Chaotification of Discrete Dynamical Systems in Banach Spaces

This paper is concerned with chaotification of discrete dynamical systems in Banach spaces via feedback control techniques. A criterion of chaos in Banach spaces is first established. This criterion extends and improves the Marotto theorem. Discussions are carried out in general and some special Banach spaces. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. As a consequence, a controlled system described in a finitedimensional real space studied by Wang and Chen is shown chaotic not only in the sense of Li–Yorke but also in the sense of Devaney. The original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control in a certain space. In addition, the Chen–Lai anti-control algorithm via feedback control with mod-operation in a finite-dimensional real space is extended to a certain infinite-dimensional Banach space, and the controlled system is shown chaotic in the sense of Devaney as well as in the sense of both Li–Yorke and Wiggins. Differing from many existing results, it is not here required that the map corresponding to the original system has a fixed point in some cases. An application of the theoretical results to a class of first-order partial difference equations is given with some numerical simulations.