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

Shadowing near nonhyperbolic fixed points

Alexey A. Petrov 1, and  Sergei Yu. Pilyugin 2,

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.
Keywords: Lyapunov function., shadowing, Dynamical system, fixed point.
Mathematics Subject Classification: Primary: 37C5.
Citation: Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete and 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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

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

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