</p><p>This leads to the asymptotic definition of 
  
    
      
        
          K
          
            c
          
        
      
    
    {\displaystyle K_{c}}
  
, the critical coupling strength, as 
  
    
      
        N
        →
        ∞
      
    
    {\displaystyle N\to \infty }
  
 and 
  
    
      
        t
        →
        ∞
      
    
    {\displaystyle t\to \infty }

</p><p>
  
    
      
        r
        =
        
          
            {
            
              
                
                  0
                  ,
                
                
                  K
                  <
                  
                    K
                    
                      c
                    
                  
                
              
              
                
                  
                    
                      1
                      −
                      (
                      
                        K
                        
                          c
                        
                      
                      
                        /
                      
                      K
                      )
                    
                  
                  ,
                
                
                  K
                  ≥
                  
                    K
                    
                      c
                    
                  
                
              
            
            
          
        
      
    
    {\displaystyle r={\begin{cases}0,&K<K_{c}\\{\sqrt {1-(K_{c}/K)}},&K\geq K_{c}\end{cases}}}

</p><p>with 
  
    
      
        
          K
          
            c
          
        
        =
        2
        γ
      
    
    {\displaystyle K_{c}=2\gamma }
  
.
</p><p>Note that 
  
    
      
        r
        =
        0
        ⇒
      
    
    {\displaystyle r=0\Rightarrow }
  
 no synchronization, and 
  
    
      
        r
        =
        
          |
        
        
          e
          
            i
            θ
          
        
        
          |
        
        =
        1
        ⇒
      
    
    {\displaystyle r=|e^{i\theta }|=1\Rightarrow }
  
 perfect synchronization.
</p><p>Beyond 
  
    
      
        
          K
          
            c
          
        
      
    
    {\displaystyle K_{c}}
  
, each oscillator will belong to one of two groups:
</p>
<ul><li>a group that is synchronized.</li>
<li>a group that will never synchronize, since their natural frequencies vary too greatly from the synchronization frequency.</li></ul>
<h2 id="network-topology">Network topology</h2>
<p>Synchronization networks may have many topologies.   <a href="/facts/Network_topology/rBYE0kke">Topology</a> may have a great deal of influence over the spread of dynamics.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p><p>Some major topologies are listed below:
</p>
<ul><li><a href="/facts/Regular_graph/j9dSoLE6">Regular networks</a>: This describes networks where every node has the same number of links.  <a href="/facts/Lattice_graph/4EHMt1b6">Lattices</a>, <a href="/facts/Network_topology/rBYE0kke">rings</a>, and fully connected networks are some examples of this topology.</li>
<li><a href="/facts/Random_graph/gfQhq1Sz">Random graphs</a>:  Developed by <a href="/facts/Paul_Erd%25C5%2591s/63LoUTe1">Erdős</a> and <a href="/facts/Alfr%25C3%25A9d_R%25C3%25A9nyi/q7Dx7fgB">Rényi</a>, these graphs are characterized by a constant probability of a link existing between any two nodes.</li>
<li><a href="/facts/Small_world_networks/dALPpt9m">Small world networks</a>:  These networks are the result of rewiring a certain number of edges in regular lattice networks.  The resulting networks have much smaller average path length than the original networks.</li>
<li><a href="/facts/Scale-free_network/2ENlcjX3">Scale-free networks</a>:  Found ubiquitously in naturally occurring systems, scale free networks are characterized by a large number of high-degree nodes.  In particular, the degree distribution follows a power-law.</li></ul>
<h2 id="history">History</h2>
<p><a href="/facts/Coupled_oscillation/6vSrm84a">Coupled oscillators</a> have been studied for many years, at least since the <a href="/facts/Wilberforce_pendulum/lKr8mLQy">Wilberforce pendulum</a> in 1896.
In particular, pulse coupled oscillators were pioneered by Peskin in 1975 with his study of cardiac cells.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
Winfree developed a <a href="/facts/Mean-field_approach/tJrwSURE">mean-field approach</a> to synchronization in 1967, which was developed further in the <a href="/facts/Kuramoto_model/kobF89Pd">Kuramoto model</a> in the 1970s and 1980s to describe large systems of coupled oscillators.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
Crawford brought the tools of manifold theory and bifurcation theory to bear on the stability of synchronization with his work in the mid-1990s.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
These works coincided with the development of a more general theory of coupled dynamical systems and popularization by Strogatz <i>et al.</i> in 1990, continuing through the early 2000s.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Kuramoto_model/kobF89Pd">Kuramoto model</a></li>
<li><a href="/facts/Complex_Networks/fxFrvRKs">Complex Networks</a></li>
<li><a href="/facts/Coupled_oscillation/6vSrm84a">Coupled Oscillators</a></li>
<li><a href="/facts/Dynamical_Systems/782gREl5">Dynamical Systems</a></li>
<li><a href="/facts/Statistical_Physics/SnlJE78a">Statistical Physics</a></li>
<li><a href="/facts/Self-organizing_system/NSGBKxUa">Self-Organizing Systems</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.ted.com/talks/steven_strogatz_on_sync">Steven Strogatz: The science of sync</a>, a <a href="/facts/TED_(conference)/Sc3JxnA1">TED talk</a></li>
<li><a href="http://tam.cornell.edu/faculty-bio.cfm?NetID=shs7">Strogatz @ Cornell</a></li>
<li><a href="http://www.eecs.harvard.edu/ssr/projects/sync/">Self Organizing Systems Research Group at Harvard</a></li>
<li><a href="https://www.amazon.com/NextGen-Network-Synchronization-Dhiman-Chowdhury-ebook/dp/B0951M3F45/ref=mp_s_a_1_3?dchild=1&keywords=nextgen+network+synchronization&qid=1623721265&sprefix=nextgen+network+synchronization&sr=8-3/">Nextgen Network Synchronization</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Steven H. Strogatz (March 2001). "Exploring complex networks". Nature 410 (6825). <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Y. Kuramoto AND I. Nishikawa, Statistical macrodynamics of large dynamical systems. Case of a
phase transition in oscillator communities, J. Statist. Phys., 49 (1987) <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Mirollo, R. E., Steven H. Strogatz (December 1990). "Synchronization of pulse-coupled biological oscillators". SIAM Journal on Applied Mathematics 50 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Steven H. Strogatz (September 2000). "From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators". Physica D: Nonlinear Phenomena 143 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Strogatz, Steven (2003). Sync : the emerging science of spontaneous order. Hyperion. ISBN 978-0-7868-6844-5. OCLC 50511177 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Nature, Vol. 393, No. 6684. (4 June 1998), pp. 440-442 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Peskin, C. S., Mathematical Aspects of Heart Physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Winfree, A. T., Biological rhythms and the behavior of populations of coupled oscillators, J. Theoret. Biol., 16 (1967) <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>J.D. Crawford, J. Statist. Phys. 74 (1994) 1047. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
</ol>

Synchronization network open-in-new

Synchronization network