On the Rational Recursive Sequence x

(1)

Volume 2007, Article ID 23618,12pages doi:10.1155/2007/23618

Research Article

On the Rational Recursive Sequence x

n+1=

(A +

^k_i₌₀

α

i

x

n₋i

)/(B +

^k_i₌₀

β

_i

x

n₋i

)

E. M. E. Zayed and M. A. El-Moneam

Received 13 August 2006; Revised 22 January 2007; Accepted 22 January 2007 Recommended by Martin J. Bohner

The main objective of this paper is to study the boundedness character, the periodic character, the convergence, and the global stability of the positive solutions of the diﬀerence equationxn+1=(A+^k_i₌₀αixn−i)/(B+^k_i₌₀β_ixn−i),n=0, 1, 2,..., whereA,B,αi,β_iand the initial conditionsx₋k,...,x₋1,x0are arbitrary positive real numbers, whilekis a positive integer number.

Copyright © 2007 E. M. E. Zayed and M. A. El-Moneam. 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 prop- erly cited.

1. Introduction

Our goal in this paper is to investigate the boundedness character, the periodic character, the convergence and the global stability of the positive solutions of the diﬀerence equation

xn+1=A+^k_i₌₀αixn−i

B+^k_i₌0βixn−i, n=0, 1, 2,..., (1.1) where A,B, αi, βi and the initial conditions x₋k,...,x₋1, x0 are arbitrary positive real numbers, whilekis a positive integer number. The case where any ofA,B,αi,βiis allowed to be zero gives different special cases of (1.1) which are studied by many authors (see, e.g., [1–14]). For the related work, see [15–26]. The study of these equations is challenging and rewarding and is still in its infancy. We believe that the nonlinear rational difference equations are paramount importance in their own right. Furthermore, the results about such equations offer prototypes for the development of the basic theory of the global behavior of nonlinear difference equations. Note that the difference equation (1.1) has

(2)

been extensively studied in the special casek=1 in the monograph [14]. So, the results presented in our paper are new.

Definition 1.1. The equilibrium pointxof the diﬀerence equation (1.1) is the point that satisfies the conditionx=F(x,x,...,x). That is, the constant sequence {xn}^∞_n₌₋_kwithxn= xfor alln≥ −kis a solution of the diﬀerence equation (1.1).

Definition 1.2. Letx∈(0,∞) be an equilibrium point of the diﬀerence equation (1.1).

Then, the following hold

(i) The equilibrium pointxof the diﬀerence equation (1.1) is called locally stable if for everyε >0 there existsδ >0 such thatx−k,...,x−1,x0∈(0,∞) with|x−k− x|+···+|x₋1−x|+|x0−x|< δ, then|xn−x|< εfor alln≥ −k.

(ii) The equilibrium pointxof the diﬀerence equation (1.1) is called locally asymptotically stable if it is locally stable and if there existsγ >0 such thatx−k,...,x−1, x0∈(0,∞) with|x₋k−x|+···+|x₋1−x|+|x0−x|< γ, thenxn→xasn→ ∞. (iii) The equilibrium pointxof the diﬀerence equation (1.1) is called global attractor

if for everyx−k,...,x−1,x0∈(0,∞) one hasxn→xasn→ ∞.

(iv) The equilibrium pointxof the equation (1.1) is called globally asymptotically stable if it is locally stable and global attractor.

(v) The equilibrium pointxof the diﬀerence equation (1.1) is called unstable if it is not locally stable.

Definition 1.3. Say that the sequence{xn}^∞_n₌₋_kis bounded and persists if there exist positive constantsmandMsuch that

m≤xn≤M ∀n≥ −k. (1.2)

Definition 1.4. A sequence{xn}^∞_n₌₋_kis said to be periodic with periodpifxn+p=xnfor alln≥ −k. A sequence{xn}^∞_n₌₋_k is said to be periodic with prime period p ifp is the smallest positive integer having this property.

Assume thata=_k

i₌0αi,a=_k

i₌0(−1)ⁱαi,b=_k

i₌0βi, andb=_k

i₌0(−1)ⁱβi. Then the equilibrium pointxof the diﬀerence equation (1.1) is the solution of the equation

x=A+ax

B+bx. (1.3)

Consequently, the positive equilibrium pointxof the diﬀerence equation (1.1) is given by

x=(a−B) +(a−B)²+ 4Ab

2b . (1.4)

LetF: (0,∞)^k+1→(0,∞) be a continuous function defined by Fu0,u1,...,uk

=A+^k_i₌₀αiui

B+^k_i₌₀βiui. (1.5)

(3)

Now, we have

yn+1= k j=0

∂F(x,...,x)

∂uj yn−j, (1.6)

and then the linearized equation is yn+1+

k j=0

bjyn−j=0, (1.7)

where

bj=βjx−αj

B+bx . (1.8)

2. The main results

In this section, we establish some results which show that the positive equilibrium point xof the diﬀerence equation (1.1) is globally asymptotically stable and every positive solution of the diﬀerence equation (1.1) is bounded and has prime period two.

Theorem 2.1 (see [4,10,13,17]). Assume thata,b∈Randk∈ {0, 1, 2,...}. Then

|a|+|b|<1 (2.1)

is a suﬃcient condition for the asymptotic stability of the diﬀerence equation

xn+1+axn+bxn−k=0, n=0, 1,.... (2.2) Remark 2.2 (see [13]). Theorem 2.1can be easily extended to a general linear diﬀerence equation of the form

xn+k+p1xn+k−1+···+pkxn=0, n=0, 1, 2,..., (2.3) wherep1,p2,...,pk∈Randk∈ {1, 2,...}. Thenequation 2.3is asymptotically stable provided that

k i=1

pi<1. (2.4)

Theorem 2.3. Assume that B >a holds. Let {xn}^∞_n₌₋_k be a solution of the diﬀerence equation (1.1) such that for somen0≥0,

eitherxn≥x forn≥n0 (2.5)

orxn≤x forn≥n0. (2.6)

Then{xn}converges toxasn→ ∞, that is,

nlim→∞xn=x. (2.7)

(4)

Proof. Assume that (2.5) holds. The case where (2.6) holds is similar and will be omitted.

Then forn≥n0+k, we deduce that xn+1=A+^k_i₌₀αixn−i

B+^k_i₌0βixn−i = k i=0

αixn−i 1 + (A/^k_i₌₀αixn−i) B+^k_i₌0βixn−i

≤ k i=0

αixn−i 1 +A/ax B+bx ⁼

k i=0

αixn−i A+ax axB+bx.

(2.8)

With the aid of (1.3), the last inequality becomes xn+1≤

k i=0

αixn−i/a, (2.9)

and so

xn+1≤max

0≤i≤k

xn−i

forn≥n0+k. (2.10)

Set

yn=max

0_≤i_≤k

xn−i

forn≥n0+k. (2.11)

Then clearly

yn≥xn+1≥x forn≥n0+k. (2.12) Next, we claim that

yn+1≤yn forn≥n0+k. (2.13)

Now, we have yn+1=max

0≤i≤k

xn+1−i

=maxxn+1, max

0≤i≤k

xn−i

≤maxxn+1,yn

=yn. (2.14) From (2.12) and (2.13), it follows that the sequence{yn}is convergent and that

y=_nlim

→∞yn≥x. (2.15)

Furthermore, we get

xn+1≤A+^k_i₌₀αixn−i

B+bx ^≤

A+a y n

B+bx. (2.16)

From this and by using (2.13) we obtain, xn+i≤A+ay n+i−1

B+bx ^≤ A+ay n

B+bx fori=1,...,k+ 1. (2.17)

(5)

Then

yn+k+1= max

1_≤i≤k+1

xn+i

≤A+ay n

B+bx, (2.18)

and by lettingn→ ∞, we obtain

y≤A+a y

B+bx. (2.19)

Consequently, we obtain

y1− a B+bx

≤ A

B+bx. (2.20)

From (1.3) and (2.20), we deduce thaty≤x, and in view of (2.15), we obtainy=x. Thus,

the proof ofTheorem 2.3is completed.

Theorem 2.4. Let{xn}^∞_n₌₋_kbe a positive solution of the diﬀerence equation (1.1) andB >1.

Then there exist positive constantsmandMsuch that

m≤xn≤M, n=0, 1,.... (2.21)

Proof. From the diﬀerence equation (1.1), we have whenB >1 xn+1≤A

B+1 B

_k

i=0

αixn−i

, n=0, 1,.... (2.22)

Consider the linear diﬀerence equation yn+1=A

B+ 1 B

_k

i=0

αiyn−i

, n=0, 1,... (2.23)

with the initial conditionsyi=xi>0,i= −k,...,−1, 0. It follows by induction that

xn≤yn. (2.24)

First of all, assume thatB >a. Then we have A/(B−a) is a particular solution of (2.23) and every solution of the homogeneous equation which is associated with (2.23) tends to zero asn→ ∞. Hence

nlim→∞yn= A

B−a. (2.25)

From this and (2.24), it follows that the sequence{xn}is bounded from above by a positive constantMsay. That is,

xn≤M, n=0, 1,.... (2.26)

(6)

Set

m= A

B+bM. (2.27)

Then we have

xn+1=A+^k_i₌₀αixn−i

B+^k_i₌₀βixn−i ≥ A

B+bM ⁼m (2.28)

and consequently, we get

m≤xn≤M, n=0, 1,..., (2.29)

which completes the proof ofTheorem 2.4whenB >a. Second, consider the case when B≤a. It suﬃces to show that {xn}is bounded from above by some positive constant.

For the sake of contradiction, assume that{xn}is unbounded. Then there exists a subsequence{xnj}such that

limj→∞nj= ∞, lim

j→∞x1+nj= ∞, x1+nj=maxxn:−k≤n≤1 +nj

, (j=0, 1, 2,...). (2.30) From (2.22), we deduce that

k i=0

αix−i+nj≥Bx1+nj−A. (2.31) Taking the limit asj→ ∞of both sides of the last inequality, we obtain

limj→∞

k i=0

αix₋i+nj= ∞. (2.32)

It is easy enough to show thatx₋i+nj≤x1+nj, (i=0, 1, 2,...,k), and then asa=_k

i=0αi, we have

k i=0

αix₋i+nj≤ax 1+nj. (2.33) From the last inequality and the diﬀerence equation (1.1), we obtain

0≤ax 1+nj− k i=0

αix−i+nj=aA +^k_i₌0αix−i+nj

a−B−_k

i=0βix−i+nj

B+^k_i₌₀βix₋i+nj

. (2.34)

Consequently, it follows that

k i=0

βix−i+nj ≤a−B. (2.35)

(7)

Then for every i=0, 1, 2,...,k for which βi is positive, the subsequence {x₋i+nj} is bounded which implies that the sequence{_k

i=0αix₋i+nj}is also bounded. This contra- dicts (2.32) and the proof ofTheorem 2.4is completed.

Theorem 2.5. Assume thatB >aholds. Then the positive equilibrium pointxof the diﬀer- ence equation (1.1) is globally asymptotically stable.

Proof. The linearized equation (1.7) with (1.8) can be written in the form

yn+1+ k j=0

βjx−αj

B+bx yn−j=0. (2.36)

AsB >a, we get

k j=0

βjx−αj

B+bx ≤

a+ bx

B+bx<1. (2.37)

Thus, byRemark 2.2, we deduce that the equilibrium pointxof the diﬀerence equation (1.1) is locally asymptotically stable. It remains to prove that the equilibrium pointx is a global attractor. To this end, setI=limn→∞infxnandS=limn→∞supxn, which by Theorem 2.4are positive numbers. Then, from the diﬀerence equation (1.1), we see that

S≤A+aS

B+bI, I≥A+aI

B+bS . (2.38)

Hence

A+ (a−B)I≤bIS≤A+ (a−B)S. (2.39) From which it follows thatI=S. Thus, the proof ofTheorem 2.5is completed.

Theorem 2.6. The necessary and suﬃcient condition for the diﬀerence equation (1.1) to have positive prime period two solutions is that both inequalities

Ab−b²−(a+a)b−b(B+a)< b(B+a)², (2.40)

B+a <0 (2.41)

are valid.

Proof. First, suppose that there exist positive prime period two solutions

...,P,Q,P,Q,... (2.42)

(8)

of the diﬀerence equation (1.1). We will prove that the condition (2.40) holds. It follows from the diﬀerence equation (1.1) that

P=A+α0Q+α1P+α2Q+α3P+···

B+β0Q+β1P+β2Q+β3P+···, Q=A+α0P+α1Q+α2P+α3Q+···

B+β0P+β1Q+β2P+β3Q+···.

(2.43)

Consequently, we obtain

A+α0Q+α1P+α2Q+α3P+··· =BP+β0PQ+β1P²+β2PQ+β3P²+···, (2.44) A+α0P+α1Q+α2P+α3Q+··· =BQ+β0PQ+β1Q²+β2PQ+β3Q²+···. (2.45) By subtracting, we deduce after some reduction that

P+Q= −(B+a)

β1+β3+···, (2.46)

while by adding we obtain

PQ=Aβ1+β3+···

−

α0+α2+···

(B+a)

bβ1+β3+··· , (2.47)

whereB+a <0. Now, it is clear from (2.46) and (2.47) that P and Q are two positive distinct real roots of the quadratic equation

t²−(P+Q)t+PQ=0. (2.48)

Thus, we deduce that −(B+a)

β1+β3+···

2

>4

Aβ1+β3+···

−

α0+α2+···

(B+a) bβ1+β3+···

. (2.49)

From (2.49), we obtain

Ab−b²−(a+a)b−b(B+a)< b(B+a)², (2.50) and hence the condition (2.40) is valid. Conversely, suppose that the condition (2.40) is valid. Then, we deduce immediately from (2.40) that the inequality (2.49) holds. Conse- quently, there exist two positive distinct real numbersPandQsuch that

P= −(B+a) 2β1+β3+···−1

2

T1, (2.51)

Q= −(B+a) 2β1+β3+···+1

2

T1, (2.52)

(9)

whereT1>0 which is given by the formula T1=

−(B+a) β1+β3+···

2

−4

Aβ1+β3+···

−α0+α2+···(B+a) bβ1+β3+···

. (2.53) Thus,PandQrepresent two positive distinct real roots of the quadratic equation (2.48).

Now, we are going to prove thatPandQare positive prime period two solutions of the diﬀerence equation (1.1). To this end, we assume that

x₋k=P, x₋k+1=Q,..., x₋1=Q, x0=P. (2.54) We wish to show that

x1=Q, x2=P. (2.55)

To this end, we deduce from the diﬀerence equation (1.1) that x1=A+α0x0+α1x₋1+···+αkx₋k

B+β0x0+β1x−1+···+βkx−k

=A+Pα0+α2+···

+Qα1+α3+···

B+Pβ0+β2+···

+Qβ1+β3+···.

(2.56)

Dividing the denominator and numerator of (2.56) by−(B+a)/(β1+β3+···) and using (2.51)–(2.53), we obtain

x1

= −2Aβ1+β3+···)/(B+a+1 +K1

α0+α2+···

+1− K1

α1+α3+···

−2Bβ1+β3+···/(B+a) +1 +K1β0+β2+···

+1−

K1β1+β3+···

=

a−2Aβ1+β3+···/(B+a)+aK1

b−2Bβ1+β3+···/(B+a)+bK1

,

(2.57) where

K1=1− Ab−b²−(a+a)b−b(B+a) b(B+a)²

, (2.58)

and from the condition (2.40), we deduce thatK1>0. Multiplying the denominator and numerator of (2.57) by

b−2Bβ1+β3+···

(B+a)

−bK1. (2.59)

(10)

We have x1=

a−2Aβ1+β3+···

/(B+a)b−2Bβ1+β3+···

/(B+a)−baK1

b−2Bβ1+β3+···

/(B+a)²−b²K1

+ba−ab −a2Bβ1+β3+···

/(B+a+b2Aβ1+β3+···

/(B+a)K1

b−2Bβ1+β3+···/(B+a²−b²K1

. (2.60) After some reduction, we deduce that

x1= −(B+a) 2β1+β3+···

×

2α1+···

β0+···

−2α0+···

β1+···

−

2β1+···

(B+a(Ab−Ba1+K1

2α1+···

β0+···

−2α0+···

β1+···

−

2β1+···

/(B+a)(Ab−Ba)

=−(B+a)1 +K1

2β1+β3+··· = −(B+a) 2β1+β3+···+1

2

T1=Q.

(2.61) Similarly, we can show that

x2=A+α0x1+α1x0+···+αkx₋(k−1)

B+β0x1+β1x0+···+βkx₋(k₋1) =A+Qα0+α2+···

+Pα1+α3+···

B+Qβ0+β2+···

+Pβ1+β3+··· =P.

(2.62) By using the mathematical induction, we have

xn=P, xn+1=Q ∀n≥ −k. (2.63)

Thus, the diﬀerence equation(1.1) has positive prime period two solutions

...,P,Q,P,Q,.... (2.64)

Hence the proof ofTheorem 2.6is completed.

Acknowledgment

The authors would like to express their deep thanks to the referee for his interesting sug- gestions and comments on this paper.

References

[1] M. T. Aboutaleb, M. A. El-Sayed, and A. E. Hamza, “Stability of the recursive sequencexn+1= (α−βxn)/(γ+xn−1),” Journal of Mathematical Analysis and Applications, vol. 261, no. 1, pp. 126–

133, 2001.

[2] R. P. Agarwal, Diﬀerence Equations and Inequalities, vol. 155 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992.

(11)

[3] A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, “On the recursive sequencexn+1= α+ (xn−1/xn),” Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 790–798, 1999.

[4] R. DeVault, W. Kosmala, G. Ladas, and S. W. Schultz, “Global behavior of yn+1=(p+ yn−k)/(qyn+yn−k),” Nonlinear Analysis, vol. 47, no. 7, pp. 4743–4751, 2001.

[5] R. DeVault, G. Ladas, and S. W. Schultz, “On the recursive sequencexn+1=(A/xn) + (1/xn−2),”

Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3257–3261, 1998.

[6] R. DeVault and S. W. Schultz, “On the dynamics ofxn+1=(βxn+γxn−1)/(Bxn+Dxn−2),” Com- munications on Applied Nonlinear Analysis, vol. 12, no. 2, pp. 35–39, 2005.

[7] H. El-Metwally, E. A. Grove, and G. Ladas, “A global convergence result with applications to periodic solutions,” Journal of Mathematical Analysis and Applications, vol. 245, no. 1, pp. 161–

170, 2000.

[8] H. El-Metwally, E. A. Grove, G. Ladas, and H. D. Voulov, “On the global attractivity and the periodic character of some diﬀerence equations,” Journal of Diﬀerence Equations and Applications, vol. 7, no. 6, pp. 837–850, 2001.

[9] H. M. El-Owaidy, A. M. Ahmed, and M. S. Mousa, “On asymptotic behaviour of the diﬀerence equationxn+1=α+ (xn−1^p /xn^p),” Journal of Applied Mathematics & Computing, vol. 12, no. 1-2, pp. 31–37, 2003.

[10] H. M. El-Owaidy, A. M. Ahmed, and Z. Elsady, “Global attractivity of the recursive sequence xn+1=(α₋βxn−k)/(γ+xn),” Journal of Applied Mathematics & Computing, vol. 16, no. 1-2, pp.

243–249, 2004.

[11] G. Karakostas, “Convergence of a diﬀerence equation via the full limiting sequences method,”

Diﬀerential Equations and Dynamical Systems, vol. 1, no. 4, pp. 289–294, 1993.

[12] G. Karakostas and S. Stevi´c, “On the recursive sequencexn+1=A+f(xn,...,xn−k+1)/xn−k,” Com- munications on Applied Nonlinear Analysis, vol. 11, no. 3, pp. 87–99, 2004.

[13] V. L. Koci´c and G. Ladas, Global Behavior of Nonlinear Diﬀerence Equations of Higher Order with Applications, vol. 256 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1993.

[14] M. R. S. Kulenovi´c and G. Ladas, Dynamics of Second Order Rational Diﬀerence Equations. With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.

[15] M. R. S. Kulenovi´c, G. Ladas, and W. S. Sizer, “On the recursive sequence xn+1=(αxn+ βxn−1)/(γxn+δxn−1),” Mathematical Sciences Research Hot-Line, vol. 2, no. 5, pp. 1–16, 1998.

[16] S. A. Kuruklis, “The asymptotic stability ofxn+1−αxn+bxn−k=0,” Journal of Mathematical Analysis and Applications, vol. 188, no. 3, pp. 719–731, 1994.

[17] C. H. Gibbons, M. R. S. Kulenovi´c, G. Ladas, and H. D. Voulov, “On the trichotomy character of xn+1=(α+βxn+γxn−1)/(A+xn),” Journal of Diﬀerence Equations and Applications, vol. 8, no. 1, pp. 75–92, 2002.

[18] C. H. Gibbons, M. R. S. Kulenović, and G. Ladas, “On the dynamics of xn+1=(α+βxn+ γxn−1)/(A+Bxn),” in New Trends in Difference Equations. Proceedings of the 5th International Conference on Difference Equations and Applications, Temuco, Chile, pp. 141–158, Taylor & Fran- cis, London, UK, 2002.

[19] E. Camouzis, G. Ladas, and H. D. Voulov, “On the dynamics ofxn+1=(α+γxn−1+δxn−2)/(A+ x_n−2),” Journal of Diﬀerence Equations and Applications, vol. 9, no. 8, pp. 731–738, 2003.

[20] G. Ladas, “On the recursive sequencexn+1=(α+βxn+γxn−1)/(A+Bxn+Cxn−1),” Journal of Diﬀerence Equations and Applications, vol. 1, no. 3, pp. 317–321, 1995.

[21] W.-T. Li and H.-R. Sun, “Global attractivity in a rational recursive sequence,” Dynamic Systems and Applications, vol. 11, no. 3, pp. 339–345, 2002.

[22] S. Stevi´c, “On the recursive sequencexn+1=xn−1/g(xn),” Taiwanese Journal of Mathematics, vol. 6, no. 3, pp. 405–414, 2002.

(12)

[23] S. Stevi´c, “The recursive sequence xn+1=g(xn,xn−1)/(A+xn),” Applied Mathematics Letters, vol. 15, no. 3, pp. 305–308, 2002.

[24] S. Stevi´c, “On the recursive sequencexn+1=α+ (xn−1^p /xn^p),” Journal of Applied Mathematics &

Computing, vol. 18, no. 1-2, pp. 229–234, 2005.

[25] E. M. E. Zayed and M. A. El-Moneam, “On the rational recursive sequencexn+1=(D+αxn+ βxn−1+γxn−2)/(Axn+Bxn−1+Cxn−2),” Communications on Applied Nonlinear Analysis, vol. 12, no. 4, pp. 15–28, 2005.

[26] E. M. E. Zayed and M. A. El-Moneam, “On the rational recursive sequencex_n+1₌(αx_n+βx_n−1+ γxn−2+δxn−3)/(Axn+Bxn−1+Cxn−2+Dxn−3),” Journal of Applied Mathematics & Computing, vol. 22, no. 1-2, pp. 247–262, 2006.

E. M. E. Zayed: Mathematics Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Current address: Mathematics Department, Faculty of Science, Taif University, P.O. Box 888, El-Taif 5700, Hawai, Saudi Arabia

Email address:[email protected]

M. A. El-Moneam: Mathematics Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Email address:[email protected]

(13)

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://

mts.hindawi.com/according to the following timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Edson Denis Leonel,Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; [email protected]

Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

Hindawi Publishing Corporation http://www.hindawi.com