Some Dynamical Properties of the Family of Tent Maps *

Transcription

1 Int. Jounal of Math. Analysis, Vol. 7, 3, no. 9, HIKARI Ltd, Some Dynamical Popeties of the Family of Tent Maps * Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub Depatment of Mathematics, Faculty of Science and Ats, Najan Univesity wfha6@yahoo.com Depatment of Mathematics, Faculty of Science and Ats, Najan Univesity mohammad_abdulkawi@yahoo.com Copyight 3 Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. Abstact In this pape, we investigated some dynamical behavio of family of tent maps. Such dynamical popeties include fied points and thei stability, peiod obits, visualize the iteations using a kind of plot called a cobweb plot and demonstate bifucation diagam fot. Futhemoe, detecting the pesence of chaos in the discete dynamical systemt is investigated. Finally, we develop Matlab compute pogams that eflect the esults intepeting such dynamical behavio. Mathematics Subject Classification: 37C5, 37E5, 34C3 Keywods: Fied points, peiodic points, peiodic obits, bifucation. INTRODUCTION Conside the paameteized tent map, which can be descibed piecewise by,if T ( ) = ( ),if <

2 434 Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub whee <. This map is continuous, linea on each of the intevals (, / ] and [ /, ) (with espect to slopes and ) and has the points (,) and (, ) in its gaph. This map is known as tent maps and often intoduced as one of the fist eamples of chaotic maps liteatue fo nonlinea discete dynamical systems. Its dynamics ehibit vaious featues ae commonly used to identify chaotic systems (see fo instance [5]). Chaos theoy descibes comple motion and the dynamics of sensitive systems. Chaotic systems ae mathematically deteministic but nealy impossible to be pedicted. Chaos is moe evident in long-tem systems than in shot-tem systems. Behavio in chaotic systems is a peiodic, in othe wods, no vaiable descibes the state of the system that undegoes a egula epetition of values. A chaotic system can actually evolve in a way that appeas to be smooth and odeed; howeve, chaos efes to the issue of whethe o not it is possible to make accuate long-tem pedictions of any system if the initial conditions ae known to an accuate degee [8]. In addition to the sensitivity condition, thee is a miing condition called tansitivity and a egulaity condition called density of peiodic points[]. Deteministic chaos can be descibed by deteministic map, such as tent map T as follows: stating fom any point between and, the tace of the map would still lie in the ange [, ] of the tent map, which is a absobing set. Also, each point inside [, ] will be often visited abitaily closely and infinitely by any typical solution, hence it is also an attacto []. The pupose of this pape is to investigate the dynamical behavio of the family of tent maps, we will limit ouselves to the case < [3]. Fo cetain paamete values, the mapping undegoes stetching and folding tansfomations and displays sensitivity to initial conditions and peiodicity (see [] and [6]). The tent map is studied in the mathematics of dynamical systems. Because of its simple shape, the tent map shape unde iteation is vey well undestood. And despite its simple shape, it has seveal inteesting popeties []. The pape is oganized as follows; The fist section is the intoduction and the second section descibes the dynamical behavio fo this discete dynamical systemt such as fied points and thei stability and peiod obits followed by section thee, which pesents cobweb plots. Section fou demonstates peiod doubling and obtains the bifucation diagam oft, fo the paamete <. The damatic bifucation that occus at = will be eplained. The esults will be obtained using Matlab compute pogams. One of these pogams is to visualize the iteations using a kind of plot called a cobweb plot. Sketching the bifucation diagam oft, fo the paamete < is by using anothe Matlab pogam. Fied points will show up as a single point, a peiodic obit as seveal points, and a chaotic obit as a band o seveal bands of points. Section five pesents the esults that ae detecting the pesence of chaos in the tent mapst and eamining sensitivity to initial conditions; it is one of the dynamical

3 Some dynamical popeties of the family of tent maps 435 popeties of stange attactos. Results and discussions will be displayed. Finally, in the last section some concluding emaks will be highlighted.. FIXED POINTS AND PERIODIC ORBITS Conside an iteation = n+ f( n) (.) A fied point, o point of peiod one, of discete dynamical system (.) is a point at which n+ = f( n) = n fo all n[7] Fo the tent map, this implies that T( n) = n, fo all n. Gaphically, the fied points can be found by identifying intesections of the map T ( ) with the line y =, see [3]. Figues.(a) though.(c) displayt fo = /7,/ and 5/6by Matlab pogam in the Appi. As inceases, the height of the gaph oft ises, because of the facto in the fomula fot. Fom this obsevation and the thee gaphs in Figue.; we deduce that if < < /, thent intesects the line y = once (at ), wheeas if /< <, then thee ae two points of intesection. We ae led to analyze sepaately the membes of{ T } fo which < < /, = / and /< <. Finally, we will studyt, which is the oiginal tent mapt and which has some vey inteesting featues, we shall follow [3]. T (), is in (, ] = Gaphical iteation of tent map:t () T (), is in (, ] = Gaphical iteation of tent map:t () Figue. (a) Figue. (b)

4 436 Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub Gaphical iteation of tent map:t () T (), is in (, ] = Figue.(c) Case. < < / The gaph in Figue.(a) shows that is the only fied point of T. Since < < /, it follows fom the definition of T that if < /, then ( ) T = < and if / <, then T ( ) = ( ) < < < { n} Consequently fo any [,], the sequence { T ( )} μ is bounded and deceasing and by n= the continuity of T, the sequence conveges to the fied point. Theefoe, is an attacting fied point whose basin of attaction is [,]. Case. = / Fist we notice that if /, then T / ( ) = (/ ) =, so that is a fied point of T / (Figue.(b)). Net, we calculate that / <, then T ( ) = (/)( ) = So that T/ ( ) = is a fied point of T /. Consequently, evey point in [,] eithe is a fied point of T / o has a fied point its fist iteate. Case 3. /< < In addition to the fied point, thee is a second fied point p that lies in ( /,], as one can see in Figue.(c). To evaluate p we solve the equation p = T ( p) = ( p) which yields

5 Some dynamical popeties of the family of tent maps 437 p = + / As inceases fom ½ towad, p inceases fom / towad /3. Because ( ) T = > on [,] ecept at /, both and p ae epelling fied points. The peiod- points of T [] ae the fied points of T, which is given by The gaph of T [] ( ) 4 fo /4 ( ) fo /4 < / = ( + ) fo /< /4 4 ( ) fo /4 [] T 8 appeas in Figue. plotted by Matlab pogam in the Appi, and suggests that fo /< <, [] the fou equations T ( ) = aising fom the definition of 4 points ae,,, and T [n] (), n= [] T has fou fied points that can be found by solving Gaphical iteation of tent map, = Figue. The fist and thid ae the two fied points of T, so it follows that is a -cycle fo T. This -cycle is epelling, because ( ) ( ) [] T. We find that the fied 4, T = 4 > wheeve the [] deivative is defined. Because the gaph of T [] is linea on the n subintevals n n [,/ ],...,[ /,], it is possible to descibe the vaious n cycles of T - all of which ae epelling.

6 438 Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub Case 4. = fo / If = then T = T, the tent map given by T( ) =. The ( ) fo /< gaph of T appeas in Figue.3 (a) obtained by Matlab pogam in the Appi. The majo diffeence between the gaph of T and the gaph of T when < is the fact that the ange of T fills out the whole inteval [,]. The mapt stetches the inteval [,/ ]ove the entie inteval [,], and folds the inteval [ /,] back ove the inteval [,]. T [n] (), n= Gaphical iteation of tent map T [n] (), n= Gaphical iteation of tent map Figue.3(a) Gaphical iteation of tent map Figue.3(b) T [n] (), n= Figue.3(c) As with all membes of{ } = T = if = / 3, [] we know that /3 is the second fied point of T. Figues.3(b)-(c) indicate that T [3] and T have, espectively, fou and eight fied points. ThusT has two peiod- points [] and si peiod-3 points, which we could evaluate by solving the equations = T ( ) [3] = T fo. and ( ) T, is a fied point oft. Since ( ) ( )

7 Some dynamical popeties of the family of tent maps 439 Poposition.. A point [,] is peiodic undet iff is a ational numbe of the fom m/ nwhee m is an even positive intege and p is an odd positive intege with m< p. n i i i+. if, Poof. If n Z + n n n, then n n T ( ) = i=,,,..., n i+ i+ i+. if, n n n Theefoe, the peiodic points undet ae the solutions of n i n i + n = =, =,,,...,. o n n fo i. i. i+ Solving fo, we get,,,,..., n = o = i= n n. In eithe case, is of the + fom m, whee m is even and p odd. Suppose that is a ational numbe in [, ] of the p m fom, whee m is even and p is odd. Since ϕ ( p) ( mod p) p ϕ ( p), whee ϕ is the Eule s ϕ p function, p ; hence thee is a natual numbet such that = p. t. That is, m m. t = =. Since m is even,. k. t p ϕ( p) = whee k <. Theefoe, ϕ( p) p k. t = k p ϕ ( p) ( ). < ϕ( p) ϕ( p). That is, is a peiodic point. Now, we will detemine the eventually peiodic points and peiodic points fot. Poposition.. Let [,]. Then is ational iff is peiodic o eventually peiodic unde T. m Poof[3]. Suppose = is a ational numbe in [, ], whee m and p ae elatively p pime. If m and p ae both odd, then. m.( p m) T( ) = o T( ) =. In eithe case, T ( ) is p p peiodic by Poposition.. If m is odd and p is even, let k be the lagest positive intege such that p = k. s, whee s is odd. A shot computation shows that T k+ ( ) is of the fom even odd, which is peiodic. If m is even and p is odd, then is peiodic by Poposition.. 3. COBWEB PLOTS In this section, we ae going to intoduce a vey helpful gaphical view of the iteation pocess called cobweb plot[9]. The cobweb plot makes use of a plot of the mapping and a ( )

8 44 Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub plot of the efeence line y =. Fom these two cuves one may constuct gaphically an iteation sequence. We constuct the plot fist and then eplain it. We do this fo the case of thee iteations. The Matlab pogam 3 (with file functent.m) which does this is listed in the Appi. Hee the stating value of is.3, the numbe of iteation is 3, epeat count fo mapping (the level of map composition) and at the paamete =. Figue 3. shows a cobweb plot of thee iteations fo the tent map. The stating value is =.3. Stating at that value on the -ais, we move vetically upwad until we hit the gaph of cobwebtent(). The y-value at that point is the value of the fist iteate. Now we move hoizontally until we hit the efeence line y =. The -coodinate of that point is the value of the iteate. Net we move vetically until we each the gaph of cobwebtent(), see pogam 3. The y-coodinate of that point is the second iteate. We epeat the steps to get the thid iteate. Now let's use pogam 3 to look at a few typical events. We begin by educing to a value of/3, fo which we have a single stable equilibium. We then constuct the cobweb plot of the appoach to this equilibium. Figue 3. shows a cobweb plot of iteations fo the tent map. The stating value is =.789. This shows nicely the appoach to the oigin. Cobweb diagam fo Tent map. =. Cobweb diagam fo Tent map. = [n] T () and n =..6.5 [n] T () and n = Figue 3. Figue 3. Now let's incease to (hee the stating value of = and iteations is ) and look at a peiodic obit. In a way we ae cheated by stating on the obit (see Figue 3.3). Let's do this again, but this time we stat off the obit (hee =, the stating value of = and the iteations is ). Now we see the tansient appoach to the obit (see Figue 3.4).

9 Some dynamical popeties of the family of tent maps 44 Cobweb diagam fo Tent map. =. Cobweb diagam fo Tent map. = [n] T () and n =..6.5 [n] T () and n = Figue 3.3 Figue 3.4 So fa we have seen eamples of equilibium and of a peiodic obit. Afte that, we choose a ational numbe fo the initial so that all calculations ae done eactly, fo eample =3457/, iteations is and =, (see Figue 3.5). [n] T () and n = Cobweb diagam fo Tent map. = Figue 3.5 This is a chaotic obit fo the tent map. This is the typical esult fo the typical initial condition when eceeds/. Fo special choices of initial conditions, we can land on one of the unstable peiodic obits. 4. BIFURCATION DIAGRAM In this section, we ae going to eploe the concept of bifucations and how Matlab can help us visualize these changes. We will begin with some basic definitions, then eamine some plots, and finally ceate a bifucation diagam. A bifucation is a fundamental change in the natue of a solution(see [3] and []). When studying dynamical systems, we often want to know when the system undegoes a bifucation.

10 44 Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub A bifucation diagam is a kind of plot that shows the value of the changing paamete, in ou case, on one ais and the solution to the system on the othe ais. Ou net goal is to poduce the bifucation diagam of the 'tent' mapst. Figue 4. Often we want to know how system behavio deps on paametes. In the case of the tent mapt, we have a single paamete, and we have aleady known that thee is a stable equilibium fo < l. We also know that fo > l, thee ae two unstable equilibia, some unstable peiodic solutions, and, as we have just seen, chaotic solutions. We can get an oveview of how all of this deps on the paamete by a bifucation diagam. That is a plot in which the abcissa is the value of, and on the odinate we plot all of the -values fom an obit fo that value of. Fied points will show up as a single point, a peiodic obit as seveal points, and a chaotic obit as a band o seveal bands of points. The pogam which does this is the Matlab Pogam 4 in the Appi. Hee (see Figue 4.) the numbe of values calculated fo each paamete value is points, and the fist points ae discaded. The values ae calculated beginning with initial point =.3. The ange of paamete vaiation is to. The numbe of paamete values fo which this is done is points while the numbe of tent map values is 8 points. Stating with the value, and consideing values in the ange {,}. 5. SENSITIVE DEPENDENCE TO INITIAL CONDIDTIONS AND CHAOS Let I be an inteval, and suppose that f : I I (signifies that the domain of f is I and the ange is contained in I ). Then f has sensitive depence on initial conditions at, if thee is anε > such that fo each δ >, thee is a y in I and a positive intege n such that [ n ] [ n] y < δ and f ( ) f ( y) > ε (5.) If f has sensitive depence on initial conditions at each I, we say that f has sensitive depence on initial conditions on I [3].

11 Some dynamical popeties of the family of tent maps 443 The initial conditions hee efe to the given, o initial, points and y. Sensitive depence on initial conditions says that f has sensitive depence if y abitaily close to any given point in the domain of f, thee is a point and an nth iteate that is fathe fom the nth iteate of than a distanceε. This has pactical significance because in such instances highe iteates of an appoimate value of may not esemble the tue iteates of. Thus compute calculations may be misleading. Accoding to the well accepted definition of Devaney[], a one-dimensional map of the fom = n+ f ( n), (5.) is chaotic if it: (i) f has sensitive depence on initial conditions (s.d.i.c.), and (ii) f has a dense set of peiodic obits, and (iii) f has at least one dense obit ( f is topologically tansitive ). The following esults detect the pesence of chaos in the discete dynamical systemt, see [3] and []. Poposition 5.. The tent mapt has a sensitive depence on initial conditions on [, ] and hence chaotic. Poof. Let [,]. Fist we will show that if v is any dyadic ational numbe (of the fom j positive intege n such that m /, in lowest tems) in [, ] and w is any iational numbe in [, ], then thee is a T [ n] [ n] () v T ( w) > = (5.) m [ ] Towad that goal, if v = j /, then T m [ m+ k ] ( v) and T ( v) = fo all k >. By contast, if w is any iational numbe in [,], then since T doubles each numbe in [ ] (, /), thee eists an n > m such that T n ( w) > /. Since n > m, it follows that () v = T n, so that (5.) is valid. Net, let δ >. Then thee eist a dyadic ational v and an iational numbe w in [, ] such that v < δ and w < δ. Theefoe (5.) implies that [ n] [ n] [ n] [ n] eithe T ( ) T ( v) > o T ( ) T ( w) > 4 4 Thus if we let = / 4, then sensitive depence on initial conditions at the abitay ε numbe, and hence on [, ], is poved. Theefoe,T is chaotic. Basically the eason thatt has a sensitive depence on initial conditions is that if /, then T ( ), so that distances between pais of numbes in (, /) o in (/, ) ae doubled byt.

12 444 Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub Poposition 5.. The set of peiodic points unde T is dense in [, ]. Poof. Le (a, b) be any open inteval in (, ). Let p an odd positive intege lage enough k a + b so that < b a.thee is a least positive intege k such that a,. If k is. p p k + even, then we ae done. If k is odd then ( a, b) will do. p Poposition 5.3. The tent maps is chaotic on [, ]. Poof. Let I = ( a, b) and J = ( c, d ) be any open intevals in [, ]. Let n be the least positive intege such that < b a. Then T n ( I ) = [,], =.. This implies that n ( T n ) ( J ) is an open inteval of I. By the density of peiodic points in [, ], thee is a peiodic point in I such that ( ) J T n. 6. CONCLUSIONS This pape has focused on discete-time dynamical system and investigated some dynamical behavio of the paameteized tent maps fo some values of its contol paamete and has pointed out the Matlab implementations of the dynamical behavio such as fied points and thei stability, peiod obits, gaphics behavio (cobweb plot) and bifucation diagams. The bifucation analysis was consideed and the pesence of chaotic behavio of the discete dynamical systemt was solved by investigating the sensitive depence to initial condition. The most comple dynamics-chaos-occu only fo lage values of the paamete ; so the pediction of a chaotic time seies could be demonstated. APPENDIX The following Matlab pogams (see[4]) intepets the dynamical behavio of the family of tent maps. Pogam function Tpaamete(,maite) % The 'tent' map is given by: % **, if <.5 % T()={ ; '' is in(, ] and '' is in [, ]. % **(-), if >=.5 % whee '' can be an element, vecto o mati of initial points. clc; clf; clea all; close all; % Default settings.

13 Some dynamical popeties of the family of tent maps 445 if (nagin < ) maite = 5; ()= ; =.8333; fsize=; % maimum numbe of iteations. % initial condition. % paamete value. % font size. % Compute the tent map. fo n = :maite a(n+) = **(n)*((n)<.5); b(n+) = *(-*(n) + )*((n)>=.5); (n+) = a(n+) + b(n+); hold on % Plot tent map. c=[.5 ]; yc=[ ]; plot(c,yc,'b'); % Plot line y=. e = [ ]; ye = [ ]; plot(e,ye,'k'); title('gaphical iteation of tent map:t_()','fontsize',fsize,'colo','b'); label('','fontsize',fsize,'colo','b'); ylabel({'t_(), is in (, ]';[' =',numst(,'%.7g'),'.']},'fontsize',fsize,'colo','b') Pogam function Titeate(,maite) % The 'tent' map is given by: % **, if <= <=.5 % T()={ ; '' belongs to (,] and '' belongs to [,]. % **(-), if.5 < <= % whee '' can be an element, vecto o mati of initial points. clc; clf; clea all; close all; % Default settings. if (nagin < ) %maite = ; maite = 4; %maite = 8; ()= ; =.8333; fsize=; % maimum numbe of iteations fo T^[]. % maimum numbe of iteations fo T^[]. % maimum numbe of iteations fo T^[3]. % initial condition. % paamete value. % font size. % Compute the tent map. fo n = :maite a(n+) = **(n)*(<=(n)<=.5); b(n+) = **(-(n)+)*(.5<(n)<=);

14 446 Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub (n+) =.5*(a(n+)+b(n+)); % Plot line 'y=' and 'tent' map e = [ ]; ye = [ ]; % 'e','ye' values fo plotting y=. %c =[.5* ]; % 'c' value fo plotting T^[^n^](), n=. c=[.5*.5.75* ]; % 'c' value fo plotting T^[^^](), n=. % 'c' value fo plotting T^[^3^](), n=3. %c = [.5*.5 75*.5.65* * ]; plot(e, ye,'k', c,,'b'); title('gaphical iteation of tent map, =.8333','Fontsize',fsize,'Colo','b'); label('','fontsize',fsize,'colo','b'); ylabel('t^[^n^](), n=','fontsize',fsize,'colo','b'); Pogam 3 function cobwebtent() % The function cobwebtent() successively iteates a given point '' with the 'tent' map as a fied set. % The 'tent' map is given by: % **_n, if <=<=.5 % T_(_n)={ ;n=,,,..., '' belongs to (,] and '' belongs to [,]. % **(-_n), if.5<=<= % whee '' can be an element, vecto o mati of initial points, and 'nma' is the maimum numbe of iteations. clea all; clc; clf; global n fsize=; % font size. () = input(' Ente the initial condition _: '); maite=input(' Ente the maimum numbe of iteations: '); n=input(' Ente epeat count fo mapping: '); =input(' Ente the paamete value: '); n=:.:; y=n; plot(n,functent(n),'b-') % Plot function in file functent.m hold on plot(n,n,'k') % Supepose plot of y= ais([ ]); % Scale pictue ais('squae'); % Ensue plot is squae fpintf('\n n Iteate \n') % Pint table heade % Compute the tent map. fo k=:maite (k+) = functent((k)); fpintf(' %3.f %.f \n', k,(k+)) %Pint iteate

15 Some dynamical popeties of the family of tent maps 447 % Daw cobweb plots of the Tent map fo k=:maite a(*k-)=(k); a(*k)=(k); b()=;b()=(); fo k=:maite b(*k-)=(k); b(*k)=(k+); fo k=:maite if k> h=[a(k-) a(k)]; yh=[b(k) b(k)]; plot(h,yh,':') v=[a(k) a(k)]; yv=[b(k) b(k+)]; plot(v,yv,'') fpintf( ' Final iteation is %.f \n', (k+)) % Has huge magnitude when iteates ae unbounded title({'cobweb diagam fo Tent map.',[' =',numst(,'%.7g'),'.']}, 'Fontsize',fsize,'Colo','k') label('','fontsize',fsize,'colo','k') ylabel({'t_^[^n^]()',[' and n=',numst(n,'%.7g'),'.']},'fontsize',fsize,'colo','k') functent.m function y=functent() % Function to etun the Tent Map iteated nepeat times % global n % Needs nepeat shaed with calling pogam n=; % In this case don't want to ovewite calling agument fo k=:n n=**(.5-abs(n-.5)); % Mapping is (n+)=**(.5-abs(_n-.5)) y=n; Pogam 4 % Matlab code to poduce the bifucation diagam of the 'tent' map: % **, if <.5 % T_()={ ; '' belongs to (,] and '' belongs to [,]. % **(-), if >=.5 clc; clea all; fsize=; %Font size. =linspace(,,); %This is points fo.

16 448 Wadia Faid Hassan Al-shamei and Mohammed Abdulkawi Mahiub =.3; %An initial condition. fpintf('going into the initial obit\n'); fo i=: y=*.*(.5-abs(-.5)); =y; fpintf('going into new obit\n'); % Stoe the esults. A=zeos(8,); fo i=:8 y=*.*(.5-abs(-.5)); A(i,:)=y; =y; plot(a','k.','makesize',4); plot(a','k.','makesize',4); title('bifucation diagam fo tent map.','fontsize',fsize,'colo','b'); label('','fontsize',fsize,'colo','b'); ylabel('t_^[^n^]()','fontsize',fsize,'colo','b'); * Acknowledgments: This wok is suppoted by the Scientific Reseach Deanship in Najan Univesity, Kingdom of Saudi Aabia unde eseach poject numbe NU7/.

17 Some dynamical popeties of the family of tent maps 449 REFERENCES [] Campin, M. and Heal, Benedict, On the Chaotic Behavio of the Tent Map, Teaching Mathematics Applications (994) 3(): [] Devaney, R. L., An intoduction to chaotic dynamical systems, nd ed., Addison Wesley, 989. [3] Gulick, D., Encountes with Chaos, McGaw-Hill, Inc., 99. [4] Hahn, B.D. and Valentine, D.T., Essential Matlab fo Enginees and Scientists, Elsevie Ltd, 7. [5] Katok, A. and Hasselblatt, B., Intoduction to the moden theoy of dynamical systems, Encyclopedia of Mathematics and its Applications, 54. Cambidge Univesity Pess, 995. [6] Li, T. Y. and Yoke, J. A., Peiod thee implies chaos, Ame. Math. Monthly 8 (975), [7] Mandelbot, B. B, The factal geomety of natue, W. H. feeman & company (8th Pinting), New Yok, 999. [8] Ott, E., Chaos in Dynamical Systems. New Yok: Cambidge Univesity Pess,993. [9] [] Received: Mach, 3