On the search for unstable periodic solutions of nonlinear dynamical systems (Q2730700)

The authors consider nonlinear autonomous dynamical systems NEWLINE\[NEWLINE \frac{dx}{dt}=f(x), \quad t \geq 0, \quad x \in R^n, \tag{1}NEWLINE\]NEWLINE where \(f(x)\) is a twice differentiable function. They seek the periodic solution of the system (1) with conditions NEWLINE\[NEWLINE x(0)=x(T)=\xi, \tag{2}NEWLINE\]NEWLINE where the starting point \(\xi,\) the period \(T\) and the periodic orbit \(x(t)\) are unknown. The solution of the problem (1)-(2) is reduced to the minimization problem NEWLINE\[NEWLINE \frac{1}{2}\int\limits_0^1 \left|\frac{1}{T} z'_{\tau}(\tau)- f(z(\tau),\xi)\right|^2 d\tau \to \min. \tag{3}NEWLINE\]NEWLINE Numerical methods of solution of the problem (3) are investigated.