Double Hopf bifurcation in a container crane model with delayed position feedback

Keywords: Neutral delayed differential equation Double Hopf bifurcation Normal form Multiple time scales Center manifold reduction a b s t r a c t In this paper, we study dynamics in a container crane model with delayed position feedback , with particular attention focused on non-resonant double Hopf bifurcation. By using multiple time scales and center manifold reduction methods, we obtain the equivalent normal forms near a double Hopf critical point in this neutral delayed differential system. Moreover, bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results. Recently, much attention has been focused on the study of delayed different ial equations, since they may exhibit complex dynamical behaviors [1–3]. Some delayed differential equations were proposed via delayed feedback scheme, such as [4–7]. In many practical problems, the changing rates of some state variables not only depend on the state values at present and earlier instances, but also are influenced by the changing rates of the state variables in the past. Thus, neutral delayed differential equation s (NDDE) have been proposed in the study of population dynamics, neural network, engineering problems, etc. [8–10]. Since then, NDDE models have attracted attentions of researchers, and some results have been obtained with focus on local stability and global asymptotic behaviors of trivial solutions [11–15]. Several interesting articles on the bifurcation theory of NDDE, such as normal form of Hopf bifurcatio n, global existence of periodic solutions, equivariant Hopf bifurcation theory, have been published [16–23]. A few papers considered the existence of positive periodic solutions in neutral delayed ecological models by using a continuatio n theorem based on coincidence degree theory or other analytical techniques [24–26]. As we all know, it is important to compute normal forms of differential equations in the study of nonlinear dynamical systems [27,28]. Multiple time scales (MTS) [29,30] and center manifold reduction (CMR) [27,28] are two useful tools for computing the normal forms of different ial equation s. Multiple time scales method is systematic and can be directly applied to the original nonlinear dynamical system, not only to ordinary different ial equation s (ODE) but also to delayed differential equations (DDE), without applicati on of center manifold theory. In fact, this approach combines the two steps involved in using center manifold theory and normal form theory into one unified procedure to obtain the normal form …