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Hopf bifurcation analysis of a general non-linear differential equation with delay. (English) Zbl 1402.34074

Summary: This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson-Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book of B. D. Hassard et al. [Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press (1981; Zbl 0474.34002)]. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincaré normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations.

MSC:

34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
37G10 Bifurcations of singular points in dynamical systems

Citations:

Zbl 0474.34002
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