This paper is concerned with anti-control (or, chaotification) of chaos for discrete dynamical systems in general and some special Banach spaces, via feedback control techniques. The controlled systems are proved to be chaotic in the sense of both Devaney and Li-Yorke. The original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control in certain Banac...

It is well known that most conventional control methods and many special techniques can be used for chaos control [Chen & Dong, 1998] whether the purpose is to reduce “bad” chaos or to introduce “good” chaos. Numerous control methodologies have been proposed, developed, tested and applied. Due to its great potential in nontraditional applications such as those found within the context of physic...

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 ...

In this paper Li–Yorke chaotic sets generated by shift and weighted shift maps are studied. The characterization of Li–Yorke chaotic sets by p-scrambled sets, maximal scrambled sets and orbit invariants are proved for the general shift maps. For weighted shift maps on infinite-dimensional spaces, the necessary and sufficient conditions for Li–Yorke chaotic are proved both in the abstract sequen...

Recent progress in symbolic dynamics of cellular automata (CA) shows that many CA exhibit rich and complicated Bernoulli-shift properties, such as positive topological entropy, topological transitivity and even mixing. Noticeably, some CA are only transitive, but not mixing on their subsystems. Yet, for one-dimensional CA, this paper proves that not only the shift transitivity guarantees the CA...

This paper is devoted to an in-depth study of cellular automaton rule 180 under the framework of symbolic dynamics. Rule 180, a member of Wolfram’s class IV and Chua’s hyper Bernoulli shift rules, defines infinite number of generalized sub-shifts. An effective method of constructing the shift invariant sets of the rule’s global map is proposed. It is noted that this method is also applicable to...

In the past decade, there have been some studies about control methods with the purpose of either reducing “bad” chaos or introducing “good” chaos [Chen & Dong, 1998]. Due to its great potential in nontraditional applications such as those found within the context of physical, chemical, mechanical, electrical, optical, and particularly, biological and medical systems [Chen et al., 2001; Schiff ...

Using the Devaney chaos theory, designed a compound chaotic mapping as the encryption function, and had proven that the track was non-periodic for the initial value of a finite decimal. We have designed an encryption algorithm successively according to this, and realized the encryption for sound documents. Afterward the simulation experiment has been set on. We realized the key sensitivity anal...

There is one-to-one correspondence between quadratic operators (mapping $${\mathbb {R}}^m$$ to itself) and cubic matrices. It known that any operator corresponding a stochastic (in fixed sense) matrix preserves the standard simplex. In this paper we find conditions on (non-stochastic) ensuring Moreover, construct several non-stochastic which generate chaotic dynamical systems These behaviors ar...