Naslov
Normalizable, integrable, and linearizable saddle points for complex quadratic systems in C2

Autori
Christopher, C. ; Mardešić, Pavao ; Rousseau, C.

Izvornik
Journal of dynamical and control systems (1079-2724) 9 (2003), 3; 311-363

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
C^2 saddle; normalizability

Sažetak
In this paper we consider complex differential systems in the neighbor- hood of a singular point with eigenvalues in the ratio 1 : − λ with λ ∈ R + ∗ . We address the questions of normalizability (i.e. convergence of the normalizing tra ns- formation), integrability and linearizability of the system. We intro duce the notion of isochronicity of a system at an integrable saddle for general λ and prove that a system is linearizable if and only if it is isochronous. We then specialize to q uadratic systems and give explicit examples of non-normalizable quadratic systems as wel l as quadratic systems which are integrable but not linearizable, for any λ satisfying a convenient diophantine condition. A distinction between normalizable and orbitally normalizable systems is also drawn along similar lines. Our main interest is the global organization of the strata of those system s for which the normalizing transformations converge, or for which we have integ rable or linearizable saddles as λ and the other parameters of the system vary. We give several tools for demonstrating normalizability, integrability and linea rizability and apply them to the detailed study of several classes of quadratic systems. Many of the results are valid in the larger context of polynomial or analytic vect or fields. We explain certain features of the bifurcation diagram , namely the existence of a continuous skeleton of integrable (linearizable systems) with sequences of holes filled with orbitally normalizable (normalizable) systems.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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