Méthodes asymptotiques-numériques pour le calcul de bifurcations de Hopf et de solutions périodiques

Abstract : In this work, we have presented a study on the ordinary differential equations which have periodic solutions or Hopf bifurcation points. For this study, we have applied an asymptotic-numerical methods that have been applied up to now only in static. We have started our test on the conservative differential equations or dissipative ones which have one degree of freedom. The domain of validity of the representation by power series is limited by a raduis of convergence. By use of the techniques discuted (approximants of Padé, projection technique and transformation of Euler), we have extended this domain up to a large value. In the second part, we have detected the Hopf bifurcation points by an asymptotic numerical algorithm. So, these points are detected through a perturbed and linear problem which depends on two real parameters. Indeed, we have introduced an Hopf bifurcation index which is expanded firstly into power series of two parameters. Then, we have caracterized the Hopf bifurcation points from this index. Since, we have showed that the index is a rational fraction. So, the series can be replaced by the approximants of Padé which lead to the exact value of the index. We have also showed that the "reduced strategies", i.e, the approximants of Padé which replace the series truncated at inferior orders, permit also to detect the Hopf bifurcation points. The efficiency, of these procedures is tested on the problems with small number of degrees of freedom. The applications on the systems with great number of degrees of freedom are the aim of others thesis in Metz
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My El Hassan Ben Saadi. Méthodes asymptotiques-numériques pour le calcul de bifurcations de Hopf et de solutions périodiques. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1995. Français. ⟨NNT : 1995METZ047S⟩. ⟨tel-01777099⟩

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