Normal Forms For A Type Of Delayed Dynamic Systems And Their Applications

Due to the ubiquitous existence of “delay” effect in nature and engineering systems, there hasbeen an increasing interest in delayed dynamic systems. A wide type of delayed dynamic systems canbe modeled by Retarded Delay Differential Equations (RDDEs) and Neutral Delay DifferentialEquations (NDDEs). Therefore, the main objects of this study are dynamical systems characterized byRDDEs and NDDEs with constant delays.The infinite dimensionality renders the analysis of Delay Differential Equations (DDEs) muchmore difficult than that of Ordinary Differential Equations (ODEs). Normal form theory is one of themost powerful tools for studying the local dynamics around a nonhyperbolic equilibrium. However,when it comes to applying the normal form theory on DDEs, complicated and time-consumingalgebra deduction is often involved. This study presents symbolic computation schemes and acorresponding Maple program for computing the normal forms of Hopf bifurcations in RDDEs andNDDEs with bifurcating parameters. The schemes have the center manifold reduction and normalform calculation processed at the same time, not like the existing ones, which firist comput the centermainfold and then derive the normal form of the dynamic equation on the center manifold. The Mapleprogram requires no knowledge of the normal form theory. It will provide the normal form of Hopfbifurcation of required order with only the basic information of the system equation being input.This study also applies two different ways of Method of Multiple Scales (MMS), which arelabeled as MMS1and MMS2, respectively, to studying the Hopf bifurcation of DDEs. By comparingthe normal forms obtained by MMS1, MMS2and the Maple program, the following conclusions canbe derived.1) The normal forms derived via MMS1are in a full agreement with that obtained by the normalform theory, which means MMS1actully projects the system on the center manifold for the purposeof computing the normal forms.2) The normal forms obtained via MMS2and the normal form theory differ from each other.However, both results describe the same dynamics around the bifurcation point. The comparisionreveals that MMS2does not project the system on the center manifold but on some moving manifold.When the terms related with delay are comparatively small and the frequency of the bifurcatedperiodic motion doesn’t vary a lot, one can use MMS2to get the global view of the Hopf bifurcation.To illustrate the power of the Maple program and MMS, this paper presents a detailed study of Hopf bifurcations of a van der Pol system with delayed state feedback (an RDDE) and a three-orderapproximated nonlinear model of a crane container with delayed displacement feedback (an NDDE).The results show that Hopf bifurcated periodic motions persist when the bifurcation parameter variesfar from the bifurcation point. The Hopf bifurcation analysis also suggests how to choose feedbackgain and time delay as the subcritical Hopf bifurcation should be avoided in the crane containersystem.Moreover, to verify the analytical analysis, some related numerical softwares, such asDDE-BIFTOOL and RADAR5, are used to conduct numerical analysis. The results indicate that boththe normal form theory and MMS are very powerful methods for analyzing local dynamics of DDEsand the Maple program is an effective and trustable tool and can be expected as the basis of a futuresymbolic software.