By Prund A. H.
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1878) M´emoire sur les ´equations diﬀ´erentielles alg´ebriques du premier ordre et du premier degr´ e. Bull. Sci. Math. S´er. 2 (2), 60-96, 123-143, 151-200. [Demazure 1989] Demazure, M. (1989) Catastrophes et Bifurcations. Ellipses, Paris. [Fenichel 1979] Fenichel, N. (1979) Geometric singular perturbation theory for ordinary diﬀerential equations. J. Diﬀ. , 31, 53-98. -M. Ginoux and B. Rosseto [Ginoux et al. ; and B. Rossetto (2006) Diﬀerential Geometry and Mechanics: applications to chaotic dynamical systems.
N j=1 Dxn (t) = anj (t)xj + fn (t, x1 , x2 , . . , xn ), subject to the initial condition, x1 (0) = c1 , x2 (0) = c2 , . . , xn (0) = cn , (2) where fi is a nonlinear function for i = 1, 2, . . , n. In view of the homotopy perturbation technique, we can construct, for i = 1, 2, . . , n, the following homotopy, n Dxi (t) − aij (t)xj = pfi (t, x1 , x2 , . . , xn ), (3) j=1 where p ∈ [0, 1]. The homotopy parameter p always changes from zero to unity. In case of p = 0, Eq. (3) becomes the linear equation, n Dxi (t) = aij (t)xj , (4) j=1 and when it is one, Eq.
When a zone becomes less interesting, ants leave it and pheromone disappear. When an area becomes of a great interest, ants colonize it by laying down pheromones that attract more ants, and the process self-ampliﬁes. We respect the metaphor here since it brings us the very interesting property of handling the dynamics. Indeed, our application continuously changes, the graph that represents it follows this, and we want our method to be able to discover new highly communicating clusters, while abandoning vertices that are no more part of a cluster.