S. Gómez, R. Jaimes-Reátegui and S. Boccaletti, Explosive first-order transition to synchrony in networked chaotic oscillators, Phys. Rev. Lett., 108 (2012), 168702, URL http://link.aps.org/doi/10.1103/PhysRevLett.108.168702. doi: 10.1103/PhysRevLett.108.168702.

[15]

E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott, P. So and T. M. Antonsen, Exact results for the Kuramoto model with a bimodal frequency distributions, Phys. Rev. E, 79 (2009), 026204, 11 pp, URL https://link.aps.org/doi/10.1103/PhysRevE.79.026204 doi: 10.1103/PhysRevE.79.026204.

[16]

A. Navas, J. A. Villacorta-Atienza, I. Leyva, J. A. Almendral, I. Sendiña Nadal and S. Boccaletti, Effective centrality and explosive synchronization in complex networks, Phys. Rev. E, 92 (2015), 062820, URL http://link.aps.org/doi/10.1103/PhysRevE.92.062820. doi: 10.1103/PhysRevE.92.062820.

[17]

D. Pazó, Thermodynamic limit of the first-order phase transition in the Kuramoto model, Phys. Rev. E, 72 (2005), 046211, 6pp, URL http://journals.aps.org/pre/abstract/10.1103/PhysRevE.72.046211. doi: 10.1103/PhysRevE.72.046211.

[18]

M. Rohden, A. Sorge, M. Timme and D. Witthaut, Self-organized synchronization in decentralized power grids, Phys. Rev. Lett., 109 (2012), 064101, URL http://link.aps.org/doi/10.1103/PhysRevLett.109.064101. doi: 10.1103/PhysRevLett.109.064101.

[19]

X. Zhang, S. Boccaletti, S. Guan and Z. Liu, Explosive synchronization in adaptive and multilayer networks, Phys. Rev. Lett., 114 (2015), 038701, URL http://link.aps.org/doi/10.1103/PhysRevLett.114.038701. doi: 10.1103/PhysRevLett.114.038701.

[20]

X. Zhang, Y. Zou, S. Boccaletti and Z. Liu, Explosive synchronization as a process of explosive percolation in dynamical phase space, Sci. Rep., 4 (2014), 5200, URL http://www.ncbi.nlm.nih.gov/pubmed/24903808. doi: 10.1038/srep05200.

show all references

References:
[1]

D. Achlioptas, R. M. D'Souza and J. Spencer, Explosive percolation in random networks, Science, 323 (2009), 1453–1455, URL http://www.sciencemag.org/content/323/5920/1453. abstract. doi: 10.1126/science.1167782.

[2]

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93–153, URL http://www.sciencedirect.com/science/ article/pii/S0370157308003384. doi: 10.1016/j.physrep.2008.09.002.

[3]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509–512, URL http://science.sciencemag.org/content/286/5439/509./ doi: 10.1126/science.286.5439.509.

[4]

S. Boccaletti, J. Almendral, S. Guan, I. Leyva, Z. Liu, I. Sendiña-Nadal, Z. Wang and Y. Zou, Explosive transitions in complex networks' structure and dynamics: Percolation and synchronization, Physics Reports, 660 (2016), 1–94, URL https://www.sciencedirect.com/ journal/physics-reports/vol/660. doi: 10.1016/j.physrep.2016.10.004.

[5]

S. Boccaletti, G. Bianconi, R. Criado, C. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang and M. Zanin, The structure and dynamics of multilayer networks, Phys. Rep., 544 (2014), 1–122, URL http://linkinghub.elsevier.com/retrieve/pii/S0370157314002105. doi: 10.1016/j.physrep.2014.07.001.

[6]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175–308, URL http://www.sciencedirect.com/science/article/pii/S037015730500462X. doi: 10.1016/j.physrep.2005.10.009.

[7]

M. M. Danziger, O. I. Moskalenko, S. A. Kurkin, X. Zhang, S. Havlin and S. Boccaletti, Explosive synchronization coexists with classical synchronization in the Kuramoto model Chaos, 26 (2016), 065307, 6 pp, URL http://scitation.aip.org/content/aip/journal/chaos/26/6/10.1063/1.4953345. doi: 10.1063/1.4953345.

[8]

P. Erdős and A. Rényi, On random graphs. i, Publicationes Mathematicae Debrecen, 6 (1959), 290-297.

[9]

J. Gómez-Gardeñes, S. Gómez, A. Arenas and Y. Moreno, Explosive synchronization transitions in scale-free networks, Phys. Rev. Lett., 106 (2011), 1–4, URL http://link.aps.org/doi/10.1103/PhysRevLett.106.128701
[10]

M. Kim, G. A. Mashour, S. Blain-Moraes, G. Vanini, V. Tarnal, E. Janke, A. G. Hudetz and U. Lee, Functional and topological conditions for explosive synchronization develop in human brain networks with the onset of anesthetic-induced unconsciousness, Front. Comput. Neurosci., 10 (2016), p1, URL https://www.ncbi.nlm.nih.gov/pubmed/26834616.

10.3389/fncom.2016.00001
[11]

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, 1984. doi: 10.1007/978-3-642-69689-3.

[12]

I. Leyva, A. Navas, I. Sendiña-Nadal, J. A. Almendral, J. M. Buldú, M. Zanin, D. Papo and S. Boccaletti, Explosive transitions to synchronization in networks of phase oscillators, Sci. Rep., 3 (2013), 1281, URL http://www.nature.com/doifinder/10.1038/srep01281. doi: 10.1038/srep01281.

[13]

I. Leyva, I. Sendiña-Nadal, J. A. Almendral, A. Navas, S. Olmi and S. Boccaletti, Explosive synchronization in weighted complex networks, Phys. Rev. E, 88 (2013), 042808, URL http://link.aps.org/doi/10.1103/PhysRevE.88.042808. doi: 10.1103/PhysRevE.88.042808.

[14]

I. Leyva, R. Sevilla-Escoboza, J. M. Buldú, I. Sendiña-Nadal, J. Gómez-Gardeñes, A. Arenas, Y. Moreno, S. Gómez, R. Jaimes-Reátegui and S. Boccaletti, Explosive first-order transition to synchrony in networked chaotic oscillators, Phys. Rev. Lett., 108 (2012), 168702, URL http://link.aps.org/doi/10.1103/PhysRevLett.108.168702. doi: 10.1103/PhysRevLett.108.168702.

[15]

E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott, P. So and T. M. Antonsen, Exact results for the Kuramoto model with a bimodal frequency distributions, Phys. Rev. E, 79 (2009), 026204, 11 pp, URL https://link.aps.org/doi/10.1103/PhysRevE.79.026204 doi: 10.1103/PhysRevE.79.026204.

[17]

D. Pazó, Thermodynamic limit of the first-order phase transition in the Kuramoto model, Phys. Rev. E, 72 (2005), 046211, 6pp, URL http://journals.aps.org/pre/abstract/10.1103/PhysRevE.72.046211. doi: 10.1103/PhysRevE.72.046211.

[18]

M. Rohden, A. Sorge, M. Timme and D. Witthaut, Self-organized synchronization in decentralized power grids, Phys. Rev. Lett., 109 (2012), 064101, URL http://link.aps.org/doi/10.1103/PhysRevLett.109.064101. doi: 10.1103/PhysRevLett.109.064101.

[19]

X. Zhang, S. Boccaletti, S. Guan and Z. Liu, Explosive synchronization in adaptive and multilayer networks, Phys. Rev. Lett., 114 (2015), 038701, URL http://link.aps.org/doi/10.1103/PhysRevLett.114.038701. doi: 10.1103/PhysRevLett.114.038701.

[20]

X. Zhang, Y. Zou, S. Boccaletti and Z. Liu, Explosive synchronization as a process of explosive percolation in dynamical phase space, Sci. Rep., 4 (2014), 5200, URL http://www.ncbi.nlm.nih.gov/pubmed/24903808. doi: 10.1038/srep05200.

Figure 1.  (a) Phase synchronization level $r$ vs the coupling strength $\lambda$, for different values of the frequency gap $\gamma$ (see. Eq. (2)) at $\langle k\rangle = 40$ with network size N = 500. Frequency distribution $g(\omega)$ is chosen to be homogeneous in the [0, 1] interval. (b) Same as in (a), but for different values of the average degree $\langle k\rangle$ at $\gamma = 0.4$. In both panels, the legends report the color and symbol codes for the different plotted curves. In (c) and (d), the degree $k_i$ that each node achieves after the network growth is completed (upper plots) and the average of the natural frequencies $\langle \omega_j\rangle$ of the neighboring nodes ($j\in {\it N}(i)$, bottom plots) are reported vs. the node's natural frequency $\omega_i$, for $\langle k \rangle = 100$ and frequency gaps $\gamma = 0.0$ (c), and $\gamma = 0.4$ (d). The red solid line in (d) is a sketch of the theoretical prediction $f(\omega)$. Panels (e) and (f) show $r$ (color coded according to the color bar) in the parameter space ($\lambda, \gamma$) for (e) $\langle k \rangle = 20$ and (f) $\langle k \rangle = 60$. The horizontal dashed lines mark the separation between the region of the parameter space where a second-order transition occurs (below the line) and that in which the transition is instead of the first order type (above the line). The yellow striped area delimits the hysteresis region
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  (Top row) $r$ vs. $\lambda$. Synchronization transition schemes for different $g(\omega)$ or network construction rules, for system size N = 500. (Bottom row) The final node degree $k_i$ as a function of $\omega_i$. (a)-(b) Rayleigh distribution for $\gamma = 0.3$. In (b), the red solid line depicts the theoretical prediction $f(\omega)$, obtained with the same method of the red solid line in panel (d) of Fig. 1; (c)-(d) uniform frequency distribution, but network constructed accordingly to a local mean field condition (see text) for $\gamma = 0$, and $\gamma = 0.4$ (see legend for color code). In panel (d) $\gamma = 0.4$. In all cases, $\langle k\rangle = 60$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  (a) Synchronization schemes ($r$ vs. $\lambda /\langle k\rangle$) for ER networks of size $N = 500$, $\langle k\rangle $ = 30, for the un-weighted case (dark blue squares), and weighted cases for several frequency distributions within the range $[0, 1]$. (b) Node strengths $s_i$ (see text for definition) vs. natural frequencies $\omega_i$, for the un-weighted (dark blue squares) and weighted (light blue circles) networks reported in (a). The solid line corresponds to the analytical prediction
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  (a) $I$ as a function of $\mu = \lambda r$ (solid curve) as given by Eq. (16). The dashed lines are those straight lines whose intersection with $I$ marks the backward ($\lambda_1^c$) and forward ($\lambda_2^c$) critical points of ES for a fully connected network and for a uniform frequency distribution. (b) The corresponding synchronization order parameter $r$ as a function of the coupling strength. Solid (dashed) curves correspond to the stable (unstable) solution. Dotted vertical lines mark the region of hysteresis defined by $\lambda_1^c$ and $\lambda_2^c$ in (a)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Plots of the matrix $r_{ij}$ (Eq. (17)) for fully connected (a)-(d) and ER (e-h) networks. Oscillators are labeled in ascending order of the frequency $\omega_i$. The critical coupling strengths where the ES takes place are $\lambda_c$ = 2.13 in $\lambda_c = 2.17$ respectively
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Comparison between synchronization centrality $\Lambda^C_i$ (black squares) and topological centrality $\Lambda^A_i$ (red dots). (a) ER networks, $\langle k \rangle$ = 50, $\Lambda^C_i$ and $\Lambda^A_i$ are reported vs. the nodes' natural frequencies $\omega_i$; (b) SF networks, $\langle k \rangle$ = 12, $\Lambda^C_i$ and $\Lambda^A_i$ are plotted vs. the node degrees $k_i$. All data refer to ensemble averages over 100 different network realizations for N = 500 nodes
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Forward (black line with squares) and backward (red line with circles) synchronization transitions for a single network with N = 1000 and f = 1. The insets report the dependence on $f$ of the average transition points $\lambda_F$ and $\lambda_B$ for ten realizations
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Analytical solutions of Eq. (23) for the order parameter $r$ vs. $\lambda$ for the ER network with $f$ = 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Synchronization parameters $r^{1}$ and $r^{2}$ vs. $\lambda$ for two inter-dependent networks with $N = 10^3$ and $f = 1$. Squares and circles (triangles and stars) denote the forward (backward) transitions, and the insets report the average width $\langle d\rangle$ of the hysteresis loop as a function of $f$. Here layer 1 is a ER network with $\langle k\rangle = 12$. From (a) to (b), layer 2 is ER network with $\langle k\rangle = 12$ and $\langle k\rangle = 6$, and $g(w_i^2)$ is a random homogeneous distribution in the range $[-1, 1]$. From (c) to (d), layer 2 is ER network and SF network with $\langle k\rangle = 12$, and $g(w_i^2)$ is Lorentzian distribution and random homogeneous fashion
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139
[2]

Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial & Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87
[3]

Paola Goatin. Traffic flow models with phase transitions on road networks. Networks & Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287
[4]

Jin-Liang Wang, Zhi-Chun Yang, Tingwen Huang, Mingqing Xiao. Local and global exponential synchronization of complex delayed dynamical networks with general topology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 393-408. doi: 10.3934/dcdsb.2011.16.393
[5]

Honghu Liu. Phase transitions of a phase field model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883
[6]

Sujit Nair, Naomi Ehrich Leonard. Stable synchronization of rigid body networks. Networks & Heterogeneous Media, 2007, 2 (4) : 597-626. doi: 10.3934/nhm.2007.2.597
[7]

Daniel M. N. Maia, Elbert E. N. Macau, Tiago Pereira, Serhiy Yanchuk. Synchronization in networks with strongly delayed couplings. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3461-3482. doi: 10.3934/dcdsb.2018234
[8]

Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure & Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545
[9]

Steffen Arnrich. Modelling phase transitions via Young measures. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 29-48. doi: 10.3934/dcdss.2012.5.29
[10]

Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163
[11]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565
[12]

Nicolai T. A. Haydn. Phase transitions in one-dimensional subshifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1965-1973. doi: 10.3934/dcds.2013.33.1965
[13]

Joseph D. Skufca, Erik M. Bollt. Communication and Synchronization in Disconnected Networks with Dynamic Topology: Moving Neighborhood Networks. Mathematical Biosciences & Engineering, 2004, 1 (2) : 347-359. doi: 10.3934/mbe.2004.1.347
[14]

Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997
[15]

Valeria Berti, Mauro Fabrizio, Diego Grandi. A phase field model for liquid-vapour phase transitions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 317-330. doi: 10.3934/dcdss.2013.6.317
[16]

Zhen Jin, Guiquan Sun, Huaiping Zhu. Epidemic models for complex networks with demographics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1295-1317. doi: 10.3934/mbe.2014.11.1295
[17]

Larry Turyn. Cellular neural networks: asymmetric templates and spatial chaos. Conference Publications, 2003, 2003 (Special) : 864-871. doi: 10.3934/proc.2003.2003.864
[18]

John Erik Fornæss. Infinite dimensional complex dynamics: Quasiconjugacies, localization and quantum chaos. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 51-60. doi: 10.3934/dcds.2000.6.51
[19]

Tingting Su, Xinsong Yang. Finite-time synchronization of competitive neural networks with mixed delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3655-3667. doi: 10.3934/dcdsb.2016115
[20]

William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971

2017 Impact Factor: 0.972

Tools

Metrics

  • PDF downloads (58)
  • HTML views (196)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2018 American Institute of Mathematical Sciences