# Synchronization in Coupled Nonlinear Oscillators

I hope with this first question I respect the standards of the forum :)

I am currently working on an ANN model for Working Memory (Short Term memory in the Brain) and as a first step I am studying a reduced dimensional system. Unfortunately, I am kind of new to the theory of Nonlinear oscillators, therefore I have some doubts about how to approach the problem.

I attach here the equations and what I found at the moment about the system and what I think it would be interesting to analyze. I have two 2-dimensional units that interacts with their outputs and synchronize on a limit cycle.

$$\dot d_1 = - d_1 -e_1 + \epsilon\sigma(d_2) \\ \tau_a \dot e_1 = g_a\sigma(d_1) - e_1 \\$$

$$\dot d_2 = - d_2 -e_2 + \epsilon\sigma(d_1) \\ \tau_a \dot e_2 = g_a\sigma(d_2) - e_2$$

$$\sigma(d_i) = 1 - \frac{2}{1+e^d}$$

From some simple simulations one can see that the system above synchronize ($$||d_1-d_2||->0$$) on a stable limit cycle. Limit_Cycle, Synch

Now we can go with the questions:

My intuition suggested me that to study the limit cycle's properties (with Poincaré-Bendixsson theorem for example) I can study the system on the synchronization manifold where $$d_1=d_2$$. On this invariant manifold with with a fairly simple bifurcation analysis, one can study how the system's behaviour changes with changing the parameters that define the dynamics

Q1) In order to study the synchronization of the two systems, what could be a good approach? I read several works but most of them are on systems that are diffusively coupled (the coupling depends on a function of the difference of the output/states of the system).

One idea could be to study the error dynamics. By defining the quantities $$x = d_1 - d_2$$ and $$y = e_1 - e_2$$ we obtain:

$$\dot x = - x - y + \epsilon(\sigma(d_2)-\sigma(d_1)) \\ \tau_a \dot y = g_a(\sigma(d_1)-\sigma(d_2)) - y \\$$

The main issue is to 'get rid of' the quantity $$(\sigma(d_2)-\sigma(d_1))$$ and replace it with some function $$f(x)$$ of the only difference. Unfortunately, this is not possible since:

$$a(d_1, d_2) = \sigma(d_2)-\sigma(d_1) = \frac{e^{d_2}-e^{d_1}}{(1+e^{d_1})(1+e^{d_2})}$$

Although the function $$a(d_1, d_2)$$ can not be expressed only as a function of the difference $$x$$ it could be interesting to see $$a(d_1, d_2)$$ as a time-varying function $$f(x, t)$$ that preserves some properties, for example $$x f(x,t)<0, \forall x\neq0$$.

At this point if one finds the right tools the GAS of the synchronization manifold could be proven. (Maybe some passivity analysis or similar for time-varying systems)

(If somebody is interested in simulating the system by themselves, parameters that could be used are: $$\tau_a = 2.7, g_a=97, \epsilon>2(1+1/\tau_a)$$)

• What is interesting to study depends a lot on what you're interested in achieving. If this is mostly for practice, I would suggest doing a bifurcation study. Finding basins of attraction (and especially how they depend on the parameters) could be a nice second step. Also, Q2 doesn't seem to make sense to me. How can it be "completely decoupled" and "coupled oscillators that synchronize" at the same time? What do you mean by completely decoupled? – Steve Heim Apr 5 at 14:22
• About the bifurcation analysys: luckily it is fairly simple to analytically compute the parameters for which the (stable) limit cycle occurs, i.e. when $\epsilon>2(1+\frac{1}{\tau_a})$. Furthermore one can also study the number of fixed points and their stability. About Q2: What I meant is that, when there is no coupling (i.e. $\epsilon=0$) both the systems converge to the origin without any 'interesting' behaviour. From the question I agree that could be quite confusing. – giangian Apr 5 at 14:33
• Using the property that $\sigma(\cdot)$ is globally Lipschitz with $|\sigma(d_1)-\sigma(d_2)|\leq \frac{1}{2}|d_1-d_2|$ might give you an aswer. – RTJ Apr 7 at 8:27
• Ok, I found some result on time-varying sector nonlinearities, but I would prefer a more "Lyapunov" approach. How could I proceed with this? – giangian Apr 13 at 13:08