Nonlinear Dynamics Of Singularly Perturbed Dynamical Systems With Delays

Many investigations have been made on singularly perturbed ordinary differential systems, which indicates there are lots of particular dynamics for these systems, such as boundary layer or interior layer, relax oscillation, slow through effect, memory effect, hazardous jump, bifurcation delay, and so on. With the development of science and technology, and with increasingly strict requirements for control speed and system performance, it is shown the time delay is unavoidable in dynamic systems. However, little efforts have been made to investigate the singularly perturbed delay differential systems. Based on the singular perturbation theory, the dynamics of the singularly perturbed delay differential systems is studied in this dissertation, the particular dynamics is focused on.Delay differential systems lose its stability often through Hopf‐bifurcation, this is the phenomenon of nonlinear dynamics. During the process of Hopf‐bifurcation, the nonlinear terms can be regarded as perturbation terms due to the Hopf‐bifurcation is local dynamics, so the classical perturbation methods can be utilized to calculate the periodic solutions resulted from the Hopf‐bifurcation. The periodic solutions resulted from the Hopf‐bifurcation in an optoelectronic system with time‐delay feedback and a class of TCP/AQM network system are calculated through the method of Lindstedt‐Poincaréand the method of multiple scales respectively, the results are very compact, and are well verified through numerical simulations.Slow‐fast system is a typical singularly perturbed system, which includes different dynamical time scales, and its typical dynamics are Pulsating and Bursting. Lots of investigations have been made on slow‐fast ordinary differential systems, but few for slow‐fast delay differential systems. It is unclear how the time delay impacts the dynamics of the slow‐fast systems. To investigate the influences of the time delay, a slow‐fast optoelectronic system is considered. Studies indicate the time delay can not only lead to local instability and a series of Hopf‐bifurcation and double Hopf‐bifurcation, but also impact the time for Pulsating and the properties of the Pulsating solution, therefore, result to complex dynamics, even chaos. A two‐parameter partition diagram about the local stability is obtained through careful calculation, which indicates this system exhibits different dynamics when the parameters locate in different regions, moreover, a formula is presented for determining the first time for Pulsating.Bifurcation is a common phenomenon for dynamical systems when some bifurcation parameters vary. However, as the bifurcation parameters vary slowly with time, an interesting phenomenon is the bifurcation will be delayed. This phenomenon happens often in slow‐fast systems, where the slow variables are regarded as the slowly varying parameters. This phenomenon has been investigated intensively in slow‐fast ordinary differential systems, but little work is done for slow‐fast delay differential systems. Based on the center manifold reduction and normal form, and the geometric singular perturbation theory, and the stability switch method, the Hopf‐bifurcation delay in slow‐fast delay differential systems is investigated. General conditions for the existence of Hopf‐bifurcation delay in slow‐fast delay differential systems are presented through utilizing some results about ordinary differential systems, moreover, the expression of the entry‐exit function is obtained. Two example studies show the analytic results can predict the behaviors of these systems exactly. In particular, the close expressions of the entry‐exit function for these two example systems are obtained through using the Lambert W function.The dynamics of the Duffing‐systems including slowly‐varying parameters are investigated, which indicates the existence of the bifurcation delay, moreover, some particular dynamics are discussed. Delay differential systems lose its stability often through Hopf‐bifurcation, foregoing investigations show the Hopf‐bifurcation can be delayed if the bifurcation parameters are varying slowly with time, this phenomenon is of benefit to improving the local stability for dynamic systems. In this dissertation, we consider whether the stability can be improved through introducing some proper slowly varying parts into the bifurcation parameters. Example studies demonstrate the validity of this idea. And it is prospective to contribute to engineering.