Multiple Stable Motions And Their Regions Of Attraction The Delay Induces In Nonlinear Dynamical Systems

Time delay often affects the dynamics of systems essentially which can not only affect the stability of systems, but also lead to the complex dynamics of systems. According to the character of time delay, we study the multiple attractors and their basins of attraction that the delay induces in several classical systems.First of all, the effects of time delay on the dynamics of a typical nonlinear autonomous system. The linear delayed position feedback is induced to a Van der Pol-Duffing system. By using the method of multiple scales, we obtain the closed form of the periodic solution and judge its stability theoretically. It is found that the delay can induces the coexistence of multiple periodic solutions. We classify the basins of attraction of the multiple periodic solutions by the numerical approach. The results show that time delay not only changes the stability of the origin, but also leads to the complex dynamics such as periodic motions, the coexistence of multiple periodic motions, quasi-periodic motions and chaotic motions.Secondly, the effects of the delay on the dynamics of a typical nonlinear non-autonomous system are investigated. We aim to study the effects of linear delayed position feedbacks on the steady motion of a parametrically excited system with quadratic and cubic nonlinearities. A new method is proposed to obtain the approximation of the periodic solution and predict the bifurcation direction and the stability of the bifurcating solutions. It follows from the theoretical results that the delay leads to the multiple attractors of the system. The basins of attraction are classified numerically. It is found that time delay changes the stability of the equilibrium, transforms the unstable motions to stable ones, controls the fractal boundary of the basin of attraction, and generates complex dynamical motions such as quasi-periodic motions and chaos.Since the delay induces multiple solutions, three types of delayed feedbacks, namely, the delayed position feedback (DPF), delayed velocity feedback (DVF) and delayed state feedback (DSF), are proposed to control the erosion of safe basins in a parametrically excited system, and the effects of the three types of delayed feedbacks are also compared. The time delay and the gain in the feedbacks are chosen as control parameters. The delay and gain in each of the three types of the delayed feedbacks play very important roles in affecting the boundaries of safe basins. For negative feedbacks, the delay in the three types of the delayed feedback control can only aggravate the erosion of safe basins. For positive feedbacks, the delay can be indeed used to reduce the erosion of safe basins. The delay in DPF can be used as an efficient "switch" to affect the safe basin. The delays in both DVF and DSF can be used as good approaches to control the erosion of safe basins. Comparatively, the effect of the delay in DPF is the best to expand the area and to control the erosion of safe basins when the delay is short, while the effect of the delay in DSF becomes the best as the delay grows large enough. The gain plays an important role in affecting the area of safe basins and preventing the chosen point from being erosion in the three delayed feedback controlled systems. In each of the three delayed systems, there is a critical value of the gain at which the maximum area appears in the system. When the gain is further increased to cross through the value, the basin is eroded rapidly until the area vanishes. Besides, when the gain grows larger and larger, the effect of the excitation in each of the three delayed systems on the safe basin will become weaker and weaker. Among the three types of delayed feedbacks, DPF is the best strategy to expand the area and reduce the erosion of safe basins at small values of the delay and the gain, while DSF becomes the best when the delay or the gain is large.Finally, the effect of the delay on the boundary of the basin of attraction in a Hopfield neural network system is studied. We obtain some sufficient conditions for the stability of the stationary points by the theoretical approach. Then we are able to characterize theoretically and numerically the evolution of the boundary separating the basins of attraction of two locally stable equilibrium points as the delay varies. It follows that the delay may affect the boundaries of the basins of attraction even if it does not affect the dynamics of the system. If there is self connection in the neural system, the evolution of the boundary with the delays is neither simple nor intuitive. The delays affect the shape of the boundary greatly. The numerical results verify the validity of the analytical predictions only if the delays in the self and neighbor connections are short. The longer the delays in the self and neighbor connections are, the smaller the range where the analytical and numerical boundaries agree with each other will be.The main innovations of this paper are shown as follows. Firstly, the paper proposes the definition of the basin of attraction of the delayed nonlinear system projected on a finite dimensional Euclid space. Secondly, the global dynamics of the delayed system, including the complex dynamics such as the coexistence of the multiple attractors, the fractal of the basin of attraction and chaos, is observed by the investigation of the evolution of the region of attraction with the variation of the delay in this paper. Thirdly, the stradegy of the delayed feedbacks to control the safe basin is proposed in this paper, and it is found that the delayed feebacks can be successfully used to control the erosion of the safe basin.Our investigation show that time delay can induces the coexistence of the multiple solutions, and change the nature of the basins. By the property, we can use the delayed feedback to control the erosion of safe basins and optimaze of the region of attraction for designing the network according to need.