In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-di...In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desired numerical results with two different continuous higher order dynamical systems that exhibit chaotic behavior, the Lorenz equations and the Rössler attractor.展开更多
Purpose-The purpose of this paper is with delay-independent stabilization of nonlinear systems with multiple time-delays and its application in chaos synchronization of Rössler system.Design/methodology/approach-...Purpose-The purpose of this paper is with delay-independent stabilization of nonlinear systems with multiple time-delays and its application in chaos synchronization of Rössler system.Design/methodology/approach-Based on linear matrix inequality and algebra Riccati matrix equation,the stabilization result is derived to guarantee asymptotically stable and applicated in chaos synchronization of Rössler chaotic system with multiple time-delays.Findings-A controller is designed and added to the nonlinear system with multiple time-delays.The stability of the nonlinear system at its zero equilibrium point is guaranteed by applying the appropriate controller signal based on linear matrix inequality and algebra Riccati matrix equation scheme.Another effective controller is also designed for the global asymptotic synchronization on the Rössler system based on the structure of delay-independent stabilization of nonlinear systems with multiple time-delays.Numerical simulations are demonstrated to verify the effectiveness of the proposed controller scheme.Originality/value-The introduced approach is interesting for delay-independent stabilization of nonlinear systems with multiple time-delays and its application in chaos synchronization of Rössler system.展开更多
Optimal control is one of the most popular decision-making tools recently in many researches and in many areas. The Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;back...Optimal control is one of the most popular decision-making tools recently in many researches and in many areas. The Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model is one of the interesting models because of the idea of consolidation of the two models<span style="font-family:Verdana;">:</span><span style="font-family:Verdana;"> Lorenz and <span style="white-space:nowrap;"><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span><span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""></span>ssler. This paper discusses the Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model from the bifurcation phenomena and the optimal control problem (OCP). The bifurcation property at the system equilibrium <img alt="" src="Edit_128925fa-e315-4db4-b9e4-9cd999342cb9.bmp" /> </span><span style="font-family:Verdana;">is studied and it is found that saddle-node and Hopf bifurcations can be holed under some conditions on the parameters. Also, the problem of the optimal control of Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model is discussed and </span><span style="font-family:Verdana;">it </span><span style="font-family:Verdana;">u</span><span style="font-family:Verdana;">ses</span><span style="font-family:Verdana;"> the Pontryagin’s Maximum Principle (PMP) to derive the optimal control inputs that achieve the optimal trajectory. Numerical 展开更多
文摘In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desired numerical results with two different continuous higher order dynamical systems that exhibit chaotic behavior, the Lorenz equations and the Rössler attractor.
基金This work was jointly supported by Research Foundation of Department of Education of Sichuan Province(Grant Nos.14ZA0203 and 14ZB0210)Open Foundation of Enterprise Informatization and Internet of Things Key Laboratory of Sichuan Province(Grant Nos.2014WYJ01 and 2013WYY06)+2 种基金Open Foundation of Artificial Intelligence Key Laboratory of Sichuan Province(Grant Nos.2014RYY02 and 2013RYJ01)the Science Foundation of Sichuan University of Science and Engineering(Grant No.2014PY14 and 2014RC11)National Natural Science Foundation of China(Grant No.6160021729).
文摘Purpose-The purpose of this paper is with delay-independent stabilization of nonlinear systems with multiple time-delays and its application in chaos synchronization of Rössler system.Design/methodology/approach-Based on linear matrix inequality and algebra Riccati matrix equation,the stabilization result is derived to guarantee asymptotically stable and applicated in chaos synchronization of Rössler chaotic system with multiple time-delays.Findings-A controller is designed and added to the nonlinear system with multiple time-delays.The stability of the nonlinear system at its zero equilibrium point is guaranteed by applying the appropriate controller signal based on linear matrix inequality and algebra Riccati matrix equation scheme.Another effective controller is also designed for the global asymptotic synchronization on the Rössler system based on the structure of delay-independent stabilization of nonlinear systems with multiple time-delays.Numerical simulations are demonstrated to verify the effectiveness of the proposed controller scheme.Originality/value-The introduced approach is interesting for delay-independent stabilization of nonlinear systems with multiple time-delays and its application in chaos synchronization of Rössler system.
文摘Optimal control is one of the most popular decision-making tools recently in many researches and in many areas. The Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model is one of the interesting models because of the idea of consolidation of the two models<span style="font-family:Verdana;">:</span><span style="font-family:Verdana;"> Lorenz and <span style="white-space:nowrap;"><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span><span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""></span>ssler. This paper discusses the Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model from the bifurcation phenomena and the optimal control problem (OCP). The bifurcation property at the system equilibrium <img alt="" src="Edit_128925fa-e315-4db4-b9e4-9cd999342cb9.bmp" /> </span><span style="font-family:Verdana;">is studied and it is found that saddle-node and Hopf bifurcations can be holed under some conditions on the parameters. Also, the problem of the optimal control of Lorenz-R<span style="FONT-FAMILY:;COLOR: #4f4f4f" font-size:14px;white-space:normal;background-color:#ffffff;?=""><span style="color:#4F4F4F;font-family:"font-size:14px;white-space:normal;background-color:#FFFFFF;">ö</span></span>ssler model is discussed and </span><span style="font-family:Verdana;">it </span><span style="font-family:Verdana;">u</span><span style="font-family:Verdana;">ses</span><span style="font-family:Verdana;"> the Pontryagin’s Maximum Principle (PMP) to derive the optimal control inputs that achieve the optimal trajectory. Numerical
