Normal Form and Stability of the Zero Solution to a Second-Order Periodic Reversible ODE with a Small Parameter

The study of the problem of the stability of the zero solution of the reversible differential equation $$\ddot {x} + {{\beta }^{2}}{{x}^{{2n - 1}}} + \varepsilon b(t){{x}^{{n - 1}}}\dot {x} + X(t,x,\dot {x}) = 0$$ has been continued under the following assumptions: n is a natural number; ε ≥ 0 is a small parameter; b(t) is an odd 2π-periodic function; the expansion of convergent series X in powers of x, $$\dot {x}$$ with continuous 2π-periodic coefficients does not include terms of orders below 2n, if the variable x has a degree 1 while the variable $$\dot {x}$$ has a degree n. Perturbation X is not changed when time is reversed (in sign). It is well-known that all the terms of series X need to be taken into account in order to solve the problem of the stability of such perturbations. A.M. Lyapunov called such cases transcendental, unlike the algebraic ones where only a finite number of terms need to be considered. In 2022, in analyzing the case where n ≥ 2 (β = 1), it was established that unperturbed motion x ≡ 0 is Lyapunov stable for a sufficiently small ε which determines the amplitude of oscillations of periodic function b(t). In this work, the case is studied where n = 1 while β is irrational (this condition may be weakened). As for n ≥ 2, this problem is solved using the methods of the Kolmogorov–Arnold–Moser (KAM) theory modified for the analysis of reversible systems. For small values of the parameter, according to this theory, in any neighborhood of the origin, there exist two-periodic invariant tori which divide the three-dimensional configuration space and overlap the time axis, which results in the nonasymptotical stability of the unperturbed motion. The method used in the proof of the stability for n = 1 has a number of significant differences, however, besides unnormalized β. Parameter ε is introduced through the addition of equation $$\dot {\varepsilon }$$ = 0 as one more variable of the same order as x, $$\dot {x}$$ ; afterwards, the obtained system is reduced to the normal form with purely imaginary constant coefficients. It turns out to be convenient to carry out the normalization process in the general form, taking into account that X = X(t, x, $$\dot {x}$$ , $$\dot {\varepsilon }$$ ) and the first term of the expansion of this series has an order which is no lower than the second order; meanwhile, the obtained formal normal form is of independent interest. In this work, the stability is proved in the nondegenerate case where the first coefficient (which is at the term not containing ε in the normal form) is nonzero (its value has been found).