Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments

Size: px

Start display at page:

Download "Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments"

Godwin Thornton
5 years ago
Views:

Malaya Jounal of Matematik ()(22) 4 8 Chaos and bifucation of discontinuous dynamical systems with piecewise constant aguments A.M.A. El-Sayed, a, and S. M. Salman b a Faculty of Science, Aleandia Univesity, Aleandia, Egypt.

1 Malaya Jounal of Matematik ()(22) 4 8 Chaos and bifucation of discontinuous dynamical systems with piecewise constant aguments A.M.A. El-Sayed, a, and S. M. Salman b a Faculty of Science, Aleandia Univesity, Aleandia, Egypt. b Faculty of Education, Aleandia Univesity, Aleandia, Egypt. Abstact In this pape we ae concened with the definition and some popeties of the discontinuous dynamical systems geneated by piecewise constant aguments. Then we study a discontinuous dynamical system of the Riccati type equation as an eample. The local stability at the fied points is studied. The bifucation analysis and chaos ae discussed. In addition, we compae ou esults with the discete dynamical system of the Riccati type equation. Keywods: Discontinuous dynamical systems, piecewise constant aguments, Riccati type equation, fied points, bifucation, chaos. 2 MSC: 39B5, 37N3, 37N2. c 22 MJM. All ights eseved. Intoduction The discontinuous dynamical systems geneated by the etaded functional equations has been defined in []-[4]. The dynamical systems with piecewise constant aguments has been studied in [5]-[7] and efeences theein. In this wok we define the discontinuous dynamical systems geneated by functional equations with piecewise constant aguments. The dynamical popeties of the discontinuous dynamical system of the Riccati type equation will be discussed. Compaison with the coesponding discete dynamical system of the Riccati type equation will be given. n = 2 n, n =, 2, 3,..., 2 Piecewise constant aguments Conside the poblem of functional equation with piecewise constant aguments whee [.] denotes the geatest intege function. Let n =, 2, 3,... and t [n, (n + )), then (t) = f(([ t ])), t >, >. (2.) () =, (2.2) (t) = f( n (n)), t [n, (n + )). Let = and take the limit as t n +, we get n+ = f( n ), n =,, 2,... Coesponding autho. addesses: amasayed@hotmail.com (A. M. A. El-Sayed) and findingsanaa@yahoo.com (S. M. Salman).

2 A. M. A. El-Sayed et al. / Chaos and bifucation... 5 This shows that the discete dynamical system n = f(n, n ), n =, 2, 3,..., T. () = o, is a special case of the poblem of functional equation with piecewise constant aguments (2.)-(2.2). Now let t [, ), then t [, ), ([ t ]) = () and the solution of (2.)-(2.2) is given by (t) = () = f(()), t [, ), with () = lim (t) = f(()). t Fo t [, 2), then t [, 2), ([ t ]) = () and the solution of (2.)-(2.2) is given by (t) = 2 (t) = f( ()), t [, 2). Repeating the pocess we can easily deduce that the solution of (2.)-(2.2) is given by (t) = (n+) (t) = f( n (n)), t [n, (n + )), which is continuous on each subinteval (k, (k + )), k =, 2, 3,..., n, but lim (k+)(t) = f( k (k)) k (k). t k + Hence the poblem (2.)-(2.2) is a discontinuous and we have poved the following theoem. Theoem 2.. The solution of the poblem of functional equation with piecewise constant aguments (2.)-(2.2) is discontinuous (sectionally continuous) even if the function f is continuous. Now let f : [, T ] R n R n and R +. Then, the following definition can be given. Definition 2.. The discontinuous dynamical system geneated by piecewise constant aguments is the poblem (t) = f(t, ([ t ]), ([t ]),..., ([ t n ])), t [, T ], (2.3) (t) =, t. (2.4) Definition 2.2. The fied points of the discontinuous dynamical system (2.3) and (2.4) ae the solution of the equation (t) = f(t,,,..., ). 3 Main Poblem Conside the discontinuous dynamical system geneated by piecewise constant aguments of Riccati type equation (t) = 2 ([ t ]), t, >, and () =. (3.) Hee we study the stability at the fied points. In ode to study bifucation and chaos we take fistly = and we compae the esults with the esults of the discete dynamical system of Riccati type diffeence equation n+ = 2 n, n =, 2, 3,..., and = o. (3.2) Secondly, we take some othe values of and T and study some eamples.

3 6 A. M. A. El-Sayed et al. / Chaos and bifucation Fied points and stability As in the case of discete dynamical systems, the fied points of the dynamical system (3.) ae the solution of the equation f() =. Thus thee ae two fied points which ae ( fied ) = + + 4, ( fied ) 2 = + 4. To study the stability of these fied points, we take into account the following theoem. Theoem 3.. [8] Let f be a smooth map on R, and assume that is a fied point of f.. If f ( ) <, then is stable. 2. If f ( ) >, then is unstable. Now since in ou case f() = 2, the fist fied point ( fied ) = + +4 is stable if that is, 4 < < 3 4. The second fied point ( fied ) 2 = <, is stable if <, which can neve happen since is always >. So, the second fied point is unstable. Figue () shows the tajectoies of (3.) when =, while Figue (2) shows the tajectoies of (3.2). Figue : Tajectoies of (3.), =. Figue 2: Tajectoies of (3.2). 4 Bifucation and Chaos In this section, the numeical epeiments show that the dynamical behavios of the discontinuous dynamical system (3.) depends completely on both and T as follows:. Take = and t [, 3], in this case the dynamical behavios of the two dynamical systems (3.) and (3.2) ae identical (Figue 4). 2. Take =.25 and t [, 2] in the dynamical system (3.) (Figue 5). 3. Take = and t [, 2] in the dynamical system (3.) (Figue 6). 4. Take =.25 and T = N = 3 in the dynamical system (3.) (Figue 7). 5. Take = and T = N = 35 in the dynamical system (3.) (Figue 3).

4 A. M. A. El-Sayed et al. / Chaos and bifucation Figue 3: Bifucation diagam of the dynamical systems (3.) with = and (3.2) whee N = T = 7. Figue 4: Bifucation diagam fo (3.), =, t = [, 3]. Figue 5: Bifucation diagam fo (3.), =.25, t = [, 3] Figue 6: Bifucation diagam fo (3.), =, T = N = 3. Figue 7: Bifucation diagam fo (3.), =.25, T = N = 3.

5 8 A. M. A. El-Sayed et al. / Chaos and bifucation... 5 Conclusion The discontinuous dynamical system models geneated by piecewise constant aguments have the same behavio as its discete vesion when =. On the othe hand, changing the paamete togethe with the time t [, T ] affects the chaos behavio of the dynamical system geneated by the piecewise constant aguments model as it is shown clealy in the above figues. Refeences [] A. M. A. El-Sayed, A. El-Mesiy and H. EL-Saka, On the factional-ode logistic equation, Applied Mathematics Lettes, 2(27), [2] A. M. A. El-Sayed and M. E. Nas, Eistence of unifomly stable solutions of nonau- tonomous discontinuous dynamical systems, J. Egypt Math. Soc., 9() (2), -6 [3] A. M. A. El-Sayed and M. E. Nas, On some dynamical popeties of discontinuous dynamical systems, Ameican Academic and Scholaly Reseach Jounal, 2()(22), [4] A. M. A. El-Sayed and M. E. Nas, On some dynamical popeties of the discontinuous dynamical system epesents the Logistic equation with diffeent delays, I-manage s Jounal on Mathematics, accepted manuscipt. [5] D. Altintan, Etension of the Logistic equation with piecewise constant aguments and population dynamics, Maste dissetation, Tukey 26. [6] M. U. Akhmet, Stability of diffeential equations with piecewise constant aguments of genealized type, accepted manuscipt. [7] M.U. Akhmet, D. Altntana and T. Egen, Chaos of the logistic equation with piecewise constant aguments, Applied Mathematis Lettes, pepint (2), 2-5. [8] S. El Aidy, An Intoducation to Diffeence Equations, Spinge, Thid Edition, 25. Received: August 28, 22; Accepted: Septembe 4, 22 UNIVERSITY PRESS

Research Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

Research Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming