Gradient trajectories for plane singular metrics I : oscillating trajectories

• Autorzy: Grandjean V.
• Języki publikacji: EN

Abstrakty

EN

In this short note, we construct an example of a real plane analytic singular metric, degenerating only at the origin, such that any gradient trajectory (respectively to this singular metric) of some well chosen function spirals around the origin. The inversion mapping carries this example into an example of a gradient spiraling dynamics at infinity.

Słowa kluczowe

EN

oscillating trajectories; non-oscillating trajectories; singular gradient; singular metric

PL

trajektorie oscylacyjne; trajektorie nie oscylujące; gradient; metryka pojedyncza

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2014
• Tom: Vol. 47, nr 1
• Strony: 69--78
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Grandjean V.

• Departamento De Matemática, UFC Av. Humberto Monte s/n, Campus do Pici Bloco 914 CEP 60.455-760, Fortaleza-Ce, Brasil

Bibliografia

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• [2] P. Fortuny, F. Sanz, Gradient vector fields do not generate twister dynamics, J. Differential Equations 174(1) (2001), 91–100.
• [3] P. Goldstein, Gradient flow of a harmonic function in R3, J. Differential Equations 247(9) (2009), 2517–2557.
• [4] V. Grandjean, Limit set at infinity of gradient of semi-algebraic functions, J. Differential Equations 233(1) (2007), 22–41.
• [5] V. Grandjean, Gradient trajectories for plane singular metrics II: singular metrics a with few limits, in preparation.
• [6] V. Grandjean, F. Sanz, On restricted analytic gradient on analytic isolated surface singularities, preprint 2011, 32 pages, available at http://arxiv.org/abs/1105. 3888.
• [7] K. Kurdyka, T. Mostowski, A. Parusinski, Proof of the gradient conjecture of R. Thom, Ann. of Math. 152 (2000), 763–792.
• [8] S. Łojasiewicz, Sur les trajectoires du gradient d’une fonction analytique, Seminari di Geometria, Bologna (1983), 115–117.
• [9] S. Łojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 449–474.
• [10] R. Moussu, Sur la dynamique des gradients. Existence de variétés invariantes, Math. Ann. 307(3) (1997), 445–460.
• [11] F. Sanz, Non-oscillating solutions of analytic gradient vector fields, Ann. Inst. Fourier (Grenoble) 48(4) (1998), 1045–1067.