Finding all periodic orbits of maps using Newton methods: Sizes of basins
The paper is devoted to finding \(k\)-periodic orbits of a diffeomorphism \(F\) on \(\mathbb{R}^2\) by applying Newton's method to the function \(G=F^k-I\), where \(I\) is the identity function. The authors actually use some variants of Newton's method (continuous Newton and quasi-Newton methods) that are more robust than the traditional Newton's method. For a \(k\)-periodic point \(p\in \mathbb{R}^2\), the most attention is focused on the size of the Newton basin \[ B(p,k)=\{x\in \mathbb{R}^2:\text{ Newton's method starting from }x \text{ converges to} p\} \] and the manner it depends on \(p\) and \(k\). It is shown that if \(p\) is an attracting point of \(F\) then there is an open neighborhood of \(p\) that is in all Newton basins \(B(p,nk)\) for \(n=1,2,\dots\). If \(p\) is a repelling point then it is possible that \(p\) is the only point which is in all of the Newton basins \(B(p,nk)\) for all \(n\). The main result states that if \(p\) is a saddle point, then the intersection of Newton basins \(B(p,nk)\) includes a segment of the local stable manifold of \(p\). Numerical data presented in the paper indicate that the area of the basin \(B(p,k)\) of a periodic saddle point \(p\) is proportional to \(\lambda ^c\), where \(\lambda\) is the unstable eigenvalue of \(DF^k(p)\) and \(c\approx -1\). From this it follows that for long periods many periodic orbits of \(F\) have \(\lambda\) so large that the basins are numerically undetectable. The results described above are illustrated by detailed numerical analysis of the Hénon and the Ikeda mappings.
scientific article; zbMATH DE number 1398660
scientific article