Geodesic
Bifurcation of the Equilibrium Point in the Critical Case of Two Pairs of Zero Characteristic Roots
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 45-60 
V. V. Basov. Bifurcation of the Equilibrium Point in the Critical Case of Two Pairs of Zero Characteristic Roots. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 45-60. http://geodesic.mathdoc.fr/item/TM_2002_236_a5/
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     title = {Bifurcation of the {Equilibrium} {Point} in the {Critical} {Case} of {Two} {Pairs} of {Zero} {Characteristic} {Roots}},
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A real autonomous system of four differential equations with a small parameter is considered. It is proved that, under a finite number of explicit conditions on the coefficients of the lower order terms in the expansion of the right-hand sides, its two-dimensional invariant torus bifurcates at infinitesimal frequencies for sufficiently small values of the parameter. Such a system describes, in particular, the oscillations of two weakly coupled oscillators with restoring forces of orders $2n-1$ and $2n+1$.

[1] Basov V. V., Bibikov Yu. N., “Bifurkatsiya polozheniya ravnovesiya sistemy differentsialnykh uravnenii v kriticheskom sluchae dvukh chisto mnimykh i dvukh nulevykh kornei kharakteristicheskogo uravneniya, I”, Dif. uravneniya, 36:1 (2000), 26–32 | MR | Zbl

[2] Basov V. V., “Bifurkatsiya polozheniya ravnovesiya sistemy differentsialnykh uravnenii v kriticheskom sluchae dvukh chisto mnimykh i dvukh nulevykh kornei kharakteristicheskogo uravneniya, II”, Dif. uravneniya, 37:4 (2001), 435–438 | MR | Zbl

[3] Basov V. V., “Ob ustoichivosti polozheniya ravnovesiya v kriticheskom sluchae dvukh chisto mnimykh i dvukh nulevykh kornei kharakteristicheskogo uravneniya”, Dif. uravneniya, 35:10 (1999), 1313–1318 | MR | Zbl

[4] Lyapunov A. M., “Issledovanie odnogo iz osobennykh sluchaev zadachi ob ustoichivosti dvizheniya”, Sobr. soch., T. 2, AN SSSR, M., L., 1956, 272–331

[5] Bibikov Yu. N., Mnogochastotnye nelineinye kolebaniya i ikh bifurkatsii, LGU, L., 1991 | MR

[6] Hale J. K., “Integral manifolds of perturbed differential systems”, Ann. Math., 73:3 (1961), 496–531 | DOI | MR | Zbl