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Regular level sets of Lyapunov graphs of nonsingular Smale flows on 3-manifolds
Coupled-expanding maps under small perturbations
1. | Department of Mathematics, Shandong University, Jinan, Shandong 250100, China, China |
2. | Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong S.A.R. |
References:
[1] |
A. A. Andronov and C. E. Chaikin, "Theory of Oscillations" (translated and adapted by S. Lefschetz),, Princeton Univ. Press, (1949).
|
[2] |
A. Andronov and L. Pontrjagin, Systèmes grossiers,, Dokl. Akad. Nauk. SSSR, 14 (1937), 247. Google Scholar |
[3] |
J. Awrejcewicz and M. M. Holicke, "Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods,", World Scientific Publishing Co. Pte. Ltd., (2007).
doi: 10.1142/9789812709103. |
[4] |
J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos,, Amer. Math. Monthly, 99 (1992), 332.
doi: 10.2307/2324899. |
[5] |
G. D. Birkhoff, "Dynamical Systems,", Amer. Math. Soc., (1927).
|
[6] |
L. Block and W. Coppel, "Dynamics in One Dimension, Lecture Notes in Math. Vol. 1513,", Springer-Verlag, (1992).
|
[7] |
L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimentional maps,, in, 819 (1980), 18.
|
[8] |
R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Addison-Wesley, (1989).
|
[9] |
M. Fečkan, "Topological Degree Approach to Bifurcation Problems,", Springer, (2008).
doi: 10.1007/978-1-4020-8724-0. |
[10] |
S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$-stability and $\Omega$-stability conjectures for flows,, Ann. of Math., 145 (1997), 81.
doi: 10.2307/2951824. |
[11] |
S. Hu, A proof of $C^1$ stability conjecture for three-dimensional flows,, Trans. Amer. Math. Soc., 342 (1994), 753.
doi: 10.2307/2154651. |
[12] |
B. P. Kitchens, "Symbolic Dynamics, One-sided, Two-sided and Countable State Markov Shifts,", Springer-Verlag, (1998).
|
[13] |
T. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985.
doi: 10.2307/2318254. |
[14] |
A. M. Liapunov, "The General Problems of the Stability of Motion" (translated by A. T. Fuller),, Taylor & Francis, (1992).
|
[15] |
R. Mane, A proof of the $C^1$ stability conjecture,, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161.
doi: 10.1007/BF02698931. |
[16] |
M. Misiurewicz, Horseshoes for mappings of the interval,, Bull. Acad. Polon Sci. Sér. Sci. Math., 27 (1979), 167.
|
[17] |
J. Palis and S. Smale, Structural stability theorems,, Global Analysis, 14 (1970), 223.
|
[18] |
K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications,", Kluwer Academic Publishers, (2000).
|
[19] |
M. Peixoto, On structural stability,, Ann. of Math., 69 (1959), 199.
doi: 10.2307/1970100. |
[20] |
M. Peixoto, Structural stability on two dimensional manifolds,, Topology, 2 (1962), 101.
doi: 10.1016/0040-9383(65)90018-2. |
[21] |
H. J. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1.
doi: 10.1007/BF02392506. |
[22] |
H. J. Poincaré, "Les Méthodes Nouvelles de la Mécanique Celeste, Vols. 1-3,", Gauthiers-Villars, (1892).
|
[23] |
J. Robbin, A structural stability theorem,, Ann. of Math., 94 (1971), 447.
doi: 10.2307/1970766. |
[24] |
C. Robinson, Structural stability of $C^1$ flows,, in, 468 (1975), 262.
|
[25] |
C. Robinson, Structural stability of $C^1$ diffeomorphisms,, J. Differential Equations, 22 (1976), 28.
doi: 10.1016/0022-0396(76)90004-8. |
[26] |
C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics and Chaos,", CRC Press, (1999).
|
[27] |
Y. Shi and G. Chen, Chaos of discrete dynamical systems in complete metric spaces,, Chaos Solit. Fract., 22 (2004), 555.
doi: 10.1016/j.chaos.2004.02.015. |
[28] |
Y. Shi and G. Chen, Discrete chaos in Banach spaces,, Science in China, 34 (2004), 595.
|
[29] |
Y. Shi and G. Chen, Some new criteria of chaos induced by coupled-expanding maps,, in, (2006), 28. Google Scholar |
[30] |
Y. Shi, H. Ju and G. Chen, Coupled-expanding maps and one-sided symbolic dynamical systems,, Chaos Solit. Fract., 39 (2009), 2138.
doi: 10.1016/j.chaos.2007.06.090. |
[31] |
Y. Shi and P. Yu, Study on chaos induced by turbulent maps in noncompact sets,, Chaos Solit. Fract., 28 (2006), 1165.
doi: 10.1016/j.chaos.2005.08.162. |
[32] |
S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.
doi: 10.1090/S0002-9904-1967-11798-1. |
[33] |
S. Wiggins, "Chaotic Transport in Dynamical Systems,", Springer-Verlag, (1992).
|
[34] |
X. Yang and Y. Tang, Horseshoes in piecewise continuous maps,, Chaos Solit. Fract., 19 (2004), 841.
doi: 10.1016/S0960-0779(03)00202-9. |
[35] |
X. Zhang and Y. Shi, Coupled-expanding maps for irreducible transition matrices,, Int. J. Bifurcation and Chaos, (). Google Scholar |
[36] |
X. Zhang, Y. Shi and G. Chen, $A$-coupled-expanding maps in compact sets,, submitted for publication., (). Google Scholar |
[37] |
Z. Zhang, "The Princinple of Differential Dynamics,", Scientific Publishing, (2003). Google Scholar |
show all references
References:
[1] |
A. A. Andronov and C. E. Chaikin, "Theory of Oscillations" (translated and adapted by S. Lefschetz),, Princeton Univ. Press, (1949).
|
[2] |
A. Andronov and L. Pontrjagin, Systèmes grossiers,, Dokl. Akad. Nauk. SSSR, 14 (1937), 247. Google Scholar |
[3] |
J. Awrejcewicz and M. M. Holicke, "Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods,", World Scientific Publishing Co. Pte. Ltd., (2007).
doi: 10.1142/9789812709103. |
[4] |
J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos,, Amer. Math. Monthly, 99 (1992), 332.
doi: 10.2307/2324899. |
[5] |
G. D. Birkhoff, "Dynamical Systems,", Amer. Math. Soc., (1927).
|
[6] |
L. Block and W. Coppel, "Dynamics in One Dimension, Lecture Notes in Math. Vol. 1513,", Springer-Verlag, (1992).
|
[7] |
L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one dimentional maps,, in, 819 (1980), 18.
|
[8] |
R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", Addison-Wesley, (1989).
|
[9] |
M. Fečkan, "Topological Degree Approach to Bifurcation Problems,", Springer, (2008).
doi: 10.1007/978-1-4020-8724-0. |
[10] |
S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$-stability and $\Omega$-stability conjectures for flows,, Ann. of Math., 145 (1997), 81.
doi: 10.2307/2951824. |
[11] |
S. Hu, A proof of $C^1$ stability conjecture for three-dimensional flows,, Trans. Amer. Math. Soc., 342 (1994), 753.
doi: 10.2307/2154651. |
[12] |
B. P. Kitchens, "Symbolic Dynamics, One-sided, Two-sided and Countable State Markov Shifts,", Springer-Verlag, (1998).
|
[13] |
T. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985.
doi: 10.2307/2318254. |
[14] |
A. M. Liapunov, "The General Problems of the Stability of Motion" (translated by A. T. Fuller),, Taylor & Francis, (1992).
|
[15] |
R. Mane, A proof of the $C^1$ stability conjecture,, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161.
doi: 10.1007/BF02698931. |
[16] |
M. Misiurewicz, Horseshoes for mappings of the interval,, Bull. Acad. Polon Sci. Sér. Sci. Math., 27 (1979), 167.
|
[17] |
J. Palis and S. Smale, Structural stability theorems,, Global Analysis, 14 (1970), 223.
|
[18] |
K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications,", Kluwer Academic Publishers, (2000).
|
[19] |
M. Peixoto, On structural stability,, Ann. of Math., 69 (1959), 199.
doi: 10.2307/1970100. |
[20] |
M. Peixoto, Structural stability on two dimensional manifolds,, Topology, 2 (1962), 101.
doi: 10.1016/0040-9383(65)90018-2. |
[21] |
H. J. Poincaré, Sur le problème des trois corps et les équations de la dynamique,, Acta Mathematica, 13 (1890), 1.
doi: 10.1007/BF02392506. |
[22] |
H. J. Poincaré, "Les Méthodes Nouvelles de la Mécanique Celeste, Vols. 1-3,", Gauthiers-Villars, (1892).
|
[23] |
J. Robbin, A structural stability theorem,, Ann. of Math., 94 (1971), 447.
doi: 10.2307/1970766. |
[24] |
C. Robinson, Structural stability of $C^1$ flows,, in, 468 (1975), 262.
|
[25] |
C. Robinson, Structural stability of $C^1$ diffeomorphisms,, J. Differential Equations, 22 (1976), 28.
doi: 10.1016/0022-0396(76)90004-8. |
[26] |
C. Robinson, "Dynamical Systems: Stability, Symbolic Dynamics and Chaos,", CRC Press, (1999).
|
[27] |
Y. Shi and G. Chen, Chaos of discrete dynamical systems in complete metric spaces,, Chaos Solit. Fract., 22 (2004), 555.
doi: 10.1016/j.chaos.2004.02.015. |
[28] |
Y. Shi and G. Chen, Discrete chaos in Banach spaces,, Science in China, 34 (2004), 595.
|
[29] |
Y. Shi and G. Chen, Some new criteria of chaos induced by coupled-expanding maps,, in, (2006), 28. Google Scholar |
[30] |
Y. Shi, H. Ju and G. Chen, Coupled-expanding maps and one-sided symbolic dynamical systems,, Chaos Solit. Fract., 39 (2009), 2138.
doi: 10.1016/j.chaos.2007.06.090. |
[31] |
Y. Shi and P. Yu, Study on chaos induced by turbulent maps in noncompact sets,, Chaos Solit. Fract., 28 (2006), 1165.
doi: 10.1016/j.chaos.2005.08.162. |
[32] |
S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.
doi: 10.1090/S0002-9904-1967-11798-1. |
[33] |
S. Wiggins, "Chaotic Transport in Dynamical Systems,", Springer-Verlag, (1992).
|
[34] |
X. Yang and Y. Tang, Horseshoes in piecewise continuous maps,, Chaos Solit. Fract., 19 (2004), 841.
doi: 10.1016/S0960-0779(03)00202-9. |
[35] |
X. Zhang and Y. Shi, Coupled-expanding maps for irreducible transition matrices,, Int. J. Bifurcation and Chaos, (). Google Scholar |
[36] |
X. Zhang, Y. Shi and G. Chen, $A$-coupled-expanding maps in compact sets,, submitted for publication., (). Google Scholar |
[37] |
Z. Zhang, "The Princinple of Differential Dynamics,", Scientific Publishing, (2003). Google Scholar |
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