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Special exotic class of dynamical systems~ -- the implicit maps~ -- isconsidered. Such maps, particularly, can appear as a result of using ofimplicit and semi-implicit iterative numerical methods. In the present work wepropose the generalization of the well-known Newton-Cayley problem. NewtonianJulia set is a fractal boundary on the complex plane, which divides areas ofconvergence to different roots of cubic nonlinear complex equation when it issolved with explicit Newton method. We consider similar problem for therelaxed, or damped, Newton method, and obtain the implicit map, which isnon-invertible both time-forward and time-backward. It is also possible toobtain the same map in the process of solving of certain nonlinear differentialequation via semi-implicit Euler method. The nontrivial phenomena, appearing insuch implicit maps, can be considered, however, not only as numericalartifacts, but also independently. From the point of view of theoreticalnonlinear dynamics they seem to be very interesting object for investigation.Earlier it was shown that implicit maps can combine properties of dissipativenon-invertible and Hamiltonian systems. In the present paper strange invariantsets and mixed dynamics of the obtained implicit map are analyzed.

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