  Two classes of two-dimensional time-periodic systems of ordinary differential
equations with a small parameter e in the perturbed part, which is continuous
and, for $\varepsilon=0$, analytic in zero, are studied. Depending on the
presence or absence of the common factor $\varepsilon$, these classes contain
the system with "fast" or "slow" time. The unperturbed part of these systems is
generated by the hamiltonian $H=2x^2-x^4+\gamma(2y^2-y^4)$ $(\gamma\in(0,1])$.
The universal method, meaning it can be applied to any hamiltonian, called the
method of the generating tori splitting (GTS method), is developed and applied
to the research of such systems. For arbitrary system of any class, this method
allows to find the sets of the initial values for the solutions of the
corresponding unperturbed system, and for each such set, to provide the
explicit conditions on the system perturbations independent of the parameter.
Any such set, that satisfies the aforementioned conditions, determines the
solution of the unperturbed system, which parametrizes the generating cycle.
The obtained cycle is a generatrix of an invariant cylindrical surface. It is
proven that the system has two-periodic invariant surface, homeomorphic to
torus, if time is factored with the respect to the period, in the small with
the respect to e neighbourhood of this surface. An example of the set of
systems with eleven invariant tori and a perturbation, which average value is a
three-term polynomial of the third degree, is constructed as a demonstration of
the practical use of the GTS method. The GTS method is an universal alternative
to the so-called method of detection functions and the Melnikov function
method, which are used in studies concerning the weakened XVI Hilbert's problem
on the evaluation of a number of limit cycles of autonomous systems with the
hamiltonian unperturbed part.
