Conclusion
Dans ce mémoire, nous avons étudié les
systèmes lent-rapides déterministes puis stochastique et on a
introduit un exemple typique de ces système sont les résonances,
nous avons donné des exemples pratiques avec simulation.
Ce mémoire est consacré à l'étude
des systèmes de FitzHugh-Nagumo stochastiques qui ont été
introduits pour modéliser la transmission de l'influx nerveux dans un
neurone. Nous avons commencé par rappeler des résultats sur le
système déterministe associé afin de trouver les valeurs
des paramètres intéressants pour notre étude. Celles-ci
correspondent aux cas où le système admet au moins un point
d'équilibre stable et où le système est excitable.
Comme perspectives, on s'intéresse à
l'étude du comportement de solutions des systèmes lent-rapides
plus compliqués dans lesquels les coefficients du système de
FitzHugh-Nagumo (a, b, c) sont aléatoires, et dans ce cas, on introduit
d'autres approches probabilistes pour étudier ce système et voir
le comportement asymptotique puis la stabilité et la stabilisation.
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