The Dynamical Properties of Modified of Bogdanov Map
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Journal of Al-Qadisiyah for Computer Science and Mathematics Vol. 13(2) 2021 , pp Math. 14–21 14 The Dynamical Properties of Modified of Bogdanov Map (1) (2) Wafaa H. Al-Hilli , Rehab Amer Kamel Department of Mathematics, College of Education University of AL-Qadisiyah, Iraq,wafaahadi23@yahoo.com Department of Mathematics, College of Education for pure Sciences, University of Babylon,Iraq, Email: re_ami_ka@yahoo.com A R T I C L E I N F O A B S T R A C T Article history: In the work, we study the general properties of modified of the Received: 26 /03/2021 Bogdanov map in the form Fa,m,k . Rrevised form: 05 /04/2021 푥 푥 + 푦 Accepted : 15 /04/2021 ( ) = ( ) 푦 푦 + 푎푦 + 푘 푐표푠푥 − 푘푥 + 푚푥푦 Available online: 07 /05/2021 Keywords: We prove it has sensitive dependence on initial conditions and positive Lyapunov exponent with some conditions . So we give estimate of Sensitive dependence on topological entropy of modified Bogdanov map . initial conditions, Lyapunov exponent, MSC. 41A25; 41A35; 41A36. Topological entropy. DOI : https://doi.org/10.29304/jqcm.2021.13.2.790 _________________________________________________________________________________________________________________________________________________________________________ 1. Introduction Chaos theory is a branch of mathematics concern with studying the properties of deterministic systems that depend on their behavior on a set of elementary conditions, making their study somewhat complex using traditional mathematical tools. Mathematicians use chaos theory to model these systems in ways Different in order to arrive at a specific mathematical description of it and its behavior depending on all possible initial conditions. The Bogdanov map provides a good approximation to the dynamics of the Poincaré map of periodically forced oscillators, first considered by Bogdanov [5]. Vicente ∗Corresponding author: Wafaa H. Al-Hilli Email addresses: wafaahadi23@yahoo.com Communicated by: Dr. Rana Jumaa Surayh aljanabi. Wafaa H. Al-Hilli et al., Vol.13(2) 2021 , pp Math. 14–21 15 Aboites & etal, studied the dynamic behavior among periodic orbits high periodicity and chaos of Bogdanov map were observed through bifurcation[6] The Bogdanov map is 2-dimension and discrete dynamical system, its form Fa,m,k 푥 푥 + 푦 ( ) = ( ), we changed this form of Bogdanov map to modified Bogdanov map by 푦 푦 + 푎푦 + 푘푥2 − 푘푥 + 푚푥푦 2 1 replacing x to cos(푥) and denoted to 퐹푎,푚,푘. We find important chaotic properties of it. One of this properties sensitive dependence on initial condition and Lyapunov expanents. 2. Basic Definitions 푘 푘 푘 Any pair ( ) for which f ( ) = 푘, 푔 ( ) = ℎ is called fixed point of 2-D ℎ ℎ ℎ 2 푥 dynamical system [3]. Let V be a subset of R and v0 = [푦] be any element in v Consider 퐹푎,푚,푘 :V→R² a map. Furthermore [4], assume that the first partials of the coordinate maps 퐹푎,푚,푘(1) and 퐹푎,푚,푘(2) of F exist at v0 is the linear map D 퐹푎,푚,푘 (v0) defined on R² by 1 1 ∂퐹푎,푚,푘1(v0) ∂퐹푎,푚,푘1(v0) ∂x ∂y D퐹푎,푚,푘 (v0 )=( 1 1 ), For all v0 in V the determine of D퐹푎,푚,푘 (v0 ) is said the Jacobin ∂퐹푎,푚,푘2(v0) ∂퐹푎,푚,푘2(v0) ∂x ∂y of 퐹푎,푚,푘 at v0 and is denoted by J=det D퐹푎,푚,푘 (v0), if |J퐹푎,푚,푘 (푣0 )|<1then 퐹푎,푚,푘 (v0) is area contracting at v0 1 and if |J퐹푎,푚,푘 (푣0 )| > 1 then 퐹푎,푚,푘 is area expanding at v0 [1]. 3. Some properties of modified Bogdanov map In the section, we study modified Bogdanov map from fixed point, diffeomorphism (one- to -one ,onto, invertible, c∞ ).Also we determined the contracting and expanding area. Proposition(3-1) If k≠0 Then Fa,m,k has unique fixed point Proof By definition of a fixed point 푥 푥 + 푦 0 F ( ) = ( ), y=0 so k 푐표푠(푥) =푘푥 then x=0 therefore p=( ) is unique a,m,k 푦 푦 + 푎푦 + 푘 푐표푠(푥) − 푘푥 + 푚푥푦 0 fixed point. Proposition(3-2) 2 2 Let 퐹: 푅 → 푅 be the Fa,m,k then the Jacobin of Fa,m,k is 1 + 푎 + 푘 Proof 푥 1 1 1 1 DF ( ) = ( ) 푠표 퐽F = 푑푒푡 ( ) = 1 + 푎 + 푘 a,m,k 푦 −푘푠푛(푥) − 푘 + 푚푦 1 + 푎 + 푚푥 a,m,k −푘 1 + 푎 Wafaa H. Al-Hilli et al., Vol.13(2) 2021 , pp Math. 14–21 16 Proposition (3-3) Fa,m,k is area contracting map if |1 + 푎 + 푘| < 1 and If |1 + 푎 + 푘| > 1 then Fa,m,k is area expanding. Proof by proposition (3-2) then |1 + 푎 + 푘| < 1 hence 푘 < 푎 < 2 + 푘 푥 therefore the Fa,m,k (푦) is an area contracting map and if 2 + 푘 < 푎 < 푘 the Fa,m,k is an area expanding. Proposition(3-4) 푥 −(푎+푏)±√(푎^2+8푎−4푘) The eigenvalues of D F ( ) 푎푡 푓푥푒푑 푝표푛푡 푠 휆 = a,m,k 푦 1,2 2 Proof 1 1 −(푎+푏)±√(푎^2+8푎−4푘) 퐷푒푡(퐷F (푣) − 퐼휆) = 푑푒푡 ( ) = (1 + 푎)(푘) = 0 then 휆 = . a,m,k −푘 1 + 푎 1,2 2 Proposition (3-5) 2 2 Let F:R → R be modified bogdanov map, then Fa,m,k is diffeomorphism. Proof 1. Let 푇(푥, 푦) = (푥 + 푦, 푦 + 푎푦 + 푘푐표푠푥 − 푘푥 + 푚푥푦) 푇(1, 0) = (1, 푘 푐표푠(1) − 푘) T(0, 1)=(1, 1+a+k) 1 푘푐표푠(1) − 푘 1 푘푐표푠(1) − 푘 Then , F ( ) 푓rom ( ) a,m,k 1 1 + 푎 + 푘 0 2푎 + 3푘 − 2푘푐표푠(1) Fa,m,k has a pivot position in every Colum then Fa,m,k is one to one and has a pivot position in every row then Fa,m,k is onto. 푥 푥 + 푦 2. F is C∞ : F ( ) = ( ) a,m,k a,m,k 푦 푦 + 푎푦 + 푘푐표푠(푥) − 푘푥 + 푚푥푦 휕푓1(푥,푦) 휕2푓1(푥,푦) 휕푛푓1(푥,푦) Note that = 1, = 0 … … … . = 0 푓표푟 푎푙푙 푛 ∈ 푁푎푛푑 푛 ≥ 2 휕푥 휕푥2 휕푥푛 휕푓1(푥, 푦) 휕2푓1(푥, 푦) 휕푛푓1(푥, 푦) = 1, = 0 … … … . = 0 푓표푟 푎푙푙 푛 ∈ 푁푎푛푑 푛 ≥ 2 휕푦 휕푦2 휕푥푛 휕푓2(푥, 푦) 휕2푓2(푥, 푦) = −푘푠푛 − 푘 + 푚푦 (푥), = 푘푐표푠(푥) 푓표푟 푎푙푙 푛 ∈ 푁 휕푥 휕푥2 휕푓2(푥, 푦) 휕2푓2(푥, 푦) 휕푛푓2(푥, 푦) = 1 + 푎 + 푚푥, = 0 … … … . = 0 푓표푟 푎푙푙 푛 ∈ 푁푎푛푑 푛 ≥ 2 휕푦 휕푦2 휕푥푛 ∞ Then the partial derivatives continuous and exist then Fa,m,k is c Wafaa H. Al-Hilli et al., Vol.13(2) 2021 , pp Math. 14–21 17 4. Sensitive Dependence on initial conditions (S.D.I) Definition (4-1) Let (X, k) be a metric space. A map F: (X, k) →(X, k) is said to be sensitive dependence on initial conditions (S. D. I) if there exit 휖 > 0 ∋ for any 푥0 ∈ 푋 and any open set U⊆ X containing 푥0 there exists + 푛 푛 푦0 ∈ 푈 and n ∈ 푍 푠푢푐ℎ 푡ℎ푎푡 푑 ( 푓 (푥0), 푓 (푦0)) >∈ that is ∃ ∈ > 0, ∀ 푥 > 0, ∃ 푦 ∈ 퐵0(푥), ∃ 푛 ∈ 푁, 푛 푛 푑(푓 (푥0), 푓 (푦0)) >∈ . 0.2 1 0 0.5 -0.2 -0.4 0 -0.6 -0.5 -0.8 -1 -1 -1.2 -1.5 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 -2 -2 -1.5 -1 -0.5 0 0.5 x1 (1) = 0.02 ;y1 (1) = 0.01;a1=-1.00005;m1= 1.20004;k1=-1.4;andx2 (1) = 0.03 ;y2 (1) = x1 (1) = 0.02 ;y1 (1) = 0.01;a1=-1.00005;m1=- 1.20004;k1=-1;and x2 (1) = 0.03 ;y2 (1) = 0.02;a2= 1.000004;m2=-1.100009;k2=-1.5; 0.02;a2=-1.000004;m2=-1.100009;k2=-1.2; 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -1.2 -1.2 -1.4 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1.4 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 x1 (1) = 0.02 ;y1 (1) = 0.01;a1=-1.25;m1=- x1 (1) = 0.02 ;y1 (1) = 0.01;a1=-1.25;m1=- 1.4003;k1=-1.24;and x2 (1) = 0.03 ;y2 (1) = 1.501;k1=-1.24;and x2 (1) = 0.03 ;y2 (1) = 0.02;a2=-1.18;m2=-1.5004;k2=-1.15; 0.02;a2=-1.18;m2=-1.203;k2=-1.1 Wafaa H. Al-Hilli et al., Vol.13(2) 2021 , pp Math. 14–21 18 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -1.2 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1.4 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 x1 (1) = 0.02 ;y1 (1) = 0.01;a1=-1.10000; m1=-1.40000; k1=1.0000024;andx2 (1) = 0.03; x1 (1) = 0.02 ;y1 (1) = 0.01;a1=-1.025; m1=- y2 (1) = 0.02; a2=-1.20000; m2=-1.60000;k2=- 1.04; k1=-1.2; and x2 (1) = 0.03 ; 1.0000015; y2(1)=0.02; a2= -1.034; m2=-1.09; k2=-1.3; 5.Lyapunov exponent The Lyapunov play role important in the field of calculus and integration and control theory, as it has many life applications in the field of Physics, Medicine, Engineering, Space Science and other Scientific fields.