Friday 29th of March 2024
 

Numerical Approach For Local Bifurcation Analysis of Nonlinear Physical System


Abdelhak Chikhaoui, Tayeb Benouaz and Fatiha Lassouani

In this paper, we propose an study that combines classical linearization method with the Routh Herwitz criterion theory of complex nonlinear systems to compute local stability boundaries and visualize such bifurcation surfaces of nonlinear dynamical systems as function of parameters set (Analytical Search for Bifurcation Surfaces in Parameter Space). Therefore, we proposed a numerical method for the bifurcation analyses, our goal is to applied the optimal derivative (based on the minimization in the least-square sense) as introduced by O. Arino and T. Benouaz. In order to gain some progress with this procedure in the term of bifurcation analysis (detection of the local bifurcation in the neighborhood of the bifurcation parameters with respect to an initial condition) . This application enables us to compare the results obtained with those found by the classical linearization (Frchet derivative (jaccobian matrix) in the equilibriums points).

Keywords: Nonlinear ordinary differential equation, optimal derivative, Classical linearization (Freshet derivative in the equilibrium point), asymptotic stability, bifurcation analysis

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ABOUT THE AUTHORS

Abdelhak Chikhaoui
Abdelhak Chikhaoui Born in sidi senouci, Tlemcen, in western algeria on January 27, 1975. He attended the primary school and attended secondary school. Graduated from Tlemcen university in physic (option optoelectronic) (Jun 1996), he received the D.E.S degree. Undertake postgraduate studies (1997) in physic option electronic and physical modeling when he received the magister (master) degree (jun 2000) from Tlemcen university under the supervision of Prof. T.BENOUAZ. Since 2003 he is a teaching member at the Department of physics, faculty of science, U.A.B. Tlemcen. he received the Doctorat degree (jun 2009) from Tlemcen university in the field of approximation and stability of nonlinear systems. He participated in several internationals conferences. His research interests to modeling and simulation of nonlinear systems (physical,electronic and electromagnetic problems,biology,ecology), The last publications: \"Numerical simulation of nonlinear pulses propagation in a nonlinear optical directional coupler\",International Journal of Physical Sciences, Volume 4, Issue 9, Septembre 2009, pp. 505-513. On the Relationship between the Optimal Derivative and Asymptotic Stability, African Diaspora Journal of Mathematics. Special Issue in Memory of Prof. Ibni Oumar Mahamat Saleh, Volume 8, Issue 2, , pp. 148–162 (2009), Computationnal Approch of the Optimal Linearization of the Nonlinear O.D.E.: Application to a Nonlinear Electronic Circuit, International Journal of Computers and Electrical Engeenering ( I.J.C.E.E.), Volume 1, Juin 2009,pp.135-141,

Tayeb Benouaz
Professor Tayeb Benouaz works as a Teaching, Department of Physics, Tlemcen university. Director of Post graduation of electronic physical modeling in the same university. His current research interest includes the computational physics, modeling and simulation of the nonlinear systems, applied mathematics. Director several research projects and has several publications in this field. The last publications: \"Numerical simulation of nonlinear pulses propagation in a nonlinear optical directional coupler\", International Journal of Physical Sciences Vol. 4 (9), pp. 505-513, September, 2009.ISSN 1992 - 1950 © 2009 Academic Journals. \"Prediction model for the diffusion lengh in silicon-based solar cells\", Journal of Semiconductors ,Vol.30, N°.5, pp. 40001-1-40001-4, May 2009. “Introduction to control of solar gain and internal temperatures by thermal insulation, proper orientation and eaves” Energy and Buildings 43 (2011) 2414–2421. “ Study of energy transfer by electron cyclotron resonance in tokamak plasma” Energy Procedia 6 (2011) 194–201.

Fatiha Lassouani
Fatiha Lassouani, works as a Teaching, Department of Physics, Preparatory School of Technical Science, PSTST Tlemcen (Algeria). She prepares his Phdthesis in the field of approximation and stability of nonlinear systems.


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