时滞诱发的非线性动力系统多稳态运动及其吸引域特性
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摘要
时滞常常引起系统动力学行为的定性改变,如影响系统稳定性,引起系统复杂性等。本文针对时滞这一特点详细研究了在几个典型系统中时滞引起的多稳态运动,及多稳态运动下的吸引盆问题。
首先,分析了时滞对一个典型非线性自治系统动力学行为的影响规律。文章对一个Van der Pol-Duffing系统引入线性时滞位移反馈控制,通过运用多尺度法得到周期解的解析形式,并判断其稳定性。发现时滞能够引发系统多稳态运动共存现象。并数值地划分了时滞引起的系统多稳态运动的吸引盆。研究发现,时滞不仅可以改变平衡点的稳定性,还可以将不稳定的运动转化为稳态运动,和使系统产生复杂运动,如多稳态运动,概周期运动和混沌运动。
其次,分析时滞对一个典型非线性非自治系统动力学行为的影响规律。文章主要研究线性时滞位移反馈对于一个有二次和三次非线性项的参振激励系统动力学行为的影响,构造了一种解析方法得到时滞引发的共振响应解的近似解,并确定了分叉解的分叉方向及稳定性,发现时滞引起系统多吸引子共存现象。从而绘制了时滞引起的多吸引子的吸引盆。结果表明,时滞可以改变平衡点的稳定性,将不稳定的运动转化为稳态运动,控制吸引盆边界分形,而且可使系统产生复杂运动,如概周期运动和混沌运动。
利用时滞引起多解等复杂性这一特点,文章接下来运用三种时滞反馈(位移,速度和状态反馈)分别对一个典型参振激励系统安全盆侵蚀进行控制,并对比三种控制效果。反馈增益系数和时滞被作为控制参数来控制调节安全盆的侵蚀。研究发现在三类时滞反馈控制中,时滞和反馈均对影响安全盆边界有着重要作用。在负反馈的情况下,三类时滞反馈控制下时滞的增大都只能导致安全盆侵蚀的加剧。在正反馈的情况下,时滞能够被用来控制安全盆的侵蚀。在时滞位移反馈下时滞可以被当成影响系统安全盆的有效开关。而在时滞速度或状态反馈下,安全盆侵蚀能够随时滞的增大成功地抑制。相比之下,当时滞量较小时,时滞位移反馈控制下时滞对控制安全盆侵蚀的效果最好;但当时滞足够大时,时滞状态反馈下时滞对安全盆侵蚀的控制效果最好。在三类时滞反馈控制系统中,增益系数在影响安全盆面积和防止固定点的侵蚀方面也起了重要作用。每种反馈下,总有一个反馈增益的临界点,使得安全盆面积达到最大,一旦增益系数越过这个临界点,安全盆就被迅速侵蚀直至面积为零。另外还发现,当反馈增益系数越来越大时,这三类系统中参数激励对于安全盆的影响也越来越小。在三类时滞反馈中,当时滞和增益系数都较小时,时滞位移反馈是控制安全盆侵蚀和增大安全盆面积的最佳策略;但当时滞或增益足够大时,时滞状态反馈则成为控制安全盆侵蚀的最佳方法。
最后,研究了时滞对于一类典型的时滞Hopfield神经网络系统平衡点吸引盆边界的影响规律。构造了一类两维多时滞Hopfield神经网络系统平衡点与时滞无关的渐进稳定的充分条件,并解析地对吸引子吸引域边界做出了估计,同时通过数值仿真比较验证。研究发现即使时滞不影响系统的动力学行为,仍会引起吸引域边界的变化。当系统存在自连接时,系统吸引域的边界随时滞的变化既不单调也不直观;自连接和他连接的时滞量严重影响吸引域边界,只有当他连接时滞非常小的时候理论预测与数值结果才比较接近;自连接和它连接时滞越大,数值与理论结果吻合的区域就越小。
本文的创新点有:提出了时滞非线性系统吸引域向有限维欧式空间投影的概念;通过研究吸引域随时滞的变化规律,观察了时滞非线性系统的全局动力学行为,其中包括多吸引子共存,吸引域分形和混沌等复杂运动;提出及运用时滞反馈进行安全盆控制的手段和方法,结果发现时滞反馈可以实现对安全盆侵蚀的控制。
本文研究发现时滞可以引起系统多吸引子共存,且能够改变吸引盆边界,甚至引起吸引盆拓扑形态的变化。利用这一特性,可以起到控制安全盆侵蚀,和优化吸引域从而按记忆要求设计网络的作用。
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.
首先,分析了时滞对一个典型非线性自治系统动力学行为的影响规律。文章对一个Van der Pol-Duffing系统引入线性时滞位移反馈控制,通过运用多尺度法得到周期解的解析形式,并判断其稳定性。发现时滞能够引发系统多稳态运动共存现象。并数值地划分了时滞引起的系统多稳态运动的吸引盆。研究发现,时滞不仅可以改变平衡点的稳定性,还可以将不稳定的运动转化为稳态运动,和使系统产生复杂运动,如多稳态运动,概周期运动和混沌运动。
其次,分析时滞对一个典型非线性非自治系统动力学行为的影响规律。文章主要研究线性时滞位移反馈对于一个有二次和三次非线性项的参振激励系统动力学行为的影响,构造了一种解析方法得到时滞引发的共振响应解的近似解,并确定了分叉解的分叉方向及稳定性,发现时滞引起系统多吸引子共存现象。从而绘制了时滞引起的多吸引子的吸引盆。结果表明,时滞可以改变平衡点的稳定性,将不稳定的运动转化为稳态运动,控制吸引盆边界分形,而且可使系统产生复杂运动,如概周期运动和混沌运动。
利用时滞引起多解等复杂性这一特点,文章接下来运用三种时滞反馈(位移,速度和状态反馈)分别对一个典型参振激励系统安全盆侵蚀进行控制,并对比三种控制效果。反馈增益系数和时滞被作为控制参数来控制调节安全盆的侵蚀。研究发现在三类时滞反馈控制中,时滞和反馈均对影响安全盆边界有着重要作用。在负反馈的情况下,三类时滞反馈控制下时滞的增大都只能导致安全盆侵蚀的加剧。在正反馈的情况下,时滞能够被用来控制安全盆的侵蚀。在时滞位移反馈下时滞可以被当成影响系统安全盆的有效开关。而在时滞速度或状态反馈下,安全盆侵蚀能够随时滞的增大成功地抑制。相比之下,当时滞量较小时,时滞位移反馈控制下时滞对控制安全盆侵蚀的效果最好;但当时滞足够大时,时滞状态反馈下时滞对安全盆侵蚀的控制效果最好。在三类时滞反馈控制系统中,增益系数在影响安全盆面积和防止固定点的侵蚀方面也起了重要作用。每种反馈下,总有一个反馈增益的临界点,使得安全盆面积达到最大,一旦增益系数越过这个临界点,安全盆就被迅速侵蚀直至面积为零。另外还发现,当反馈增益系数越来越大时,这三类系统中参数激励对于安全盆的影响也越来越小。在三类时滞反馈中,当时滞和增益系数都较小时,时滞位移反馈是控制安全盆侵蚀和增大安全盆面积的最佳策略;但当时滞或增益足够大时,时滞状态反馈则成为控制安全盆侵蚀的最佳方法。
最后,研究了时滞对于一类典型的时滞Hopfield神经网络系统平衡点吸引盆边界的影响规律。构造了一类两维多时滞Hopfield神经网络系统平衡点与时滞无关的渐进稳定的充分条件,并解析地对吸引子吸引域边界做出了估计,同时通过数值仿真比较验证。研究发现即使时滞不影响系统的动力学行为,仍会引起吸引域边界的变化。当系统存在自连接时,系统吸引域的边界随时滞的变化既不单调也不直观;自连接和他连接的时滞量严重影响吸引域边界,只有当他连接时滞非常小的时候理论预测与数值结果才比较接近;自连接和它连接时滞越大,数值与理论结果吻合的区域就越小。
本文的创新点有:提出了时滞非线性系统吸引域向有限维欧式空间投影的概念;通过研究吸引域随时滞的变化规律,观察了时滞非线性系统的全局动力学行为,其中包括多吸引子共存,吸引域分形和混沌等复杂运动;提出及运用时滞反馈进行安全盆控制的手段和方法,结果发现时滞反馈可以实现对安全盆侵蚀的控制。
本文研究发现时滞可以引起系统多吸引子共存,且能够改变吸引盆边界,甚至引起吸引盆拓扑形态的变化。利用这一特性,可以起到控制安全盆侵蚀,和优化吸引域从而按记忆要求设计网络的作用。
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.
引文
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