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https://ptsldigital.ukm.my/jspui/handle/123456789/499684Full metadata record
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.advisor | Prof. Ishak Hashim. | - |
| dc.contributor.author | Saida M. K. Atawna (P59383) | - |
| dc.date.accessioned | 2023-10-13T09:33:45Z | - |
| dc.date.available | 2023-10-13T09:33:45Z | - |
| dc.date.issued | 2014-05-08 | - |
| dc.identifier.other | ukmvital:81106 | - |
| dc.identifier.uri | https://ptsldigital.ukm.my/jspui/handle/123456789/499684 | - |
| dc.description | Difference equations have been proven to be effective in modeling and analyzing discrete dynamical systems that arise in many applications such as signal processing, population dynamics, health sciences and economics. They also arise naturally in studying iterative numerical schemes. Furthermore, they appear when solving differential equations using series solution methods or studying them qualitatively using, for example, Poincare maps. Rational difference equations, particularly bilinear ones, have attracted the attention of many researchers recently. As it turns out, many models, such as population models in Mathematical Biology, are members of the family of second-order rational difference equations. The behavior of solutions of higher order rational difference equations can provide prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one. Hence, the study of this family of equations is important both from a theoretical point of view and the point of view of applications. In this thesis, the higher order nonlinear rational difference equation, xn+1 = α+βxn+γxn-k / A+Bxn+Cxn-k , n = 0, 1, 2, . . . where the parameters α, β, γ, A, B, C and the initial condition x-k, . . . , x-1, x0 are positive real numbers, k ∈ {1, 2, . . .}, is considered. The main focus is on the periodic character of the positive solutions of the equation. A necessary and sufficient condition for the equation to have a prime period-two solution is given. Furthermore, it is shown that the prime period-two solution of the equation is locally asymptotically stable. Specifically, for the case k = 1, the second order nonlinear rational difference equation, xn+1 = α+βxn+γxn-1 / A+Bxn+Cxn-1 , n = 0, 1, 2, . . . where the parameters α, β, γ, A, B, C and the initial condition x-1, x0 are positive real numbers, is also investigated in this thesis. This equation is an important stepping stone in understanding the global dynamics of the higher order rational difference equation of the form mentioned above. A complete qualitative description of the global behavior of all positive solutions of the equation is given. It is shown that all positive solutions of the equation converge to the positive equilibrium or to a prime period-two solution. In particular, the global asymptotic stability of the hyperbolic and non-hyperbolic equilibrium solutions of the equation is investigated. It is shown that for the equilibrium solution of this equation, local asymptotic stability implies its global attractivity. In addition, a necessary and sufficient condition for the equation to have a prime period-two solution is given. It is shown that the prime period-two solution of the equation is locally asymptotically stable.,Persamaan beza telah terbukti berkesan dalam pemodelan dan menganalisis sistem dinamik diskret yang wujud dalam banyak penggunaan seperti pemprosesan isyarat, dinamik penduduk, sains kesihatan dan ekonomi. Ia juga muncul secara bersahaja dalam kajian skim berangka lelaran. Tambahan pula, ia muncul apabila menyelesaikan persamaan pembezaan menggunakan kaedah penyelesaian siri atau dalam kajian secara kualitatif seperti peta Poincare. Persamaan beza nisbah, terutamanya bilinear, telah menarik perhatian ramai penyelidik baru-baru ini. Ternyata, banyak model, seperti model populasi dalam Biologi Matematik, merupakan ahli famili kepada persamaa beza nisbah peringkat dua. Telatah penyelesaian persamaan beza nisbah peringkat tinggi boleh memberikan suatu prototaip ke arah pembangunan teori asas tingkah laku global bagi penyelesaian persamaan beza tak linear dengan peringkat lebih daripada satu. Oleh itu, kajian persamaan famili ini adalah penting jika dilihat dari kedua-dua sudut pandangan teori dan penggunaan. Dalam tesis ini, persamaan beza nisba tak linear peringkat tinggi xn+1 = α+βxn+γxn-k / A+Bxn+Cxn-k , n = 0, 1, 2, . . . yang α, β, γ, A, B, C adalah parameter dengan syarat awal x-k, . . . , x-1, x0 kesemuanya nombor nyata positif dan k ∈ {1, 2, . . .}, dipertimbangkan. Fokus utama adalah pada sifat kalaan bagi penyelesaian positif daripada persamaan tersebut. Suatu syarat cukup dan perlu bagi persamaan untuk memiliki penyelesaian perdana kalaan-dua diberikan. Tambahan pula, ia menunjukkan bahawa penyelesaian perdana kalaan-dua adalah stabil secara asimptot setempat. Khususnya, untuk kes k = 1, persamaan beza nisbah peringkat kedua xn+1 = α+βxn+γxn-1 / A+Bxn+Cxn-1 , n = 0, 1, 2, . . . yang α, β, γ, A, B, C adalah parameter dengan syarat awal x-1, x0 kesemuanya nombor nyata positif juga dikaji dalam tesis ini. Persamaan ini merupakan suatu batu loncatan yang penting dalam memahami dinamik global persamaan beza nisbah peringkat tinggi dalam bentuk seperti yang dinyatakan di atas. Keterangan kualitatif lengkap tingkah laku global bagi semua penyelesaian positif persamaan itu diberikan. Ia menunjukkan bahawa semua penyelesaian positif persamaan tersebut menumpu kepada keseimbangan positif atau kepada penyelesaian perdana kalaan-dua. Khususnya, penyelesaian kestabilan asimptot global bagi penyelesaian hiperbola dan bukan hiperbola persamaan tersebut diselidiki. Ia menunjukkan bahawa penyelesaian keseimbangan bagi persamaan ini, kestabilan asimptot setempat mengimplikasikan ketarikan secara global. Di samping itu, syarat cukup dan perlu bagi persamaan untuk memiliki penyelesaian perdana kalaan-dua diberikan. Ditunjukkan bahawa penyelesaian perdana kalaan-dua adalah stabil secara asimptot setempat.,Ph.D | - |
| dc.language.iso | eng | - |
| dc.publisher | UKM, Bangi | - |
| dc.relation | Faculty of Science and Technology / Fakulti Sains dan Teknologi | - |
| dc.rights | UKM | - |
| dc.subject | Difference equations | - |
| dc.subject | Discrete dynamical systems | - |
| dc.subject | Prime period-two solution | - |
| dc.subject | Dissertations, Academic -- Malaysia | - |
| dc.title | Stability of period-two cycles of higher order rational difference equation | - |
| dc.type | Theses | - |
| dc.format.pages | 125 | - |
| dc.identifier.callno | QA371.3 .A934 2014 | - |
| dc.identifier.barcode | 001327 | - |
| Appears in Collections: | Faculty of Science and Technology / Fakulti Sains dan Teknologi | |
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|---|---|---|---|---|
| ukmvital_81106+SOURCE1+SOURCE1.0.PDF Restricted Access | 1.74 MB | Adobe PDF |  View/Open |
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