Complex Dynamics of the Implicit Maps Derived from Iteration of Newton and Euler Method

TL;DR

This paper explores the complex dynamics of implicit maps derived from iterative numerical methods like Newton and Euler, revealing non-invertible behaviors and strange invariant sets with implications for nonlinear dynamics.

Contribution

It generalizes the Newton-Cayley problem to damped Newton and semi-implicit Euler methods, analyzing the resulting implicit maps' dynamics and properties.

Findings

Implicit maps can be non-invertible in both directions.

Strange invariant sets and mixed dynamics are observed.

Implicit maps exhibit properties of both dissipative and Hamiltonian systems.

Abstract

Special exotic class of dynamical systems~ -- the implicit maps~ -- is considered. Such maps, particularly, can appear as a result of using of implicit and semi-implicit iterative numerical methods. In the present work we propose the generalization of the well-known Newton-Cayley problem. Newtonian Julia set is a fractal boundary on the complex plane, which divides areas of convergence to different roots of cubic nonlinear complex equation when it is solved with explicit Newton method. We consider similar problem for the relaxed, or damped, Newton method, and obtain the implicit map, which is non-invertible both time-forward and time-backward. It is also possible to obtain the same map in the process of solving of certain nonlinear differential equation via semi-implicit Euler method. The nontrivial phenomena, appearing in such implicit maps, can be considered, however, not only as numerical artifacts, but also independently. From the point of view of theoretical nonlinear dynamics they seem to be very interesting object for investigation. Earlier it was shown that implicit maps can combine properties of dissipative non-invertible and Hamiltonian systems. In the present paper strange invariant sets and mixed dynamics of the obtained implicit map are analyzed.