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# ordinary differential equations - Introducing noise and time lag between two coupled Rössler systems

I have two Rössler systems mutually coupled by the second component. I want to introduce some small noise and a slight time lag of the coupling between the systems.

I'm not sure
1. what the best choice for the noise parameter would be and were to apply it, e.g. gaussian noise with mean 0 standard deviation 1 and apply as additive noise at the first component of each system or only at the first or the second system? Or should I choose a different mean and std? and
2. How to introduce the time lag between the coupling?
Also I'm wondering what "unit" the time lag will be, say I want a time lag of half a second or a second. Would it just be 0.5 and 1?

$$\frac{dx_1}{dt} = -(1+v)x_2-x_3$$ $$\frac{dx_2}{dt} = (1+v)x_1+ax_2+u(y_2-x_2)$$ $$\frac{dx_3}{dt} = b+x_3(x_1-c)$$

and

$$\frac{dy_1}{dt} = -(1-v)y_2 -y_3$$ $$\frac{dy_2}{dt} = (1-v)y_1+ay_2+u(x_2-y_2)$$ $$\frac{dy_3}{dt} = b+y_3(y_1-c)$$

with $a=0.2925, b=0.1, c=8.5, v=0.02$

1. 1) What do you want the noise to do? If you want to check whether you noiseless systems behaves the same way as the one with noise -- introduce zero mean white noise times small number $\epsilon$. Where to put (additive noise or parameter noise) it is your choice, a priori you can get different answers for different places.
2) Depends again on what do you want this time lag to represent. The most naive option is just to replace $x_2(t)$ by $x_2(t+constTimeLag)$ in the second system of equations (and the similar for $y_2$ in the first system). Of course you will not be able to use standard ode solvers because it is a systems of ODE with delay which is more difficult to solve (and needs heavy mathematics to be accurately described).