Stability of differential systems
The author uses reflective functions (RF) [\textit{V. I. Mironenko}, Reflective function and periodic solution of the differential system, Minsk University Press (1986; Zbl 0607.34038)] to study properties of \(2\omega\)-periodic differential systems \[ x'= X(t,x)\tag{1} \] with the Poincaré mapping. A continuously differentiable vector function \(F(t,x)\) on \(\mathbb{R}\times\mathbb{R}^n\) is called RF if it is a solution of Cauchy problem \[ F_t(t,x)+ F_x(t,x) X(t,x)+ X(-t, F(t,x))= 0,\quad F(0,x)= x. \] The RF of the linear system \[ x'= A(t)x\tag{2} \] is also linear, i.e., \(F(t,x)= F(t)x\), where the matrix \(F(t)\) is called reflective matrix (RM). This matrix is the solution of Cauchy problem \[ F'+ FA(t)+ A(-t)F= 0,\quad F(0)= E,\tag{3} \] where \(E\) is the identity matrix, \(A(t)\) is a continuously differentiable \(n\times n\) matrix function on \(\mathbb{R}\), \(x= (x_1,x_2,\dots, x_n)^T\). If \(\phi(t)\) is a fundamental matrix of (2), then RM of (2) is \(F(t)= \phi(-t)\phi^{-1}(t)\), and so \(\text{det\,}F(t)= e^{-\int^t_{-t}\text{tr\,}A(\tau)\,d\tau}\). If \(F(t,x)= F(t)\) is RF of (2), then it also is the RF of the system \[ x'= A(t)x+ F^{-1}(t) S(t, x)- S(-t, F(t)x),\tag{4} \] where \(S(t,x)\) is an arbitrary continuously differentiable function on \(\mathbb{R}\times \mathbb{R}^n\). The author discusses when the RM of system (2) satisfies the relation \(F(t) F^T(t)= \alpha^2(t)E\) and applies the result to discuss the stability of (2) and (4). Here, \(\alpha(t)\) is a continuously differentiable scalar function on \(\mathbb{R}\), \(E\) is an \(n\times n\) identity matrix.
scientific article; zbMATH DE number 5077208