Study Of Chaos And Hopf Bifurcation Based On Simulated Circuit

Chaos and bifurcation are typical phenomena in nonlinear dynamic systems. Researchers commonly adopt nonfigurative numerical analysis as a method of study in this field. Because almost kinds of dynamic systems can be perfectly demonstrated by nonlinear circuits, the nonlinear circuits naturally came to be a popular tool for analysis of chaos and bifurcation problems in nonlinear dynamics. With the developments of numerical simulation technology, the circuit simulation platform and experimental circuit were also introduced to observe phenomenon of chaos and bifurcation conveniently.For the multi-parameter and high-order nonlinear discrete dynamic system, the critical parameter value of Hopf bifurcation is commonly obtained through calculation of eigenvalues of Jacobian matrix in classical theories. It is required to confirm whether a pair of conjugate complex eigenvalue crosses the unit circle at a non-speed and the other eigenvalues remain inside the unit disk or not when parameters change. Thus, this procedure costs large amount of computational time in practice. Subject to this bottleneck, an algebra criterion derived form Schur-Cohn principle is applied for determing the critical parameter value of Hopf bifurcation in this study. The corresponding amount of computational time could be decreased by the algebra criterion. Additionally, the stability and direction of Hopf bifurcation is analyized by projection method. Based on the theoretical analysis, analogical electronic of Hénon map and Lorenz system was simulated by the software PSpice, finally, the practical circuit for Hopf bifurcation of Hénon map is constructed for validating of corresponding theories.This paper is composed of five chapters.1^st chapter: Divinations of chaos and bifurcation and according theories were introduced first. Additionally, developments and literatures in this field were also summarized.2^rd chapter: The traditional circuit simulation system PSpice was simply introduced. Some simulation experiments such as chaos of Lorenz system and hyperchaos of three-order Hénon map were performed based on the PSpice simulation platform. The corresponding results of simulation were match with analytic solutions.3^th chapter: An algebra criterion of Hopf bifurcation was employed for a discrete problem. The critical parameter of Hopf bifurcation of three-order Hénon map was studied theoretically by the algebra algorithm, at same time projection method was applied to analysis stability and simulation was also performed on PSpice simulation platform.4^th Chapter: Experiment of three-order Hénon system was completed on hardware platform, details of corresponding electronic components was also introduced. The real signal from oscillograph agreed with the corresponding analytic results. Therefore, PSpice simulation provides vision evidences for validation successfully.5^th Chapter: Status feedback controller module was built according to the pole placement method in Lorenz system producing phenomenon of Hopf bifurcation, and the controllable circuit based on Lorenz theory was also constructed, consequently, the corresponding simulation model was established and obtained Hopf bifurcation phenomenon, this results were in accordance with the theoretical prediction.The results from this investigation suggest that:An algebra criterion of Hopf bifurcation derived from on Schur-Cohn theory is effective and simple for nonlinear discrete dynamic system.The simulation of chaos and bifurcations phenomena based on PSpice not only provides reliable evidences for analysis,but also saves amount of work due to construction of practical electronic circuits. Therefore, it is an effective means of verification for corresponding theories of chaos and bifurcations.The practical circuits were constructed and observed for probing deep and adding to our understanding of nonlinear dynamic theories and corresponding applications in practical engineering.Feedback controller based on pole placement was designed to make behaviors of system satisfy expectation performance indices prior. Consequently, the continuous Lorenz system was controlled to produce Hopf bifurcation dynamic behavior by developed controller.