The reflection function of the differential system \[ \frac{dx}{dt}= X(t,x), t\in \mathbb{R},x\in D\subset \mathbb{R}^{n}, \tag{*} \] whose right-hand side is continuously differentiable and \( D \) is a domain in \(\mathbb{R}^{n}\) , is a function \( F(t,x) \) that has the following properties: for each solution \( x(t) \) of (*), is defined on an interval \( (-\alpha,\alpha) \) and is symmetric with respect to zero, \( F(t,x(t))\equiv x(-t) \) for all \( t\in (-\alpha,\alpha), \) and \( F(0,x)\equiv x. \)
The notion of reflecting function of a differential system first appeared in the paper by V. I. Mironenko [Differ. Uravn. 20, No. 9, 1635–1638 (1984; Zbl 0568.34029)]. Subsequently, the theory of reflecting function was studied in the monograph by V. I. Mironenko [Отражающая функция и периодические решения дифференциал’ных уравнений (Russian). Minsk: Izdatel’stvo “Universitetskoe” (1986; Zbl 0607.34038)] and Z. Zhou [The theory of reflecting function of differential equations and applications. Beijing (2014)].
A first integral of (*) is generally defined as a differentiable function \( U(t,x) \) that is constant on each solution \( x(t) \) of (*). Any differentiable function \( U(t,x) \) that is not identically constant is called a generaized first integral of (*) if the function \( U(t,x(t)) \) is even for any solution \( x(t) \) of (*) and is defined on a zero-symmetric interval \( (-\alpha, \alpha) \).
In this paper, the authors trace the relationship between the notion of generalized first integral and the notion of reflecting function and Poincaré map (periodic map) for periodic differential systems. The notion of generalized first integral is used to study questions of the existence and stability of periodic solutions of periodic differential systems. Here the centre focus problem is also analyzed.