Weakened XVI Hilbert's Problem: Invariant Tori of the Periodic Systems with the Nine Equilibrium Points in Hamiltonian Unperturbed Part

TL;DR

This paper introduces the GTS method for analyzing invariant tori in perturbed Hamiltonian systems, demonstrating its effectiveness in identifying invariant surfaces and limit cycles, and providing explicit conditions for system perturbations.

Contribution

The paper develops the universal GTS method for studying invariant tori in Hamiltonian systems, offering an alternative to traditional detection and Melnikov methods.

Findings

Proves existence of two-periodic invariant tori in perturbed systems.

Constructs examples with eleven invariant tori.

Demonstrates practical application of the GTS method.

Abstract

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.