September  2014, 34(9): 3761-3772. doi: 10.3934/dcds.2014.34.3761

Shadowing near nonhyperbolic fixed points

1. 

Faculty of Mathematics and Mechanics St. Petersburg State University, University av., 28, 198504, St. Petersburg, Russian Federation

2. 

Faculty of Mathematics and Mechanics, St. Petersburg State University, University av. 28, 198504, St. Petersburg, Russian Federation

Received  February 2013 Revised  December 2013 Published  March 2014

We use Lyapunov type functions to find conditions of finite shadowing in a neighborhood of a nonhyperbolic fixed point of a one-dimensional or two-dimensional homeomorphism or diffeomorphism. A new concept of shadowing in which we control the size of one-step errors is introduced in the case of a nonisolated fixed point.
Citation: Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761
References:
[1]

S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999.  Google Scholar

[2]

K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Mathematics and its Applications, 501. Kluwer Academic Publishers, Dordrecht, 2000.  Google Scholar

[3]

S. Y. Pilyugin, Theory of pseudo-orbit shadowing in dynamical systems, Differential Equations, 47 (2011), 1929-1938. doi: 10.1134/S0012266111130040.  Google Scholar

[4]

S. M. Hammel, J. A. Yorke and C. Grebogi, Numerical orbits of chaotic dynamical processes represent true orbits, Bull. Amer. Math. Soc., 19 (1988), 465-469. doi: 10.1090/S0273-0979-1988-15701-1.  Google Scholar

[5]

J. Kennedy, James A. Yorke, Shadowing in higher dimensions, Progress in Nonlinear Differential Equations and Their Applications Volume, 75 (2008), 241-246. doi: 10.1007/978-3-7643-8482-1_19.  Google Scholar

[6]

J. Lewowicz, Lyapunov functions and topological stability, J. Differential Equations, 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.  Google Scholar

[7]

A. A. Petrov and S. Y. Pilyugin, Lyapunov functions, shadowing, and topological stability, Topol. Methods Nonlin. Anal. (2014). Google Scholar

[8]

S. Tikhomirov, Holder Shadowing on Finite Intervals,, , ().   Google Scholar

show all references

References:
[1]

S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999.  Google Scholar

[2]

K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Mathematics and its Applications, 501. Kluwer Academic Publishers, Dordrecht, 2000.  Google Scholar

[3]

S. Y. Pilyugin, Theory of pseudo-orbit shadowing in dynamical systems, Differential Equations, 47 (2011), 1929-1938. doi: 10.1134/S0012266111130040.  Google Scholar

[4]

S. M. Hammel, J. A. Yorke and C. Grebogi, Numerical orbits of chaotic dynamical processes represent true orbits, Bull. Amer. Math. Soc., 19 (1988), 465-469. doi: 10.1090/S0273-0979-1988-15701-1.  Google Scholar

[5]

J. Kennedy, James A. Yorke, Shadowing in higher dimensions, Progress in Nonlinear Differential Equations and Their Applications Volume, 75 (2008), 241-246. doi: 10.1007/978-3-7643-8482-1_19.  Google Scholar

[6]

J. Lewowicz, Lyapunov functions and topological stability, J. Differential Equations, 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.  Google Scholar

[7]

A. A. Petrov and S. Y. Pilyugin, Lyapunov functions, shadowing, and topological stability, Topol. Methods Nonlin. Anal. (2014). Google Scholar

[8]

S. Tikhomirov, Holder Shadowing on Finite Intervals,, , ().   Google Scholar

[1]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

[2]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297

[3]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017

[4]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709

[5]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271

[6]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692

[7]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979

[8]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381

[9]

Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115

[10]

Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467

[11]

Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621

[12]

Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure & Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645

[13]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248

[14]

Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273

[15]

Ruhua Wang, Senjian An, Wanquan Liu, Ling Li. Fixed-point algorithms for inverse of residual rectifier neural networks. Mathematical Foundations of Computing, 2021, 4 (1) : 31-44. doi: 10.3934/mfc.2020024

[16]

Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial & Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487

[17]

Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793

[18]

Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843

[19]

Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881

[20]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4927-4962. doi: 10.3934/dcdsb.2020320

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]