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Sopra l’equazione di A. Lienard delle oscillazioni di rilassamento. (Italian) Zbl 0037.19001


MSC:

34C25 Periodic solutions to ordinary differential equations
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References:

[1] Cartan, H. Et É., Note sur la génération des oscillations entretenues, Annales des Postes, Télégraphes et Téléphone, 14, 1196-1207 (1925)
[2] Van Der Pol, B., Sur les oscillations de relaxations, The philosophical Magazine, 2, 7, 978-992 (1926) · JFM 52.0450.05
[3] Van Der Pol, B., The non linear theory of electric oscillations, Proc. Inst. Radio Engr., 22, 1051-1086 (1934) · Zbl 0009.42107
[4] Levinson, N.; Smith, O. K., A general equations for relaxtion oscillations, Duke Math. Journ., 9, 382-403 (1942) · Zbl 0061.18908 · doi:10.1215/S0012-7094-42-00928-1
[5] Graffi, D., Sopra alcune equazioni differenziali non lineari della Fisica-Matematica, Mem. della R. Acc. delle S. dell’Istituto di Bologna, 7, 9, 121-129 (1940) · Zbl 0026.12102
[6] Levinson-O, N.; Smith, K., A general equations for relaxtion oscillations, Duke Math. Journ., 9, 384-384 (1942)
[7] Levinson-O., N.; Smith, K., A general equations for relaxtion oscillations, Duke Math. Journ., 9, 397-397 (1942)
[8] Levinson-O., N.; Smith, K., A general equations for relaxtion oscillations, Duke Math. Journ., 9, 399-399 (1942)
[9] I., Bendixson, Sur les courbes défines par les équations différentielle, Acta Math., 24, 1-88 (1901) · JFM 31.0328.03
[10] Cfr.A. Liénard, a) (Etude des oscillations entretenues, « Revue Général de lÉlectricité », 23 (1928); b)Oscillations auto-entretenues, « Proc. of the Third Int. Congr. of Applied Mechanich (Stockholm, 1930), 3 (1930), pp. 173-177. Lo studio asintotico degli integrali dell’equazione (d^2i/dt^2)+ωɛ^−1f(i)(di/dt)+ω^2i=0, ove ɛ è un parametro infinitesimo trovasi inJ. Haag, a)Étude asymptotique des oscillations, « Ann, Sc. Éc. Norm. Sup. », (3), 60 (1943), pp. 35-111; b)Exemples concrets d’étude asymptotique d’oscillations de relaxations, ibidem, (3), 61 (1944), pp. 73-117, Per altri studi sulle soluzioni periodiche della medesima equazione quando ɛ →+∞, oppure ɛ → 0, cfr.J. Shohat, a)A new analytical types for solving van der Pol’s …, « Journ. Appl. Phys. », 14 (1943), pp. 40-48; b)On van der Pol’s and non linear differential equations, ibidem, 15 (1944), pp. 568-574.
[11] All’equazione diH. edÉ. Cartan può dunque applicarsi questo teorema di esistenza.
[12] Cfr.G. Sansone,Equazioni differenziali nel Campo Reale, I (2^a ed. Bologna, 1948), p. 15.
[13] Grafei, D., A general equations for relaxtion oscillations, Duke Math. Journ., 9, 83-84 (1942)
[14] Cfr. ad es.G. Sansone, op. cit. in (11), I, Cap. V, § 4, n. 4.
[15] Cfr.C. Carathéodory,Vorlesungen üher Reeller Funktionen, (zweite Auflage, 1927) p. 678. Cfr. ancheG. Sansone, op. cit. in (11), p. 48.
[16] Haag, J., A general equations for relaxtion oscillations, Duke Math. Journ., 9, 36-36 (1942)
[17] Liênard, A., A general equations for relaxtion oscillations, Duke Math. Journ., 9, 5, 36-36 (1942)
[18] Levinson, N.; Smith, O. K., A general equations for relaxtion oscillations, Duke Math. Journ., 9, 391-398 (1942)
[19] Cfr.G. Sansone,Equazioni differenziali nel Campo Reale, 1 (2^a ed. Bologna, 1948), p. 33.
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