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where A, > 0 for all If we assume for a moment that H3 and all the higher-order terms vanish identically, we can define the normal modes via the invariant eigenspaces, on which the flow is circular. If we fix an energy surface H2(x, y) = h, we immediately obtain n circular solutions, each of energy h. These are the normal modes, and their frequencies of vibration are the fundamental frequencies. If the higher-order terms do not vanish, the construction above can no longer be performed. In the case of low energy and positive definite H2(x, y), Weinstein [23] has proven the existence of periodic solutions continuing the modes. In the case of arbitrary energy the only results available seem to be the ones of Seifert [21], Rosenberg [14, 17] and Gordon [7], but the solutions so obtained may or may not be "modes" in any recognizable sense. Following Rosenberg [10, 12], we shall adopt a definition of "normal mode" as being that of a solution (jc,(r), x2{t), • • •, x„(t), y,(t), • ■ -, yM) of

Normal modes, uncoupling, and stability for a class of nonlinear oscillators