Geodesic
Bifurcation of the Equilibrium Point in the Critical Case of Two Pairs of Zero Characteristic Roots
Informatics and Automation, Differential equations and dynamical systems, Tome 236 (2002), pp. 45-60

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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 $2n+1$. 
@article{TRSPY_2002_236_a5,
     author = {V. V. Basov},
     title = {Bifurcation of the {Equilibrium} {Point} in the {Critical} {Case} of {Two} {Pairs} of {Zero} {Characteristic} {Roots}},
     journal = {Informatics and Automation},
     pages = {45--60},
     publisher = {mathdoc},
     volume = {236},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2002_236_a5/}
}
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V. V. Basov. Bifurcation of the Equilibrium Point in the Critical Case of Two Pairs of Zero Characteristic Roots. Informatics and Automation, Differential equations and dynamical systems, Tome 236 (2002), pp. 45-60. http://geodesic.mathdoc.fr/item/TRSPY_2002_236_a5/