Preliminary Remarks. Auxiliary Statements

We want to find an approximate periodic solution of the system (1.1) in the form of a trigonometric polynomial 

The coefficients of this polynomial are determined from the system of algebraic equations 

The trigonometric polynomial x (r) satisfying (1.4) is called Bubnov-Galerkin s approximation of the mth order. The system (1.4) can be rewritten as follows 

We now prove the theorem which establishes the criterion on the solubility of the system of equations (1.5) 

Theorem 2.1. (Urabe, 1966). Consider a real system of equations 

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