Self-oscillations in second-order nonlinear dynamic systems: divergent conditions for their nonexistence (Q1881866)

The author studies self-oscillations in nonlinear dynamical systems. He establishes some criteria of Bendixson's type for the nonexistence of self-oscillations in \(\mathbb{R}^2\). A typical result of the paper is given in the next theorem: Asymptotical orbitally stable limit cycles corresponding to self-oscillations do not exist in a domain \(Q\subseteq G\subseteq\mathbb{R}^2\) if: (1) \text{div}\(f(x)\geq0\) (or \(\equiv0\)) in a 1-connected or disconnected domain \(Q\) or (2) \text{div}\(f(x)\leq0\) (\(\not\equiv0\)) in a 1-connected domain \(Q\). Some examples demonstrating impossibility to extend these results to \(\mathbb{R}^n\) for \(n\geq3\) are given.