Team:Paris/Network analysis and design/Core system/Mathematical analysis and simulations/Mathematical analysis


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Understanding the dynamics

  • Theory introduction

The classical approach consists in finding the equilibrium state, by setting


Then, we evaluate the jacobian matrix, so as to put the system under its linearized form:

Matrix truebis.jpg

Where [EnvZ]* represents the equilibirum state of [EnvZ] and :

Def fbis.jpg

which gives :


Then, we want to find the eigenvalues of the jacobian matrix, because they make us understand the behavior of the system. Here is the theoretical explanation :


The λ and γ coefficients make the convergence. In our case they are negative terms, which explains the fact that we have a quick convergence for the system. Ideally, it could be convenient to find which parameter influence this coefficient, so as to play with it and be able to propose a better control of the convergence.
The μ coefficient is responsible for the oscillations. In fact, in another base, with only real matrix, this would beget cosinuses and sinuses terms. Furthermore, it is strongly linked to the period of the oscillations.

  • What about the periods
We can evaluate a value of the period we would have obtained without damping. Indeed, we have:
With the parameters presented in the array before, we have evaluated the equilibrium values by solving

We obtained three solutions :

-0.2382 - 0.6139i

-0.2382 + 0.6139i


We are only interested in positive real solutions. We can note that 0.4764 corresponds well to the experimental equilibrium value. Then we evaluated the eigenvalues for the jacobian :

-1.0159 + 0.5042i

-1.0159 - 0.5042i


We can note that the λ and γ coefficient are negative, which corroborates the fact that we obtain a convergence.

Finally for
since μ=0.5042 we find a theoretical period of Ttheoretical=12.4617.
  • Understanding the attenuation

Furthermore, this theoretical study helps us understand that reducing the attenuation is tough. In fact, whatever the base in which you consider a matrix, the trace is conserved. Here, we obtain :

Trace matrix.jpg

Since λ and γ are assumed (thanks to simulations) to be negative, we cannot avoid to have their sum to be (with absolute value) superior to 2. The only chance we have would be that the weight goes in the λ parameter because this parameter plays a role in the third dimension, in which we are not interested. However, for every simulation we made, we found the opposite. Trying to evaluate with non-numerical calculus does not provide relevant material.

However, we now know that βFliA influences the attenuation. We ran a simulation with βFliA=0.02

Damped BliA.jpg

This confirms what we saw previously. In this second case, the system is therefore less attenuated, and some very tiny damped oscillations can be observed.

Last but not least, one should not forget that we chose a normalized interpretation of the system. Thus, even though the attenuation phenomenon remains unchanged, the tiny oscillations we see may indeed be observable.