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Attracting orbits in Newton’s method. (English) Zbl 0632.65059
In recent years several authors have studied the discrete dynamical system $x\sb{n+1}=(NF)(x\sb n)=x-f(x\sb n)/f'(x\sb n),$ generated by Newton’s method applied to a given function $F: R\to R$. This paper considers a parameter dependent family of smooth functions $F\sb{\mu}(x)=g(x)+\mu$ and addresses the question how the dynamics of $Nf\sb{\mu}$ changes when $\mu$ varies. In particular, if at $\mu =\mu\sb 0$ the number of real roots of f changes, then under certain non- degeneracy conditions it is shown that there are values $\mu$ near $\mu\sb 0$ for which $Nf\sb{\mu}$ has nontrivial periodic attractors.
Reviewer: W.C.Rheinboldt

##### MSC:
 65H10 Systems of nonlinear equations (numerical methods) 65L05 Initial value problems for ODE (numerical methods) 37-99 Dynamic systems and ergodic theory (MSC2000) 37C70 Attractors and repellers, topological structure
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