Abstract
Chaotification problems of partial difference equations are studied. Two chaotification schemes are established by utilizing the snap-back repeller theory of general discrete dynamical systems, and all the systems are proved to be chaotic in the sense of both Li-Yorke and Devaney. An example is provided to illustrate the theoretical results with computer simulations.
1. Introduction
Consider the following first-order partial difference equation: where is time step, is the lattice point with , and is a map.
Equation (1) is a discretization of the partial differential equation where is time variable, is spatial variable, and is a map. Equation (1) often appears in imaging and spatial dynamical systems and so forth [1, 2]. Chen and Liu studied the chaos for (1) in by constructing spatial periodic orbits in 2003 [3]. Chen et al. [4] reformulated (1) to a discrete system: Applying this approach, the second author of the present paper gave several criteria of chaos for (1) [5]. 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]. However, only a few papers study the chaotification problems of (1) except for [6–8]. In this paper, the chaotification of (1) is studied.
This paper is organized as follows. First, (1) is reformulated to a discrete system, and several concepts and lemmas are listed. Then, we give two chaotification schemes for (1) via controllers and prove that all the systems are chaotic in the sense of both Li-Yorke and Devaney. Finally, we give one example with computer simulation result to verify the theoretical predictions.
2. Preliminaries
Consider the following boundary condition for (1): where is a map. For the initial condition where satisfies (4), (1) has a unique solution , and it can be easily proved by iterations.
Let then (1) with (4) can be rewritten in the following form: where is said to be the induced map by and , and (7) is called the induced system by (1) with (4).
Definition 1 (see [9]). Let be a metric space and let be a map. A subset of is called a scrambled set of if for any two different points , The map is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set of .
Definition 2 (see [10]). A map is said to be chaotic on in the sense of Devaney if (i)is topologically transitive in ;(ii)the periodic points of in are dense in ;(iii) has sensitive dependence on initial conditions in .
Chaos of Devaney is stronger than that of Li-Yorke in some conditions [11].
Definition 3 (see [6]). A point is called a fixed point of (1) with (4) if ; that is, it is a fixed point of its induced system (7).
It follows from Definition 3 that is a fixed point of (1) with (4) if and only if it satisfies
Definition 4 (see [6]). Equation (1) with (4) is said to be chaotic in the sense of Li-Yorke (or Devaney) on if its induced system (7) is chaotic in the sense of Li-Yorke (or Devaney) on .
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 .
Lemma 5 (see [7]). Consider the controlled system
in . Assume that (i) is a fixed point of and there exist positive constants and such that , , and for any ;(ii) satisfies the following conditions:(iia) and with ;(iib) is a fixed point of and there exists a point such that ;(iic) is an invertible linear operator for each and there exists a positive constant such that for any ,
Then, for any constant satisfying
where , and for any neighborhood of , there exist a positive integer and a Cantor set such that is topologically conjugate to the symbolic dynamical system , where . Consequently, there exists a compact and perfect invariant set containing a Cantor set such that the controlled system is chaotic on in the sense of both Devaney and Li-Yorke.
A map is said to be an invertible linear map if it is a bounded linear map and bijective and if it has a bounded linear inverse map [6].
Lemma 6 (see [7]). Consider the controlled system where . Assume that (i)assumption (i) in Lemma 5 holds;(ii) satisfies the following conditions: (iia) and with ;(iib) is a fixed point of and there exists a point such that ;(iic) is an invertible linear operator for each and there exists a positive constant such that (12) holds for any . Then, for each constant satisfying all the results in Lemma 5 hold for therein.
3. Chaotification Problems for (1)
Assume that for . Let and be the first-order partial derivatives of for the 1st and the 2nd variables at . Let
Theorem 7. Consider the following system: with (4), where is a map and is a constant. Assume that (i) and ;(ii), and for any , where ;(iii) and there is a point satisfying ;(iv) and . Then, for and for any neighborhood of , there exist a Cantor set and a perfect as well as compact invariant set containing such that system (17) with (4) is chaotic on in the sense of both Li-Yorke and Devaney, where , is given in (16), and .
Proof. Assume that in the proof. System (17) with (4) can be rewritten as
where is defined by (8), and
By assumptions (i), (iv), and Definition 3, is a fixed point of (1) with (4), and then , for , and . Further, for any ,where for with . So, for ,
Therefore,
Now, we prove that satisfies condition (ii) in Lemma 5. By (iii), , where . Furthermore, it follows from condition (ii) that and
Obviously, is an invertible map, and it follows from condition (ii) that its inverse is
Hence, for any , one can obtain that
Therefore, is an invertible linear map. Hence,
In summary, both and meet all the conditions in Lemma 5. So this theorem holds.
Theorem 8. Assume that (1) and ;(2), for , and , for any ;(3) and there is a point satisfying ;(4) and . Then, for and for any neighborhood of , there exist a Cantor set and a perfect and compact invariant set containing such that with (4) being chaotic on in the sense of both Li-Yorke and Devaney, where ; is defined by (16), and .
Proof. The system induced by system (29) is where and are defined by (8) and (20), respectively. Similar to the proof of Theorem 7, it can be proved that and meet all the conditions of Lemma 6. Hence, Theorem 8 holds by Lemma 6.
4. An Example
Consider the controlled system (29) with (4), which is a special case of the discrete heat equation (see in [12]): where denotes the temperature at time and position of the rod. In system (29), By Corollary 5.1 [6], the original system is stable near the origin (see Figure 1(a)). In addition, , , and satisfy all the conditions of Theorem 8 with , , , , , , . Therefore, it follows from Theorem 8 that system (29) with (4) is chaotic in the sense of both Li-Yorke and Devaney for .
(a)
(b)
We take , for computer simulation. The simulation result is shown in Figure 1(b), which indicates that system (29) with (4) has a dense orbit around the origin and then has very complicated dynamical behaviors near the origin.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the RFDP of Higher Education of China (Grant 20100131110024), the NSFC (Grants 11126120, 11101246), the NNSF of Shandong Province (Grant ZR2011AM002), and the RFDP of Henan Polytechnic University (Grant B2011-032).