We establish an algebraic method and an integral method to compute the Liapunov constants and Hopf cyclicity for a general Lienard system on the plane.
In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the...In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the critical point of such nonpolynomial planar vector felds to be the focus or center. Then, using Dulac criterion, we establish some sufcient conditions for the nonexistence of limit cycles of this nonpolynomial planar vector felds. And then, according to Hopf bifurcation theory, we analyze some sufcient conditions for bifurcating limit cycles from the origin. Finally, by transforming the nonpolynomial planar vector felds into the generalized Li′enard planar vector felds, we discuss the existence, uniqueness and stability of limit cycles for the former and latter planar vector felds. Some examples are also given to illustrate the efectiveness of our theoretical results.展开更多
文摘We establish an algebraic method and an integral method to compute the Liapunov constants and Hopf cyclicity for a general Lienard system on the plane.
基金Supported by the Natural Science Foundation of Anhui Education Committee(KJ2007A003)the"211 Project"for Academic Innovative Teams of Anhui University(KJTD002B)+4 种基金the Doctoral Scientific Research Project for Anhui Medical University(XJ201022)the Key Project for Hefei Normal University(2010kj04zd)the Provincial Excellent Young Talents Foundation for Colleges and Universities of Anhui Province(2011SQRL126)the Academic Innovative Scientific Research Project of Postgraduates for Anhui University(yfc100020yfc100028)
基金Supported by the Natural Science Foundation of Anhui Education Committee(KJ2007A003)the"211 Project"for Academic Innovative Teams of Anhui University(KJTD002B)+3 种基金the Doctoral Scientifc Research Project for Anhui Medical University(XJ201022)the Key Project for Hefei Normal University(2010kj04zd)the Provincial Excellent Young Talents Foundation for Colleges and Universities of Anhui Province(2011SQRL126)the Academic Innovative Scientifc Research Project of Postgraduates for Anhui University(yfc100020,yfc100028)
文摘In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the critical point of such nonpolynomial planar vector felds to be the focus or center. Then, using Dulac criterion, we establish some sufcient conditions for the nonexistence of limit cycles of this nonpolynomial planar vector felds. And then, according to Hopf bifurcation theory, we analyze some sufcient conditions for bifurcating limit cycles from the origin. Finally, by transforming the nonpolynomial planar vector felds into the generalized Li′enard planar vector felds, we discuss the existence, uniqueness and stability of limit cycles for the former and latter planar vector felds. Some examples are also given to illustrate the efectiveness of our theoretical results.
