Discontinuous dynamical system represents the Logistic retarded functional equation with two different delays

A. M. A. El-Sayed; M. E. Nasr

doi:10.26637/mjm101/006

Abstract

In this work we are concerned with the discontinuous dynamical system representing the problem of the logistic retarded functional equation with two different delays,
$\begin{aligned} & x(t)=\rho x\left(t-r_1\right)\left[1-x\left(t-r_2\right)\right], \quad t \in(0, T], \\ & x(t)=x_0, \quad t \leq 0 . \end{aligned}$
The existence of a unique solution $x \in L^1[0, T]$ which is continuously dependence on the initial data, will be proved. The local stability at the equilibrium points will be studied. The bifurcation analysis and chaos will be discussed.

Keywords:

Logistic functional equation, existence, uniqueness, equilibrium points, local stability, Chaos and Bifurcation

Mathematics Subject Classification:

39A05, 39A28, 39A30

Authors Imprint References Funding Related Articles Downloads

A. M. A. El-Sayed Department of Mathematics, Faculty of Science, Alexandria University,Alexandria, Egypt.
M. E. Nasr Department of Mathematics,Faculty of Science, Benha University, Benha 13518, Egypt.

Pages: 50-56
Date Published: 01-01-2013
Vol. 1 No. 01 (2013): Malaya Journal of Matematik (MJM)

L. Berezansky and E. Braverman, On stability of some linear and nonlinear delay differential equations, J. Math. Anal. Appl., 314(2006), 9-15. DOI: https://doi.org/10.1016/j.jmaa.2005.03.103

A. El-Sayed, A. El-Mesiry and H. EL-Saka, On the fractional-order logistic equation, Applied Mathematics Letters, 20(2007), 817-823. DOI: https://doi.org/10.1016/j.aml.2006.08.013

A. M. A. El-Sayed and M. E. Nasr, Existence of uniformly stable solutions of nonautonomous discontinuous dynamical systems, J. Egypt Math. Soc.,19(1)(2011),10-16. DOI: https://doi.org/10.1016/j.joems.2011.09.006

A. M. A. El-Sayed and M. E. Nasr, On some dynamical properties of discontinuous dynamical systems, American Academic & Scholarly Research Journal, 2(1)(2012), 28-32. DOI: https://doi.org/10.26634/jmat.1.1.1667

A. M. A. El-Sayed and M. E. Nasr, Dynamic properties of the predator-prey discontinuous dynamical system, J. Z. Naturforsch. A, 67a(2012), 57-60. DOI: https://doi.org/10.5560/zna.2011-0051

H. EL-Saka, E. Ahmed, M. I. Shehata, A. M. A. El-Sayed, On stability, persistence, and Hopf bifurcation in fractional order dynamical systems, Nonlinear Dyn, 56(2009), 121-126. DOI: https://doi.org/10.1007/s11071-008-9383-x

J. Hale and S.M. Verduyn Lunel, Introduction to functional differential equations, springrverlag, New york, 1993. DOI: https://doi.org/10.1007/978-1-4612-4342-7

Y. Hamaya and A. Redkina, On global asymptotic stability of nonlinear stochastic difference equations with delays, Internat. J.Diff., 1(2006), 101-118.

Y. Kuang, Dely differential equations with applications in population dynamics, Academic Press, New York, 1993.

J.D. Murray, Mathematical Biology I: An Introduction, Springer-Verlag, Berlin, 3rd edition, 2002.