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

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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 $2n+1$. 
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     author = {V. V. Basov},
     title = {Bifurcation of the {Equilibrium} {Point} in the {Critical} {Case} of {Two} {Pairs} of {Zero} {Characteristic} {Roots}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     publisher = {mathdoc},
     volume = {236},
     year = {2002},
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     url = {http://geodesic.mathdoc.fr/item/TM_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. 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/