The shape of the basin of attraction and transients via transformation to normal form

A method is derived to approximate the basin and investigate the transients of an attracting fixed point with complex multipliers for an analytic invertible mapping of the real plane. It consists of constructing approximate but explicit expressions for the normal transformation about the fixed point. This transformation has the whole basin as domain, and transforms the nonlinear mapping restricted to the basin, to a linear one. With this transformation a non-negative functional for the basin is defined whose value decreases along an orbit. Explicit expressions are derived for a sequence of contours approximating a level line of this functional. At large values of the functional a level line approximates the basin boundary. Calculation of a sequence of level lines yields estimates for that part of the basin where regular (monotonic) convergence to the attracting fixed point is to be expected. The shape of the level lines demonstrates the occurrence of transient-periodic behavior. The way in which the unstable manifold of the saddle, born together with the attractor from a tangent bifurcation, intersects the level lines yields a criterion for a saddle-sink connection. In that case the stable manifold as part of the basin boundary has a simple structure, in contrast to the fractal structure that occurs when there is a homoclinic orbit, which causes final state sensitivity. The calculations are carried out for the Hénon mapping.