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In 2000, Bolotin and Treshchev proposed an invariant definition of the hyperbolic torus, generalizing the traditional coordinate definition. Simultaneously, they conjectured that, under standard assumptions on its Diophantine properties, nondegeneracy, and analyticity, the hyperbolic torus is conserved in the case of small perturbations. This conjecture generalizes Graff’s theorem. In the present...
We consider the stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom whose unperturbed part describes oscillators with restoring force of odd order greater than 1. It is proved that if the exponents of the restoring force of the oscillators are not equal, then the equilibrium position is Lyapunov stable. If the exponents are equal,...
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