A synchronization network is a network of coupled dynamical systems. It consists of a network connecting oscillators, where oscillators are nodes that emit a signal with somewhat regular (possibly variable) frequency, and are also capable of receiving a signal. Particularly interesting is the phase transition where the entire network (or a very large percentage) of oscillators begins pulsing at the same frequency, known as synchronization. The synchronization network then becomes the substrate through which synchronization of these oscillators travels. Since there is no central authority organizing nodes, this is a form of self organizing system.
Definition Generally, oscillators can be biological, electronic, or physical. Some examples are fireflies, crickets, heart cells, lasers, microwave oscillators, and neurons. Further example can be found in many domains. In a particular system, oscillators may be identical or non-identical. That is, either the network is made up of homogeneous or heterogeneous nodes. Properties of oscillators include: frequency, phase and natural frequency. Network edges describe couplings between oscillators. Couplings may be physical attachment, or consist of some proximity measure through a medium such as air or space. Networks have several properties, including: number of nodes (oscillators), network topology, and coupling strength between oscillators.
Kuramoto model
Kuramoto developed a major analytical framework for coupled dynamical systems, as follows:
A network of oscillators with varied natural frequencies will be incoherent while the coupling strength is weak. Letting θ i ( t ) {\displaystyle \theta _{i}(t)} be the phase of the i {\displaystyle i} th oscillator and ω i {\displaystyle \omega _{i}} be its natural frequency, randomly selected from a Cauchy-Lorentz distribution as follows,
g ( ω ) = γ π [ γ 2 + ( ω − ω 0 ) 2 ) {\displaystyle g(\omega )={\frac {\gamma }{\pi [\gamma ^{2}+(\omega -\omega _{0})^{2})}}} , having width γ {\displaystyle \gamma } and central value ω 0 {\displaystyle \omega _{0}} , we obtain a description of collective synchronization:
d θ i d t = ω i + 1 N ∑ j = 1 N K i j sin ( θ j − θ i ) , i = 1 , . . . , N {\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+{\frac {1}{N}}\sum _{j=1}^{N}K_{ij}\sin(\theta _{j}-\theta _{i}),i=1,...,N} , where N {\displaystyle N} is the number of nodes (oscillators), and K i j {\displaystyle K_{ij}} is the coupling strength between nodes i {\displaystyle i} and j {\displaystyle j} . Kuramoto has also developed an "order parameter", which measures synchronization between nodes:
r ( t ) = | 1 N ∑ j = 1 N e i θ j ( t ) | {\displaystyle r(t)={\bigg |}{\frac {1}{N}}\sum _{j=1}^{N}e^{i\theta _{j}(t)}{\bigg |}}
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 }
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