Discontinuous dynamical system represents the Logistic retarded functional equation with two different delays
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DOI:
https://doi.org/10.26637/mjm101/006Abstract
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 BifurcationMathematics Subject Classification:
39A05, 39A28, 39A30- 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.
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