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⊂R2→R is a map.

Equation (1) is a discretization of the partial differential equation
(2)wt(t,s)=f~(w(t,s),ws(t,s)),
where t≥0 is time variable, s is spatial variable, and f~:D~⊂R2→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 R3 by constructing spatial periodic orbits in 2003 [3]. Chen et al. [4] reformulated (1) to a discrete system:
(3)xn+1=h(xn),n≥0.
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):
(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)xn=(x(n,0),x(n,1),…,x(n,k))T∈Rk+1,n≥0;
then (1) with (4) can be rewritten in the following form:
(7)xn+1=F(xn),n≥0,
where
(8)F(xn)=((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)liminfn→∞d(Fn(x),Fn(y))=0,limsupn→∞d(Fn(x),Fn(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

Fis 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 [11].

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

A point x∈Rk+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=0k 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⊂Rk+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 [7]; we list them as follows. For convenience, let Ck(U,Rn) be the set of all the maps f:U⊂Rn→Rn that are k times continuously differentiable in U.

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

Consider the controlled system
(11)xn+1=f(xn)+g(μxn),n≥0,
in Yk(k≤∞). Assume that

x*=0 is a fixed point of f and there exist positive constants r and L such that f∈C0([-r,r]k,Yk), f∈C1((-r,r)k,Yk), and ∥Df(x)∥≤L for any x∈(-r,r)k;

g satisfies the following conditions:

g∈C0([-r,r]k∪[a,b]k,Yk) and g∈C1((-r,r)k∪(a,b)k,Yk) 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;

Dg(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>μ0:=max{br,Lr+bNr,LbN(∥ξ∥0-a),LbN(b-∥ξ∥)},
where ∥ξ∥0=min{|ξi|: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 Fμn:Λ→Λ is topologically conjugate to the symbolic dynamical system σ:Σ2+→Σ2+, where Fμ(x)=f(x)+g(μx). 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 [6].

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

Consider the controlled system
(14)xn+1=f(xn)+μg(xn),n≥0,xn∈Yk,
where k≤∞. Assume that

assumption (i) in Lemma 5 holds;

g satisfies the following conditions:

g∈C0([-a,a]k∪[b,r]k,Yk) and g∈C1((-a,a)k∪(b,r)k,Yk) 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;

Dg(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>μ0:=max{La+rNa,LrN(∥ξ∥0-b),LrN(r-∥ξ∥)},
all the results in Lemma 5 hold for Fμ(x)=f(x)+μg(x) therein.3. Chaotification Problems for (<xref ref-type="disp-formula" rid="EEq1">1</xref>)

Assume that f∈C1([-r,r]2,R) for r>0. Let fx(x,y) and fy(x,y) be the first-order partial derivatives of f for the 1st and the 2nd variables at (x,y). Let
(16)L:=max{|fx(x,y)|+|fy(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(μx(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∈C1([-r,r]2,R) and f(0,0)=0;

g∈C1([-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∈C1([-r,r,-r,r]) and φ(0)=0.

Then, for
(18)μ>μ0:=max{br,Mr+bNr,MbN(ξ-a),MbN(b-ξ)}
and for any neighborhood U of x=0, there exist a Cantor set Λ⊂Uk+1 and a perfect as well as compact invariant set E⊂Rk+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,|fx(x,y)|+|fy(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>μ0 in the proof. System (17) with (4) can be rewritten as
(19)xn+1=F(xn)+G(μxn),n≥0,
where F is defined by (8), and
(20)G(xn)=(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∈Rk+1, and F∈C1([-r,r]k+1,Rk+1). Further, for any x={x(j)}i=0k∈[-r,r]k+1,(21)DF(x)=(fx(α(0))fy(α(0))0⋯00fx(α(1))fy(α(1))⋯000fx(α(2))⋯0⋯⋯⋯⋯⋯fy(α(k))φ′(x(0))00⋯fx(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=0k∈Rk+1,
(22)DF(x)z=(fx(α(k))ω(k)+fy(α(k))φ′(x(0))ω(0))fx(α(0))ω(0)+fy(α(0))ω(1),fDF(x)z=ffx(α(1))ω(1)+fy(α(1))ω(2),…,fDF(x)z=ffx(α(k))ω(k)+fy(α(k))φ′(x(0))ω(0))T.
Therefore,
(23)∥DF(x)∥=max{∥DF(x)ω∥:ω∈Rk+1,∥ω∥=1}≤max{|fx(α(k))|+|fy(α(k))φ′(x(0))|}|fx(α(j))|+|fy(α(j))|,0≤j≤k-1,dddddfdd|fx(α(k))|+|fy(α(k))φ′(x(0))|}≤max{L,|fx(α(k))|+|fy(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∈C1([-r,r]k+1∪[a,b]k+1,Rk+1) and
(24)DG(x)=(g′(x(0))0⋯00g′(x(1))⋯0⋯⋯⋯⋯00⋯g′(x(k))).
Obviously, DG(x) is an invertible map, and it follows from condition (ii) that its inverse is
(25)(DG(x))-1=((g′(x(0)))-10⋯00(g′(x(1)))-1⋯0⋯⋯⋯⋯00⋯(g′(x(k)))-1).
Hence, for any x∈[-r,r]k+1∪[a,b]k+1, one can obtain that
(26)∥(DG(x))-1∥≤1N.
Therefore, DG(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∈C1([-r,r]2,R);

g∈C1([-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∈C1([-r,r,-r,r]) and φ(0)=0.

Then, for
(28)μ>μ0:=max{Ma+rNa,MrN(ξ-b),MrN(r-ξ)}
and for any neighborhood U of x=0, there exist a Cantor set Λ⊂Uk+1 and a perfect and compact invariant set E⊂Rk+1 containing Λ such that
(29)x(n+1,m)=f(x(n,m),x(n,m+1))+μg(x(n,m))0000000000000000000000000000n≥0,0≤m≤k,
with (4) being chaotic on E in the sense of both Li-Yorke and Devaney, where M=max{L,|fx(x,y)|+|fy(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)xn+1=F(xn)+μG(xn),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 [12]):
(31)u(n+1,m)=αu(n,m-1)+βu(n,m)u(n+1,m)=+γu(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)=112x+112y,φ(x)=x2,g(x)={2x,x∈[-13,13],x-45,x∈[12,1],13cosx,otherwise.
By Corollary 5.1 [6], the original system
(33)x(n+1,m)=112x(n,m)+112x(n,m+1),0000000000000000000000n≥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>μ0=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).

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