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The Dynamics of the Fixed Points to Duffing Map Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.7, 2013 The Dynamics of the Fixed Points to Duffing Map Iftichar M.T.AL-Shara'a 1 May Alaa Abdul-khaleq AL-Yaseen 2* Mathematics Department ,University of Babylon ,Hilla , Babylon, Iraq * E-mail of the corresponding author : [email protected] Abstract We study the fixed points of Duffing map and the general properties of them , we find the contracting and expanding area of this map .So, we divide the parameter space for many regions to determinate the fixed points attracting , repelling or saddle . Also we determinate the bifurcation points of the parameter space . Keywords: Duffing map , fixed point , bifurcation point , attracting- expanding area. 1. Introduction Chaos Theory is a synonym for dynamical systems theory, a branch of mathematics. Dynamical systems come in three flavors: flows (continuous dynamical systems), cascades (discrete, reversible, dynamical systems), and semi-cascades (discrete, irreversible, dynamical systems).The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future.In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour. Bifurcations occur in both continuous systems (described by ordinary differential equations , delay differential equations, or partial differential equations), and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them. The Duffing map is a 2 – D discrete dynamical system , its form is . It has fractal y F, = attractor studied by Paul Bourke 1998 .Some literatures studied its chaotic properties−bx + , ayMishra − y and Mankar [3] constructed encryption method using Duffing map, this method is desined using message embedded scheme and is analyzed for its validity, for plaintext sensitivity , key sensitivity, known plaintext and brute-force attacks. Tarai and Khan[2] applied backstepping method for projective synchronization of Duffing map ,in this case the error goes to zero through oscillation , the numerical simulation results indicate that their approach works very well .In this research we show that the Duffing map is diffeomorphism map and it has three fixed points , in our research we divide the parameter space into six regions to determinate types of the fixed points, and to find the bifurcation parameter points . 2. Preliminaries ∞ Let be a map .We say F is C , if its mixed k-th partial derivatives exist and are continuous for all ∞ ∞ F , and→ it is called a diffeomorphism if it is one-to-one, onto, C and its inverse is C . Let V be a subset of , and be any element in .Consider be a map. Furthermore assume that the first partials k of the coordinate maps f and g of F exist at .The differential → of F at is the linear map defined v v DF v !" !" on by , for all in .The determinant of is called the Jacobian !# v !$ v DF v = !% !% & v DF v of F at and it is denoted!# v by !$ v , so F is said to be area-contracting at if , F is vsaid to be area-expandingJF vat = if det+ . Let be an matrix .The real,det+ number, -is called eigenvalue of A if there exists a non, zero vector , X in such that , every non zero vector X . det+ /. 4 0 n 2 n 3 satisfying this equation is called an eigen vector of A associated with the eigenvalue05 = 35 .The point is called 7 3 6 9 fixed point if , it is an attracting fixed point, the eigenvalues of DF at this point if 8 and are 7 7 : less than one in 6absolute89 = 68 value9 , and it is repelling fixed point, if and are less than one in absolute3 value3 . with , is called hyperbolic matrix 3if: 3 where are the eigenvalues of 0A .Gulick GL 2, [3] defineddet the 0bifurcation = >. point at one dimension, we generalized,3?, @. this definition3? in two dimensions. A 41 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.7, 2013 ′ parameterized family has a bifurcation of parameter if the number or nature (attracting vs. repelling ) C′ A,B ′ ′ of periodic point of the family changes as the parameter a Dor b passes through . In this case, is said to C′ C′ be a bifurcation point .We denote to the inverse of A. E: D D 0 3.The General Properties of Duffing Map In this section , we find the fixed point and we study the general properties of Duffing map , for example : is ∞ , one to one , onto , C and invertible so it is diffeomorphism, and we find the parameter which hasF area contracting or expanding . A,B Proposition 3-1 : Let be a Duffing map , if then has three fixed points : and 0 C − D − . A,B: → a / D + . A,B 6 9 , H 0 . HC − D − . − C − D − . H Proof−HC : − By D −definition . of fixed point , then and 7 8 7 A,B 6 9 = = 6 9 7 = 8 8 = I D7 + C8 − , so , therefore 8 −D7 + C8 − 8then 8 is the first fixed point , and 0 8 7 = −D7 + C7 − 7 7−D +,then C − 7 − . = 0 J and= 6 9 therefore 0 I D + C − 7 − . = 0 that…….( is 3.1)7 = and C − D − . 7 = >HC −……( D − 3.2) . are8 the = second >HC − and D − third . fixed points . C − D − . E − C − D − . J = H J = H ¶ HC − D − . −HC − D − . Proposition 3-2 : Let be the Duffing map then the Jacobian of is b . A,B Proof: : → = , so . ¶ 7 0 . 7 0 . +A,B 6 9 OA,B 6 9 = PQR = D Proposition 3-38 : −D C − N8 8 −D C − N8 The eigenvalues of are λ T T T ; . x ES$ EU>V$ E EW x DF, 6 9 :, = X 6 9 Proof : If is eigen value ofy then must be satisfied the characteristicy equation 7 3 −. 3 +A,B 6 9 Y Y = 8 D 3 − C + N8 then and the solutions of this equation are where λ T T T . ES$ EU>V$ E EW ¶0 3 − C3 + N38 + D = 0 3:, :, = Remark : 1. It is clear that if , then the eigenvalues of are real . x , ,Ny − a, / 2Hb b / 0 DF 6y9 2. The eigenvalues of at fixed points are T for fixed point and 7 A>VA EWB A,B :, + 689 3 = P 3 = T for fixed point and λ T for fixed points . EAEBE>VAEBE EWB E EEE>VEE EW E Proposition 3-4 : P :, = P Let be a Duffing map , If then is diffeomorphism . ProofF ,: To: prove→ is diffeomorphism ,b we @ 0must proveF, : 1) is one-to-oneF, map : Let such that then x: 7 7: 7 A,B 6 9 , 6 9 F, 6 9 = F, 6 9 y: so8 and 8: 8 8: 8 : : : : = hence 8 = 8 and so , that is , is −D7 + C8 − 8 −D7 + C8 − 8 one-to-one.−bx: + ay : − y: = −bx + ay − y −bx: = −bx b @ 0 x: = x F, 2) is ∞ : then all first partial derivatives exist and continuous. Note that 7 8 A,B [ 6 9 = ] 8 −D7 +] C8 − 8 ] ] \ ^_,` , and \ ^_,` , \ c_,` , , and \ c_,` , .We get that all its\_ ] = ixed 0 Xa k-th bpartial derivatives \`] = exist 0 and\_ ]continuou= 0 sXa for d all 2 k. From\` definition] = 0 ofXa diffeomorfism d e , is F∞, . F, [ 42 Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.3, No.7, 2013 i 3) is onto : Let any element in such that and i . Then there exist gh EAh gh EAh A,B 6 9 8 = 7 = EB j EB k f .Then is onto . 7 l A,B 6 9 = 6 9 F, 8 f i 4) has an inverse : Let gh EAh such that . Note E: E: E: A,B A,B 6 9 = j EB k A,B m A,B 6 9 = A,B m A,B 6 9 = 6 9 f f f f i i that , EBhAgEg g EAg , and E: E: f E: A,B m A,B 6 9 = A,B 6 9 = j EB k = 6 9 A,B m A,B 6 9 = f −D + Cf − f f f i f gh EAh . Then has an inverse and it is invertible . ¶ i A,B EB gh EAh , j k = j−D 6 9 + C − k = 6 9 F EB f It is clear that if or then is a hyperbolic matrix , the question what about 7 : A,B : , these proposition3 3 answer= D @ this . question−. .+ 689 3 3 = Remark. 3-5 : If then , so then is not one to one, that y 7 D = 0 A, = nQoSA,BU = p6 9 : 7 q A, is , is not diffeomorphism . ay − y 8 Proposition A, 3-6 : 1. is non hyperbolic matrix if and . Also , it is non hyperbolic matrix if 7 AE A,B and+ 689 .
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