Oriented Shadowing Property and \(\Omega \)-Stability for Vector Fields
- 247 Downloads
We call that a vector field has the oriented shadowing property if for any \(\varepsilon >0\) there is \(d>0\) such that each \(d\)-pseudo orbit is \(\varepsilon \)-oriented shadowed by some real orbit. In this paper, we show that the \(C^1\)-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the \(\Omega \)-stability.
KeywordsVector fields Oriented shadowing \(\Omega \)-stability
Mathematics Subject Classification37C50
Shaobo Gan is supported by 973 project 2011CB808002, NSFC 11025101 and 11231001. Ming Li is partially supported by the National Science Foundation of China (Grant No. 11201244), the Specialized Research Fund for the Doctoral Program of Higher Education of China (SRFDP, Grant No. 20120031120024), and the LPMC of Nankai University. Sergey Tikhomirov is partially supported by Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026, JSC “Gazprom neft”, by the Saint-Petersburg State University research grant 188.8.131.524 and by the German-Russian Interdisciplinary Science Center (G-RISC) funded by the German Federal Foreign Office via the German Academic Exchange Service (DAAD).
- 2.Arbieto, A., Reis, J.E., Ribeiro, R.: On Various Types of Shadowing for Geometric Lorenz Flows, http://arxiv.org/abs/1306.2061
- 8.Guchenheimer, J.: A strange, strange attractor. The Hopf bifurcation theorems and its applications. Applied Mathematical Series. Springer, Berlin (1976)Google Scholar
- 10.Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and Its Applications, vol. 54. Cambridge University Press, Cambridge (1995)Google Scholar
- 19.Morimoto, A.: The method of pseudo-orbit tracing and stability of dynamical systems. Sem. Note, Tokyo University, vol. 39 (1979)Google Scholar
- 23.Pilyugin, S.Y.: Shadowing in Dynamical Systems. Lecture Notes in Math, vol. 1706. Springer, Berlin (1999)Google Scholar
- 33.Shilnikov, L. P., Shilnikov, A. L., Taruve, D. V., Chua, L. O.: Methods of qualitative theory in nonlinear dynamics. World Scientific Series A, vol. 5Google Scholar
- 36.Tikhomirov, S.: An example of a vector field with the oriented shadowing property. http://arxiv.org/abs/1403.7378