Basins of attraction and paired Hopf bifurcations for delay differential equations with bistable nonlinearity and delay-dependent coefficient

Abstract

We consider a class of delay differential equations with bistable nonlinearity, in which the trivial equilibrium may coexist with two positive equilibria. Despite the difficulty caused by delay and bistable nonlinearity, we give a rather complete description on the dynamics including global stability, semi-stability, bistability and Hopf bifurcation. For the case where the stable trivial equilibrium coexists with a stable positive equilibrium, we obtain two delay-dependent intervals as subsets of basins of attraction of two stable equilibria. These subsets are sharp in some sense. Using delay as the bifurcation parameter, we analytically show that the number of local Hopf bifurcation values is finite and these local Hopf bifurcation values are neatly paired. A Nicholson's blowflies equation with Allee effect is used to illustrate our general results. Through this example, we show that delay can induce stability switches, symmetric transitions among multitype bistability and robust phase-transitions for long transients.