On the Dynamics of the Logistic Delay Differential Equation with ‎Two Different Delays

Document Type : Research Paper


1 Faculty of Science, Alexandria University, Alexandria, Egypt

2 Faculty of Education, Alexandria University, Alexandria, Egypt


Here, we study the logistic delay differential equation with two different delays. First of all, we disscuse the local stability and Hopf bifurcation conditons. The method of steps is used to get a discretized analogue of the original system. Local stability and bifurcation analysis of the discretized system is investigated. Finally, we carry out some numerical simulations such as bifurcation diagram, Lyapunov exponent and phase portraits to verify the theoretical results and to illustrate complex dynamics of the considered system. 


Main Subjects

[1] Tunç, C., Tunç, O., Qualitative analysis for a variable delay system of differential equations of second order, Journal of Taibah University for Science, 13(1), 2019, 468–477.
[2] Tunç, C., A note on the stability and boundedness of solutions to non-linear differential systems of second order, Journal of the Association of Arab Universities for Basic and Applied Sciences, 24, 2017, 169-175.
[3] Tunç, C., Tunç, O., On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order, Journal of Advanced Research, 7(1), 2016, 165-168.
[4] Lin, X., Wang, H., Stability analysis of delay differential equations with two discrete delays, Canadian Applied Mathematics Quarterly, 20(4), 2012, 519-533.
[5] MacDonald, N., Two delays may not destabilize although either can delay, Mathematical Biosciences, 82(2), 2006, 127-140.
[6] Biswas, D., Banerjee, T., Time-delayed chaotic dynamical systems from theory to electronic experiment, Springer, New York, 2018.
[7] Ruany, S., Weiz, J., On the zeros of transcendental functions with application to stability of differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 10(6), 2003, 863-874.
[8] El-Sayed, A.M.A., Nasr, M.E., Discontinuous dynamical system represents the Logistic retarded functional equation with two different delays, Malaya Journal of Matematik, 1(1), 2013, 50–56.
[9] Beuter, A., Larocque, D., Glass, L., Complex oscillations in a human motor system, Journal of Motor Behavior, 10, 1989, 277-289.
[10] Braddock, R.D., van den Driessche, P., On a two lag differential delay equation, Australian Mathematical Society, Ser. B, 24, 1983, 292-317.
[11] Gopalsamy, K., Global stability in the delay-logistic equation with discrete delays, Houston Journal of Mathematics, 16(3), 1990, 347-356.
[12] Hale, J. K., Huang, W., Global geometry of the stable regions for two delay differential equations, Journal of Mathematical Analysis and Application, 178(2), 1993, 344-362.
[13] Hale, J. K., Tanaka, S. M., Square and pulse waves with two delays, Journal of Dynamics and Differential Equations, 12, 2000, 1-30.
[14] Hassard, B. D., Counting roots of the characteristic equation for linear delay-differential systems, Journal of Differential Equations, 136(2), 1997, 222-235.
[15] Li, X., Ruan, S., Wei, J., Stability and bifurcation in delay-differential equations with two delays, Journal of Mathematical Analysis and Application, 236(2), 1999, 254-280.
[16] Ragazzo, C. G., Malta, C. P., Singularity structure of the Hopf bifurcation surface of a differential equation with two delays, Journal of Dynamics and Differential Equations, 4, 1992, 617-650.
[17] Almusharrf, A.H., Delay differential equations and the logistic equation with two delays, Ph.D. Thesis, Oakland University, Rochester, Michigan, 2017.
[18] Jiang, M., Shena, Y., Jian, J., Liao, X., Stability, bifurcation and a new chaos in the logistic differential equation with delay, Physics Letters A, 350, 2006, 221–227.
[19] Parrott, L., Complexity and the limits of ecological engineering, American Society of Agricultural Engineers, 45(5), 2002, 0001–2351.
[20] Mitsch, W. J., Ecological Engineering: A New Paradigm for Engineers and Ecologists, in: Engineering Within Ecological Constraints, Edited by Schulze, P.C., the National Academy of Sciences, 1996, 111-129.
[21] Focke, W. W., Van der Westhuizen, I., Musee, N., Loots, M. T., Kinetic interpretation of log-logistic dose-time response curves, Scientific Reports, 7, 2234, 2017.
[22] Luenberger, D.G., Introduction to dynamic systems: Theory, models and applications, John Wiley Sons, New York, 1979.
[23] Hale, J. K., Verduyn Lunel, S. M., Introduction to functional differential equations, Springer, New York, 1993.
[24] Driver, R. D., Ordinary and delay differential equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1977.
[25] Kuznetsov, Y., Elements of Applied Bifurcation Theory, Third ed., Springer-Verlag, New York, 2004.
[26] Albert , C. J. L., Regularity and complexity in dynamical systems, Springer, New York, 2012.