Conformable Kesirsel Mertebeden Tam Değer Fonksiyonlu Lojistik Modelin Kararlılık ve Çatallanma Analizi

Year 2020, Volume: 9 Issue: 3, 1080 - 1090, 26.09.2020

Güven Kaya , Senol Kartal

https://doi.org/10.17798/bitlisfen.665517

Cited By: 1

Abstract

Bu çalışmada, conformable kesirsel mertebeden tam değer fonksiyonlu lojistik model ele alınmıştır. Modele tam değer fonksiyonlarının kullanılmasına dayalı bir ayrıklaştırma işlemi uygulanılarak bir fark denklem sistemi elde edilmiştir. Elde edilen bu fark denklem sisteminin pozitif denge noktasının yerel asimptotik kararlı olmasını sağlayan cebirsel koşullar Schur-Cohn kriterlerinin kullanılmasıyla elde edilmiştir. Yine çatallanma analizi ile sistemde r parametresinin değişimine bağlı olarak Neimark-Sacker çatallanmasının oluştuğu gösterilmiştir. Ayrıca kesirsel mertebeden türev parametresi ( α ) ve kesiklileştirme parametresi ( h ) nin sistemin dinamik yapısı üzerindeki etkisi araştırılmıştır. Elde edilen tüm teorik sonuçlar nümerik simülasyonlarla desteklenmiştir.

Keywords

Conformable kesirli mertebeden türev, tam değer fonksiyonu, kararlılık, Neimark-Sacker çatallanma

References

• 1. Abbas S., Banerje, M., Momani,S. 2011. Dynamical Analysis of Fractional-Order Modified Logistic Model. Compu.Math.Appl., 62: 1098-1104. 2. Parra G. G., Arenas A. J., Chen-Charpentier B. M. 2014. A Fractional Order epidemic Model for the Simulation of Outbreaks of Influenza A(H1N1). Math. Method. Appl. Sci., 37: 2218-2226. 3. Khalil R., Al Horani M., Yousef, A. 2014. Sababheh M., A new definition of fractional derivative. J. Comput. Appl. Math., 264: 65-70. 4. Abdeljawad T. 2015. On conformable fractional calculus. J. Comput. Appl. Math., 279: 57--66. 5. Perez J.E.S., Gomez-Aguilar J.F., Baleanu, D., ve ark. 2018. Chaotic Attractors with Fractional Conformable Derivatives in the Liouville-Caputo Sense and Its Dynamical Behaviors. Entropy, 20:384. 6. Balcı E., Öztürk İ., Kartal Ş. 2019. Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative. Chaos. Soliton. Fract., 123: 43-51. 7. Gürcan F., Kaya G., Kartal Ş. 2019. Conformable fractional order Lotka-Volterra predator-prey model: Discretization, stability and bifurcation. J. Comput. Nonlin. Dyn., 14(11): 111007. 8. Kartal S., Gurcan F. 2019. Discretization of conformable fractional differential equations by a piecewise constant approximation. Int. J. Comput Math., 96: 1849-1860. 9. Ye H., Ding Y. 2009. Nonlinear Dynamics and Chaos in a Fractional-Order HIV Model. Math. Probl. Eng., 2009: 1-12. 10. Ertürk V.S., Odibat Z.M., Momani S. 2011. An Approximate Solution of a Fractional Order Differential Equation Model of Human T-Cell Iymphotropic Virus I (HTLV-I) Infection of CD4+ T-Cells. Comput. Math. Appl., 62: 996-1002. 11. Zeng C., Yang Q. 2010. A Fractional Order HIV Internal Viral Dynamics Model. Comput. Model. Eng.Sci., 59: 65-77. 12. Özalp N., Demirci E. 2011. A Fractional Order SEIR Model with Vertical Transmission. Math. Comput. Model., 54: 1-6. 13. Pinto C.M.A., Machado J.A.T. 2013. Fractional Model for Malaria Transmission Under Control Strategies. Comput. Math. Appl., 66: 908-916. 14. Ahmed E., Elgazzar A.S. 2007. On Fractional Order Differential Equations Model for Nonlocal Epidemics. Physicia A, 379: 607-614. 15. Rihan F.A. 2013. Numerical Modelling of Fractional-Order Biological Systems, abstr. Appl. Anal., 2013: 1-11. 16. Varalta N., Gomes A.V., Camargo R.F. 2014. A Prelude to the Fractional Calculus Applied to Tumor Dynamics. Tendencias em Matematica Aplicade e Computacional., 15: 211-221. 17. Ahmed E., Hashis A.H., Rihan F.A. 2012. On Fractional Order Cancer Model, J.Fract.calc.Appl., 3: 1-6. 18. Shanbazi M., Erjaee G.H., Erjaee H. 2014. Dynamical Analysis of chemotherapy Optimal Control Using Mathematical Model Presented by Fractional Differential Equations, Describing Efector Immune and Cancer Cells Interactions. Journal of Pharmacy and Pharmaceutical Sciences., 3: 5-17. 19. Bozkurt F. 2014. Stability Anaysis of a Fractional Order Differential equation System of a GBM-IS Interaction Depending on the Density. Appl. Math. Inf.Sci., 8: 1021-1028. 20. Jun D., Jun Z.G., Yong X., Hong Y., Jue W. 2014. Dynamic Behavior Analysis of Fractional-Order Hindmarsh-Rose Neuronal Model. Cong. Neurodyn., 8: 167-175. 21. El-Raheem Z.F., Salman S.M. 2014. On a Discretization Process of Fractional- Order Logistic Differential Equation, J.Egyptian. Math.Soc., 22: 407-412. 22. Arafa A.A.M., Rida S.Z., Khalil M. 2013. The Effect of Anti-Viral drug Treatment of Human Immunodeficiencey Virus Type 1 (HIV-1) Described by a Fractional Order Model. Appl. Math. Model., 37: 2189-2196. 23. Arafa A.A.M., Rida S.Z., Khalil M. 2014. A Fractional-Order Model of HIV Infection with Drug Therapy Effect. J.Egyptian. Math. Soc., 22: 538-543. 24. Yan Y., Kou C. 2012. Stability Analysis for a Fractional Differential Model of HIV Infection of CD4+ T-Cells with Time Delay. Math. Comput. Simulat., 82: 1572-1585. 25. Gökdoğan A., Yıldırım A. 2011. Merdan, M., Solving a Fractional Order Model of HIV Infection of CD4+ T-Cells. Math. Cumput. Model., 54: 2182-2138. 26. Matouk A.E. 2009. Stability Conditions, Hyperchaos and Control in a Novel Fractional Order Hyperchaotic System. Phys. Lett. A., 373: 2166-2173. 27. Jafari H., Daftardar-Gejji V. 2006. Solving a System of Nonlinear Fractional Differential equations Using Adomian Decomposition. J. Math. Anal. Appl., 196: 644-651. 28. Odibat Z., Momani S. 2008. Numerical Methods for Nonlinear Partial Differential Equations of Fractional Order. Appl. Math. Modelling., 32(1): 28-39. 29. Ajou A.E., Odibat Z., Momani S., Alawneh A. 2010. Construction of Analytical solutions to Fractional Differential Equations Using Homotopy Analysis Method. IAENG Int. J. Appl. Math., 40(2). 30. Ünlü C., Jafari H., Baleanu D. 2013. Revised Variational Iteration Method for Solving systems of Nonlinear fractional Order differential Equations. Bstr. Appl. Anal., 2013: 1-7. 31. Amen I., Novati P. 2017. The Solution of Fractional Order epidemic Model by Implicit Adams Methods. Aplied Mathematical Modelling, 43: 78-84. 32. Ertürk V.S., Momani S. 2008. Solving systems of Fractional Differential Equations Using differential Transform Method. J. Comput. Appl. Math., 215: 142-151.