Generalized iteration, catastrophes and generalized Sharkovsky's ordering

Abstract

We define iteration of functions that map n-dimensional vector spaces into m-dimensional vector spaces (m at most equal to n). It happens that usual iteration and Fibonacci iterative methods become special cases of this generalized iteration. Mathematical objects such as orbits, bifurcations, chaos, Feigenbaum constant, (generalized) Sharkovsky ordering, (generalized) Julia and Mandelbrot sets and a new kind of catastrophe can be found and studied in this enlarged context.

10 pages

References in corpus (1)

• A collection of simple proofs of Sharkovsky's theorem

Cited by in corpus (3)

• General definitions of chaos for continuous and discrete-time processes
• About Stable Periodic Helixes, L-iteration and Chaos Generated by Unbounded Functions
• Bifurcations, Schwarzian derivatives and Feigenbaum constant revisited