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Journal of Modern Dynamics (JMD)
 

On the singular-hyperbolicity of star flows

Pages: 191 - 219, Issue 2, June 2014      doi:10.3934/jmd.2014.8.191

 
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Yi Shi - School of Mathematical Sciences, Peking University, Beijing 100871, China and Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21000, France (email)
Shaobo Gan - School of Mathematical Sciences, Peking University, Beijing 100871, China (email)
Lan Wen - School of Mathematic Sciences, Peking University, Beijing, 100871, China (email)

Abstract: We prove for a generic star vector field $X$ that if, for every chain recurrent class $C$ of $X$, all singularities in $C$ have the same index, then the chain recurrent set of $X$ is singular-hyperbolic. We also prove that every Lyapunov stable chain recurrent class of a generic star vector field is singular-hyperbolic. As a corollary, we prove that the chain recurrent set of a generic 4-dimensional star flow is singular-hyperbolic.

Keywords:  Singular-hyperbolicity, star flow, Lyapunov stable class, shadowing.
Mathematics Subject Classification:  Primary: 37D30; Secondary: 37D50.

Received: September 2013;      Available Online: November 2014.

 References