Chaotification for a Class of First-Order Partial Difference Equations

Abstract

This paper is concerned with chaotification for a class of first-order partial difference equations, in which the system size is finite or infinite. Nine new chaotification schemes for the class of first-order partial difference equations with general controllers, mod-operation, and sawtooth functions are established, respectively. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. In addition, five new chaotification schemes for general discrete dynamical systems in finite-dimensional real spaces and l∞are established. Two illustrative examples are provided with computer simulations.

Existence of chaos for partial difference equations via tangent and cotangent functions

“…By (9) and (10) G ε is continuous in [-r, r] k+1 . By the intermediate value theorem and (11)- (14) we have…”

“…Recently, chaotification problems for Eq. (1) with general controllers, sawtooth functions, and mod operations were studied, respectively, and all the controlled systems were proved to be chaotic in the sense of both Devaney and Li-Yorke [7][8][9]. In [10], two chaotification schemes of Eq.…”

Chaotification for Partial Difference Equations via Controllers

“…Recently, some chaotification schemes of the discrete system (3) were established in [7]; we list them as follows. For convenience, let ( , ) be the set of all the maps : ⊂ → that are times continuously differentiable in .…”

“…Recently, Li studied the chaotification for delay difference equations [8]. However, only a few papers study the chaotification problems of (1) except for [6][7][8]. In this paper, the chaotification of (1) is studied.…”

“…She with her coauthors established some chaotification schemes for (1) and proved all the systems are chaotic [6,7]. Recently, Li studied the chaotification for delay difference equations [8].…”