Calculation of The Sensitivity and transitivity
? ??
? ?? ?
1||
? a y +bx
of
f (x, y) =
?x
SAMAH ABD ALHADI
Babylon University College of Science For Women
Dept. Of Computer
: صلختسملا
?
?? ةلادلل يوضوفلا كولسلا ةسارد ىلع انلمع ثحبلا اذھ يف ?
? ?? ?
? a y +bx x
1||
ل??خ ن?م ك?لذو
f (x, y) =
(Matlab) جما??نرب ضر??غلا اذ?ھل انمدخت??ساو يد?عتلاو ة???ئادتب?ا طور??شلا ى?لع ةد??متعملا ة??ساسحلا يت???صاخ ة?سارد
امد?نع ق?قحتم ة?لادلل يد?عتلا نإ اند?جو ث??ح م??قلا ملا?عم ان?سردو
م??قلا هذ?ھ د?نعو -1.2?b?0.2 و 1.2?a?1.7
.ة?دعتم نوكت ? ةساسح ةلادلا اھ?ف نوكت يتلا م?قلاو ةساسح نوكت ?و ة?دعتم ةلادلا نوكت
Abstract
In this work ,We study the chaotic behavior for f (x, y) =
? ?? ?
? ?? ?
? a y +bx x
1||
through
employment sensitivity dependent on initial condition and transitive by using the software (Matlap) these are implement by varying the parameter of system. We found the parameters which make f(x,y) sensitive does not make him transitive and vise versa.
?1-Introduction :
A dynamical system is chaotic on a given invariant set X for a flow ? when it satisfied certain properties. Thus to apply this concept we must first identify an invariant set of course X could be a very small set in the phase space. And then the assertion of chaos on X would not necessarily be of much practical important thus, a chaotic flow mixes things up and is hard to predict, although this definition of chaotic is reasonably useful, it is also important to note that the term "chaotic" in the literature is used with many definition some researches simply use the loose sense we first discussed and some require stronger condition than sensitive dependent, while it is clear from the quotes that Poincare and Lorenz had clear notions of sensitive dependent Li and Yorke first gave a mathematical definition of chaos in 1975 the definition that we use is due to AVS Lander and Yorke (1980). However all definition include element comparable to transitivity and sensitive dependent [Devaney 1986; Robinson 1999; Wiggins 2003]. While another definition include sensitive or positive Lypanov satisfies [Gulick, 1992]. The study of discrete map such as piece wise Linear map [Devaney, 1984; Lozi 1978; Cao and Liu 1998; Aharonov and Devaney and Elias 1997; Ashwin and Fu 2002] is an interesting contribution to the development of theory of dynamical system , with many possible application in science and engineering [scheizer and Hasler 1996; Abel and Bauer and Kerber and Schwarz 1997]. Discrete mathematical models arise directly from experiment or by the use of the Poincare for the study of continues models two of these models are the Henon [Henon 1976] and the Lozi [Lozi 1978] maps given by :
? ?? ?
?1? ax + y ??
? bx
( , )=
H xy
ab
,
And
? ?? ?
? ?? ?
1? a bx
+y
x
( , )=
L xy
ab
,
The Ha,b map gives a chaotic attractor called the Hennon attractor , which is obtain for a=1.4 and b=0.3 as shown in fig (a) . there are many poper that discuss the original Henon and Lozi map such as [maorotto, 1979l Gao and Liv 1998] , moreover, it is
?possible to change the form of the Henon mapping H to obtain other chaotic attractor.[lozi, 1978; Aziz alaovi and Carl Robert and Gelso Grebogi 2001, Zeraoulia Elhadj 2005].
Application of These maps include secure communication using the notions of chaos [Scheizer and Hasler, 1996; Abel and Bouer and Kerber and Schwarz 1997 ].
The Lozi map L is a 2-D non–invertible iterated map that given chaotic attractor called the Lozi attractor which is obtained for a=1.4 and b=0.3 as show in fig (1b) [Zeraoulia and Protte, 2007 ] introduce a new simple 2-D piecewise linear map given by :
? ??
? ?
1||
? a y +bx
??…………(3)
f (x, y) =
?x
Where a and b are parameter and equation (3) is an interesting system has simple form, similar to the Lozi map [ Lozi, 1978].
Fig. (1): (a) The original Hénon chaotic attractor obtained from the H mapping with its basin of attraction (white) for a = 1.4 and b = 0.3. (b) The original Lozi chaotic attractor obtained from the L mapping with its basin of attraction (white) for a = 1.4 and b = 0.3.
?2-Elementary Fundamental definition :
In this section we refered to many definition which we needed in this work
Definition . 2-1 [meiss , 2007]:
A set A is invariant under a rule ft if ft(A)=A for all t . that is for each x?A, ft (X)?A for any t.
Definition. 2-2 [meiss, 2007]
A set A is forward invariant if ft(A) ? A for all t > 0 Definition. 2-3 [meiss, 2007]
A point X* is an equilibrium of X-=f(x), x(0) = X0 if f(X*) = 0 Definition. 2-4 [meiss, 2007]
Let p be in the domain of f then p is fixed point of f if f(p)=p
Def. 2-5 [Meiss, 2007]
An equilibrium is attractor if all the eigen value has negative real part.
Def. 2-6 : [Meiss, 2007]