Estimations of domains with cycles
The author derives estimates for the locus of possible cycles of \(n\)-dimensional nonlinear autonomous systems (1) \(\dot x=f(x)\). This is accomplished by means of some very useful theorems, such as the following one: For any function \(\psi\in C^\infty(\mathbb{R}^n)\) denote by \(S_\psi\) the set of zeros of the scalar product \(f\cdot\text{grad }\psi\), and let \(\psi|_{S_\psi}\) be the restriction of \(\psi\) to \(S_\psi\). Then any cycle of (1) is contained in the set \(\{x\in\mathbb{R}^n:\inf(\psi|_{S_\psi})\leq \psi(x)\leq \sup(\psi|_{S_\psi})\}\). For applicability of such results which are generalizations of the classical direct method of Lyapunov, it is important to have some knowledge about the boundedness properties of the sets \(S_\psi\) and \(\{x\in\mathbb{R}^n:\psi(x)=\text{const}\}\); corresponding criteria are presented in case \(f\) and \(\psi\) are of polynomial type. The results are then applied to the Lorenz system and the Rössler system, for which suitable quadratic polynomials \(\psi\) are constructed. In this way, effective estimates for the position of possible cycles of the Lorenz system with arbitrary parameter are finally established.
\(n\)-dimensional nonlinear autonomous systems
scientific article; zbMATH DE number 1107127