3D quadratic ODE systems with hidden oscillations

A Brief theory of MRF

In this appendix we give a brief information on the MRF theory from [11].

Consider the ODE system

The MRF of system (A.1) is F ⁢ ( t , x ) := φ ⁢ ( - t ; t , x ) , where x = φ ⁢ ( t ; t 0 , x 0 ) is the general solution in the Cauchy form of system (A.1). Although the MRF is formally defined by the general solution of the system under consideration, the MRF can sometimes be found even for systems that are not integrable by quadratures.

If some function F ⁢ ( t , x ) is continuously differentiable and satisfies the condition F ⁢ ( - t , F ⁢ ( t , x ) ) ≡ F ⁢ ( 0 , x ) ≡ x , then it is an MRF of the set of systems in the form

where S ⁢ ( t , x ) is an arbitrary vector-function for which the solutions of system (A.2) are uniquely determined by the initial conditions.

Moreover, all systems from this set have the same translation operator (see [5]) on any interval ( - β , β ) . Consequently, all 2 ⁢ ω -periodic systems from this set have the same mapping over the period [ - ω , ω ] (Poincaré map).

For S ⁢ ( t , x ) ≡ 0 , system (A.2) is called simple [10]. Note that any autonomous system is simple, the opposite assertion is not true (see [10, 15]).

Let the 2 ⁢ ω -periodic system (A.1) and the system

have the same MRF F ⁢ ( t , x ) . Then if the solution φ ⁢ ( t ; - ω , x ) of system (A.1) and the solution ψ ⁢ ( t ; - ω , x ) of system (A.3) are extendible to the interval [ - ω , ω ] , then the mapping over period [ - ω , ω ] for system (A.1) is φ ⁢ ( ω ; - ω , x ) ≡ F ⁢ ( - ω , x ) ≡ ψ ⁢ ( ω ; - ω , x ) , although system (A.3) may be non-periodic. That is, there is a one-to-one correspondence between 2 ⁢ ω -periodic solutions of system (A.1) and solutions of the two-point boundary value problem y ⁢ ( - ω ) = y ⁢ ( ω ) for system (A.3).