JDM Journal of Discrete Mathematics 2090-9845 2090-9837 Hindawi Publishing Corporation 538423 10.1155/2014/538423 538423 Research Article Chaotification for Partial Difference Equations via Controllers http://orcid.org/0000-0002-6140-993X Liang Wei 1 Shi Yuming 2 Li Zongcheng 3 Zhou Zhan 1 School of Mathematics and Information Science Henan Polytechnic University Jiaozuo, Henan 454000 China hpu.edu.cn 2 Department of Mathematics Shandong University, Jinan Shandong 250100 China sdu.edu.cn 3 Department of Mathematics Shandong Jianzhu University Jinan, Shandong 250100 China sdjzu.edu.cn 2014 1332014 2014 01 12 2013 22 01 2014 13 3 2014 2014 Copyright © 2014 Wei Liang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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: (1) x ( n + 1 , m ) = f ( x ( n , m ) , x ( n , m + 1 ) ) , where n 0 is time step, m is the lattice point with 0 m k < + , and f : D R 2 R is a map.

Equation (1) is a discretization of the partial differential equation (2) w t ( t , s ) = f ~ ( w ( t , s ) , w s ( t , s ) ) , where t 0 is time variable, s is spatial variable, and f ~ : D ~ R 2 R 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 R 3 by constructing spatial periodic orbits in 2003 . Chen et al.  reformulated (1) to a discrete system: (3) x n + 1 = h ( x n ) , n 0 . Applying this approach, the second author of the present paper gave several criteria of chaos for (1) . 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 . However, only a few papers study the chaotification problems of (1) except for . 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): (4) x ( n , k + 1 ) = φ ( x ( n , 0 ) ) , n 0 , where φ : I R R is a map. For the initial condition (5) x ( 0 , m ) = ϕ ( m ) , 0 m k + 1 , where ϕ satisfies (4), (1) has a unique solution { x ( n , m ) : n 0 , 0 m k } , and it can be easily proved by iterations.

Let (6) x n = ( x ( n , 0 ) , x ( n , 1 ) , , x ( n , k ) ) T R k + 1 , n 0 ; then (1) with (4) can be rewritten in the following form: (7) x n + 1 = F ( x n ) , n 0 , where (8) F ( x n ) = ( ( x ( n , k ) , φ ( x ( n , 0 ) ) ) f ( x ( n , 0 ) , x ( n , 1 ) ) , f ( x ( n , 1 ) , x ( n , 2 ) ) , , f ( x ( n , k ) , φ ( x ( n , 0 ) ) ) ) T . F is said to be the induced map by f and φ , and (7) is called the induced system by (1) with (4).

Definition 1 (see [<xref ref-type="bibr" rid="B9">9</xref>]).

Let ( X , d ) be a metric space and let F : X X be a map. A subset S of X is called a scrambled set of F if for any two different points x , y S , (9) liminf n d ( F n ( x ) , F n ( y ) ) = 0 , limsup n d ( F n ( x ) , F n ( y ) ) > 0 . The map F is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set S of F .

Definition 2 (see [<xref ref-type="bibr" rid="B10">10</xref>]).

A map F : V X V is said to be chaotic on V in the sense of Devaney if

F    is topologically transitive in V ;

the periodic points of F in V are dense in V ;

F has sensitive dependence on initial conditions in V .

Chaos of Devaney is stronger than that of Li-Yorke in some conditions .

Definition 3 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

A point x R k + 1 is called a fixed point of (1) with (4) if F ( x ) = x ; that is, it is a fixed point of its induced system (7).

It follows from Definition 3 that x = { x ( m ) } m = 0 k is a fixed point of (1) with (4) if and only if it satisfies (10) x ( m ) = f ( x ( m ) , x ( m + 1 ) ) , 0 m k - 1 , x ( k ) = f ( x ( k ) , φ ( x ( 0 ) ) ) .

Definition 4 (see [<xref ref-type="bibr" rid="B6">6</xref>]).

Equation (1) with (4) is said to be chaotic in the sense of Li-Yorke (or Devaney) on V R k + 1 if its induced system (7) is chaotic in the sense of Li-Yorke (or Devaney) on V .

Recently, some chaotification schemes of the discrete system (3) were established in ; we list them as follows. For convenience, let C k ( U , R n ) be the set of all the maps f : U R n R n that are k times continuously differentiable in U .

Lemma 5 (see [<xref ref-type="bibr" rid="B7">7</xref>]).

Consider the controlled system (11) x n + 1 = f ( x n ) + g ( μ x n ) , n 0 , in Y k    ( k ) . Assume that

x * = 0 is a fixed point of f and there exist positive constants r and L such that f C 0 ( [ - r , r ] k , Y k ) , f C 1 ( ( - r , r ) k , Y k ) , and D f ( x ) L for any x ( - r , r ) k ;

g satisfies the following conditions:

g C 0 ( [ - r , r ] k [ a , b ] k , Y k ) and g C 1 ( ( - r , r ) k ( a , b ) k , Y k ) with r < a < b ;

x * = 0 is a fixed point of g and there exists a point ξ ( a , b ) k such that g ( ξ ) = 0 ;

D g ( x ) is an invertible linear operator for each x ( - r , r ) k ( a , b ) k and there exists a positive constant N such that for any x , y [ - r , r ] k [ a , b ] k , (12) g ( x ) - g ( y ) N x - y .

Then, for any constant μ satisfying (13) μ > μ 0 : = max { b r , L r + b N r , L b N ( ξ 0 - a ) , L b N ( b - ξ ) } , where ξ 0 = min { | ξ i | : 0 i k } , and for any neighborhood U of x * = 0 , there exist a positive integer n > 2 and a Cantor set Λ U such that F μ n : Λ Λ is topologically conjugate to the symbolic dynamical system σ : Σ 2 + Σ 2 + , where F μ ( x ) = f ( x ) + g ( μ x ) . Consequently, there exists a compact and perfect invariant set D X containing a Cantor set such that the controlled system is chaotic on D 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 .

Lemma 6 (see [<xref ref-type="bibr" rid="B7">7</xref>]).

Consider the controlled system (14) x n + 1 = f ( x n ) + μ g ( x n ) , n 0 , x n Y k , where k . Assume that

assumption (i) in Lemma 5 holds;

g satisfies the following conditions:

g C 0 ( [ - a , a ] k [ b , r ] k , Y k ) and g C 1 ( ( - a , a ) k ( b , r ) k , Y k ) with 0 < a < b < r ;

x * = 0 is a fixed point of g and there exists a point ξ ( b , r ) k such that g ( ξ ) = 0 ;

D g ( x ) is an invertible linear operator for each x ( - a , a ) k ( b , r ) k and there exists a positive constant N such that (12) holds for any x , y [ - a , a ] k [ b , r ] k .

Then, for each constant μ satisfying (15) μ > μ 0 : = max { L a + r N a , L r N ( ξ 0 - b ) , L r N ( r - ξ ) } , all the results in Lemma 5 hold for F μ ( x ) = f ( x ) + μ g ( x ) therein.

3. Chaotification Problems for (<xref ref-type="disp-formula" rid="EEq1">1</xref>)

Assume that f C 1 ( [ - r , r ] 2 , R ) for r > 0 . Let f x ( x , y ) and f y ( x , y ) be the first-order partial derivatives of f for the 1st and the 2nd variables at ( x , y ) . Let (16) L : = max { | f x ( x , y ) | + | f y ( x , y ) | : x , y [ - r , r ] } .

Theorem 7.

Consider the following system: (17) x ( n + 1 , m ) = f ( x ( n , m ) , x ( n , m + 1 ) ) x ( n + 1 , m ) = + g ( μ x ( n , m ) ) , n 0 , 0 m k , with (4), where g : R R is a map and μ > 0 is a constant. Assume that

f C 1 ( [ - r , r ] 2 , R ) and f ( 0,0 ) = 0 ;

g C 1 ( [ - r , r ] [ a , b ] , R ) , and g ( x ) 0 for any x [ - r , r ] [ a , b ] , where r < a < b ;

g ( 0 ) = 0 and there is a point ξ ( a , b ) satisfying g ( ξ ) = 0 ;

φ C 1 ( [ - r , r , - r , r ] ) and φ ( 0 ) = 0 .

Then, for (18) μ > μ 0 : = max { b r , M r + b N r , M b N ( ξ - a ) , M b N ( b - ξ ) } and for any neighborhood U of x = 0 , there exist a Cantor set Λ U k + 1 and a perfect as well as compact invariant set E R k + 1 containing Λ such that system (17) with (4) is chaotic on E in the sense of both Li-Yorke and Devaney, where M = max { L , | f x ( x , y ) | + | f y ( x , y ) φ ( x ( 0 ) ) | : x , y , x ( 0 ) [ - r , r ] } , L is given in (16), and N = min { | g ( x ) | : x [ - r , r ] [ a , b ] } .

Proof.

Assume that μ > μ 0 in the proof. System (17) with (4) can be rewritten as (19) x n + 1 = F ( x n ) + G ( μ x n ) , n 0 , where F is defined by (8), and (20) G ( x n ) = ( g ( x ( n , 0 ) ) , g ( x ( n , 1 ) ) , , g ( x ( n , k ) ) ) T .

By assumptions (i), (iv), and Definition 3, { x * ( m ) = 0 : 0 m k } is a fixed point of (1) with (4), and then F ( x * ) = x * , for x * : = 0 R k + 1 , and F C 1 ( [ - r , r ] k + 1 , R k + 1 ) . Further, for any x = { x ( j ) } i = 0 k [ - r , r ] k + 1 , (21) D F ( x ) = ( f x ( α ( 0 ) ) f y ( α ( 0 ) ) 0 0 0 f x ( α ( 1 ) ) f y ( α ( 1 ) ) 0 0 0 f x ( α ( 2 ) ) 0 f y ( α ( k ) ) φ ( x ( 0 ) ) 0 0 f x ( α ( k ) ) ) ( k + 1 ) × ( k + 1 ) , where α ( i ) = ( x ( i ) , x ( i + 1 ) ) for 0 i k with x ( k + 1 ) = φ ( x ( 0 ) ) . So, for ω = { ω ( i ) } i = 0 k R k + 1 , (22) D F ( x ) z = ( f x ( α ( k ) ) ω ( k ) + f y ( α ( k ) ) φ ( x ( 0 ) ) ω ( 0 ) ) f x ( α ( 0 ) ) ω ( 0 ) + f y ( α ( 0 ) ) ω ( 1 ) , f D F ( x ) z = f f x ( α ( 1 ) ) ω ( 1 ) + f y ( α ( 1 ) ) ω ( 2 ) , , f D F ( x ) z = f f x ( α ( k ) ) ω ( k ) + f y ( α ( k ) ) φ ( x ( 0 ) ) ω ( 0 ) ) T . Therefore, (23) D F ( x ) = max { D F ( x ) ω : ω R k + 1 , ω = 1 } max { | f x ( α ( k ) ) | + | f y ( α ( k ) ) φ ( x ( 0 ) ) | } | f x ( α ( j ) ) | + | f y ( α ( j ) ) | , 0 j k - 1 , d d d d d f d d | f x ( α ( k ) ) | + | f y ( α ( k ) ) φ ( x ( 0 ) ) | } max { L , | f x ( α ( k ) ) | + | f y ( α ( k ) ) φ ( x ( 0 ) ) | } = M .

Now, we prove that G ( x ) satisfies condition (ii) in Lemma 5. By (iii), G ( 0 ) = G ( ξ ¯ ) = 0 , where ξ ¯ : = ( ξ , ξ , , ξ k + 1 ) T ( a , b ) k + 1 . Furthermore, it follows from condition (ii) that G C 1 ( [ - r , r ] k + 1 [ a , b ] k + 1 , R k + 1 ) and (24) D G ( x ) = ( g ( x ( 0 ) ) 0 0 0 g ( x ( 1 ) ) 0 0 0 g ( x ( k ) ) ) . Obviously, D G ( x ) is an invertible map, and it follows from condition (ii) that its inverse is (25) ( D G ( x ) ) - 1 = ( ( g ( x ( 0 ) ) ) - 1 0 0 0 ( g ( x ( 1 ) ) ) - 1 0 0 0 ( g ( x ( k ) ) ) - 1 ) . Hence, for any x [ - r , r ] k + 1    [ a , b ] k + 1 , one can obtain that (26) ( D G ( x ) ) - 1 1 N . Therefore, D G ( x ) is an invertible linear map. Hence, (27) G ( x ) - G ( y ) = max { | g ( x ( i ) ) - g ( y ( i ) ) | : 0 i k } N x - y , x , y [ - r , r ] k + 1 [ a , b ] k + 1 .

In summary, both F and G meet all the conditions in Lemma 5. So this theorem holds.

Theorem 8.

Assume that

f ( 0,0 ) = 0 and f C 1 ( [ - r , r ] 2 , R ) ;

g C 1 ( [ - a , a ] [ b , r ] , R ) , for 0 < a < b < r , and g ( x ) 0 , for any x [ - a , a ] [ b , r ] ;

g ( 0 ) = 0 and there is a point ξ ( b , r ) satisfying g ( ξ ) = 0 ;

φ C 1 ( [ - r , r , - r , r ] ) and φ ( 0 ) = 0 .

Then, for (28) μ > μ 0 : = max { M a + r N a , M r N ( ξ - b ) , M r N ( r - ξ ) } and for any neighborhood U of x = 0 , there exist a Cantor set Λ U k + 1 and a perfect and compact invariant set E R k + 1 containing Λ such that (29) x ( n + 1 , m ) = f ( x ( n , m ) , x ( n , m + 1 ) ) + μ g ( x ( n , m ) ) 0000000000000000000000000000 n 0 , 0 m k , with (4) being chaotic on E in the sense of both Li-Yorke and Devaney, where M = max { L , | f x ( x , y ) | + | f y ( x , y ) φ ( x ( 0 ) ) | : x , y , x ( 0 ) [ - r , r ] } ; L is defined by (16), and N = min { | g ( x ) | : x [ - a , a ] [ b , r ] } .

Proof.

The system induced by system (29) is (30) x n + 1 = F ( x n ) + μ G ( x n ) , n 0 , where F and G are defined by (8) and (20), respectively. Similar to the proof of Theorem 7, it can be proved that F and G 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 ( 1.3 ) in ): (31) u ( n + 1 , m ) = α u ( n , m - 1 ) + β u ( n , m ) u ( n + 1 , m ) = + γ u ( n , m + 1 ) , α , β , γ R , where u ( n , m ) denotes the temperature at time n and position m of the rod. In system (29), (32) f ( x , y ) = 1 12 x + 1 12 y , φ ( x ) = x 2 , g ( x ) = { 2 x , x [ - 1 3 , 1 3 ] , x - 4 5 , x [ 1 2 , 1 ] , 1 3 cos x , otherwise . By Corollary 5.1 , the original system (33) x ( n + 1 , m ) = 1 12 x ( n , m ) + 1 12 x ( n , m + 1 ) , 0000000000000000000000 n 0 , 0 m k , is stable near the origin (see Figure 1(a)). In addition, f , g , and φ satisfy all the conditions of Theorem 8 with r = 1 , a = 1 / 3 , b = 1 / 2 , ξ = 4 / 5 , L = 1 / 6 , N = 1 , M = 1 / 4 . Therefore, it follows from Theorem 8 that system (29) with (4) is chaotic in the sense of both Li-Yorke and Devaney for μ > μ 0 = 13 / 4 .

Computer simulation results, where k = 2 , n = 0,1 , , 20000 , and the initial value is x ( 0,0 ) = 0.3 , x ( 0,1 ) = 0.1 , and x ( 0,2 ) = 0.8 . (a) Simple dynamical behaviors for the original system (33); (b) simulation results of the system (29) for μ = 4 , which shows that there is a dense orbit around the origin and then there are complex dynamical behaviors in (29).

We take k = 2 , μ = 4 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).

Gang H. Qu Zhilin Q. Z. Controlling spatiotemporal chaos in coupled map lattice systems Physical Review Letters 1994 72 1 68 71 2-s2.0-12044257605 10.1103/PhysRevLett.72.68 Willeboordse F. H. Time-delayed map as a model for open fluid flow Chaos 1992 2 3 423 426 10.1063/1.165885 MR1184486 ZBL1055.76512 Chen G. Liu S. T. On spatial periodic orbits and spatial chaos International Journal of Bifurcation and Chaos 2003 13 4 935 941 10.1142/S0218127403006935 MR1980775 ZBL1055.76512 Chen G. Tian C. Shi Y. Stability and chaos in 2-D discrete systems Chaos, Solitons & Fractals 2005 25 3 637 647 10.1016/j.chaos.2004.11.058 MR2132363 ZBL1071.37018 Shi Y. Chaos in first-order partial difference equations Journal of Difference Equations and Applications 2008 14 2 109 126 10.1080/10236190701503074 MR2382999 ZBL1144.39002 Shi Y. Yu P. Chen G. Chaotification of discrete dynamical systems in Banach spaces International Journal of Bifurcation and Chaos 2006 16 9 2615 2636 10.1142/S021812740601629X MR2273472 ZBL1185.37084 Liang W. Shi Y. Zhang C. Chaotification for a class of first-order partial difference equations International Journal of Bifurcation and Chaos 2008 18 3 717 733 10.1142/S0218127408020604 MR2415864 ZBL1147.37325 Li Z. Chaotification for linear delay difference equations Advances in Difference Equations 2013 2013, article 459 11 10.1186/1687-1847-2013-59 MR3040994 Li T. Y. Yorke J. A. Period three implies chaos The American Mathematical Monthly 1975 82 10 985 992 MR0385028 10.2307/2318254 ZBL0351.92021 Devaney R. L. An Introduction to Chaotic Dynamical Systems 1989 2nd Redwood City, Calif, USA Addison-Wesley xviii+336 Addison-Wesley Studies in Nonlinearity MR1046376 Huang W. Ye X. Devaney's chaos or 2-scattering implies Li-Yorke's chaos Topology and Its Applications 2002 117 3 259 272 10.1016/S0166-8641(01)00025-6 MR1874089 ZBL0997.54061 Cheng S. S. Partial Difference Equations 2003 3 London, UK Taylor & Francis xii+267 Advances in Discrete Mathematics and Applications 10.1201/9781420023688 MR2193620 ZBL0997.54061