CDS - Syncronization

Remember:

Syncronization happens in particular networks. Syncronization is one of the simpler example of “self-organization”.

Take two equation representing two distinct mass-spring systems:
$\begin{array}{l}\{\begin{array}{l}{x}_1=x_3\\ x_3=-{\omega }_1^2{x}_1\end{array}\\ \{\begin{array}{l}{x}_2=x_4\\ x_4=-{\omega }_2^2{x}_2\end{array}\end{array}$Leaving the first equaiton untuched (since it serves only to define $\ddot{x}$ in terms of ODE equations), we can add a term called “coupling parameter” that depends on a variable of the second system:$\begin{array}{l}\{\begin{array}{l}{x}_1=x_3\\ x_3=-{\omega }_1^2{x}_1+k(x_2-x_1)\end{array}\\ \{\begin{array}{l}{x}_2=x_4\\ x_4=-{\omega }_2^2{x}_2+k(x_1-x_2)\end{array}\end{array}$NOTE: for $x_2=x_1$ this term is equal to $0$ ⇒ This term is active only when these two variables are different.

Let’s see how the graph of these two systems change when we increase the $k$ or “coupling parameter”.

• For $k=0$:
  
  Here’s the base example, without coupling.
• For $k=1$:
  
  We begin to see some changes, the continous lines are the real systems (with coupling), the dotted line are are the systems without coupling (seen before).
• $k=4$:
  
• $k=10$:
  
• $k=100$, here we have an almost perfect matching:
  

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• These metronomes started with different initial conditions, will syncronize, due how the “system is built”.
• Generally we can say that to create a syncronization between multimple systems, we need to create a connection between them, in this example the connection is the plank suspended on two cans of soda.
• Syncronization happens in particular networks.
• Syncronization is one of the simpler example of “self-organization”.

• Simple harmonic oscillator

• Represents for example a mass-spring system without friction.

• Consider annother equivalent system (it will have different initial conditions, but same differential equations)

• Leaving the first equaiton untuched (since it serves only to define $\ddot{x}$ in terms of ODE equations), we can add a term called “coupling parameter” that depends on a variable of the second system.
• I rewrite here the equations:$\begin{array}{l}\{\begin{array}{l}{x}_1=x_3\\ x_3=-{\omega }_1^2{x}_1+k(x_2-x_1)\end{array}\\ \{\begin{array}{l}{x}_2=x_4\\ x_4=-{\omega }_2^2{x}_2+k(x_1-x_2)\end{array}\end{array}$
• NOTE: for $x_2=x_1$ this term is equal to $0$ ⇒ This term is active only when these two variables are different.

• $c$ is the coupling term, previosly called $k$.

• Positions of the two systems.
• This graphs are without coupling, $c=0$.

• $c=1$, now there is coupling.

• The continous lines are the real systems (with coupling), the dotted line are are the systems without coupling (seen before).

• $c=4$.

• $c=10$, we can see that the two lines are much more similar.

• $c=100$, almost perfect marching.