The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations

Waldo Arriagada-Silva; Christiane Rousseau

Displaying similar documents to “The modulus of analytic classification for the unfolding of the codimension-one flip and Hopf bifurcations”

The trivial locus of an analytic map germ

H. Hauser, G. Muller (1989)

Annales de l'institut Fourier

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We prove: For a local analytic family ${X_{s}}_{s \in S}$ of analytic space germs there is a largest subspace $T$ in $S$ such that the family is trivial over $T$ . Moreover the reduction of $T$ equals the germ of those points $s$ in $S$ for which $X_{s}$ is isomorphic to the special fibre $X_{0}$ .

Failure of analytic hypoellipticity in a class of differential operators

Ovidiu Costin, Rodica D. Costin (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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For the hypoelliptic differential operators $P = \partial_{x}^{2} + {(x^{k} \partial_{y} - x^{l} \partial_{t})}^{2}$ introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of $k$ and $l$ left open in the analysis, the operators $P$ also fail to be analytic hypoelliptic (except for $(k, l) = (0, 1)$ ), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of eigenvalues of a non-self-adjoint operator.

Overstability and resonance

Augustin Fruchard, Reinhard Schäfke (2003)

Annales de l’institut Fourier

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We consider a singularity perturbed nonlinear differential equation $ε u^{'} = f (x) u + + ε P (x, u, ε)$ which we suppose real analytic for $x$ near some interval $[a, b]$ and small $| u |$ , $| ε |$ . We furthermore suppose that 0 is a turning point, namely that $x f (x)$ is positive if $x \neq 0$ . We prove that the existence of nicely behaved (as $ϵ \to 0$ ) local (at $x = 0$ ) or global, real analytic or $C^{\infty}$ solutions is equivalent to the existence of a formal series solution $\sum u_{n} (x) ε^{n}$ with $u_{n}$ analytic at $x = 0$ . The main tool of a proof is a new “principle of analytic continuation” for...

Analytic curves in power series rings

Herwig Hauser, Gerd Müller (1990)

Compositio Mathematica

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On the analyticity of generalized eigenfunctions (case of real variables)

Eberhard Gerlach (1968)

Annales de l'institut Fourier

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On démontre que, dans les espaces fonctionnels propres de Hilbert (avec un noyau reproduisant), formés de fonctions analytiques de $n$ variables dans un domaine $G$ , pour tout opérateur auto-adjoint, les fonctions propres généralisées sont des fonctions réelles-analytiques dans $G$ .