Figure 1.
Sequences $\alpha, \alpha'\in\Phi(K,L,\varepsilon)$ restricted to the set $\text{dom}_\varepsilon(\alpha)\cap\text{dom}_\varepsilon(\alpha')=[0,n]_Z$ , $n\in{\bar{\mathbb{N}}}$
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November
2017, 22(9): 3575-3589.
doi: 10.3934/dcdsb.2017180
Expansivity implies existence of Hölder continuous Lyapunov function
Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland |
The Lyapunov function is a very useful tool in the theory of dynamical systems, in particular in the study of the stability of an equilibrium point. In this paper we construct a locally Hölder continuous Lyapunov function for a uniformly expansive set for a map f in a metric space X. In the construction a basic role is played by the functions defining the stable and unstable cone-fields. As a tool we also use the approximately quasiconvex functions.
Mathematics Subject Classification:
Primary:37D20.
Citation:
Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete and Continuous Dynamical Systems - B,
2017, 22
(9)
: 3575-3589.
doi: 10.3934/dcdsb.2017180
![]()
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Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 1188-1197.
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Persistence of semi-trajectories, Journal of Dynamics and Differential Equations, 18 (2006), 1095-1102.
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C. C. McCluskey,
Using Lyapunov functions to construct Lyapunov functionals for delay differential equations, SIAM Journal on Applied Dynamical Systems, 14 (2015), 1-24.
doi: 10.1137/140971683. |
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M. Mrozek, Topological dynamics: Rigorous numerics ia cubical homology, In Advances in Applied and Computational Topology: Proc. Symp. Amer. Math. Soc, volume 70, pages 41-73. American Mathematical Society, Providence, RI, 2012.
doi: 10.1090/psapm/070/588. |
| [19] |
S. Newhouse,
Cone-fields, domination, and hyperbolicity, Modern dynamical systems and applications, (2004), 419-432.
|
| [20] |
Ł. Struski and Ja. Tabor,
Expansivity and cone-fields in metric spaces, Journal of Dynamics and Differential Equations, 26 (2014), 517-527.
doi: 10.1007/s10884-014-9373-2. |
| [21] |
Ł. Struski, Ja. Tabor and T. Kulaga,
Cone-fields without constant orbit core dimension, Discrete and Continuous Dynamical Systems, 32 (2012), 3651-3664.
doi: 10.3934/dcds.2012.32.3651. |
| [22] |
Ja. Tabor,
Differential equations in metric spaces, Proceedings of Equadiff 10, 127 (2002), 353-360.
|
| [23] |
Ja. Tabor, Jó. Tabor and M. Żoldak,
On ω-strongly quasiconvex and ω-strongly quasiconcave sequences, Aequationes mathematicae, 82 (2011), 255-268.
doi: 10.1007/s00010-011-0094-x. |
| [24] |
J. Tolosa,
The method of Lyapunov functions of two variables, Contemporary Mathematics, 440 (2007), 243-271.
doi: 10.1090/conm/440/08489. |
| [25] |
D. Wilczak and P. Zgliczyński,
Computer assisted proof of the existence of homoclinic tangency for the Hénon map and for the forced damped pendulum, SIAM Journal on Applied Dynamical Systems, 8 (2009), 1632-1663.
doi: 10.1137/090759975. |
show all references
References:
| [1] |
E. ~Akin,
The General Topology of Dynamical Systems, volume 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993. |
| [2] |
D. Angeli,
A Lyapunov approach to the incremental stability properties, IEEE Trans. Automat. Control, 47 (2002), 410-421.
doi: 10.1109/9.989067. |
| [3] |
J. P. Aubin,
Mutational equations in metric spaces, Set-Valued Analysis, 1 (1993), 3-46.
doi: 10.1007/BF01039289. |
| [4] |
F. Forni and R. Sepulchre,
A differential lyapunov framework for contraction analysis, IEEE Trans. Automat. Control, 59 (2014), 614-628.
doi: 10.1109/TAC.2013.2285771. |
| [5] |
P. Giesl and S. Hafstein,
Review on computational methods for Lyapunov functions, Discrete and Continuous Dynamical Systems-Series B, 20 (2015), 2291-2331.
doi: 10.3934/dcdsb.2015.20.2291. |
| [6] |
J. Harjani and K. Sadarangani,
Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 1188-1197.
doi: 10.1016/j.na.2009.08.003. |
| [7] |
T. Kaczynski, K. Mischaikow and M. Mrozek,
Computational Homology, volume 157, Springer-Verlag, New York, 2004.
doi: 10.1007/b97315. |
| [8] |
A. Katok and B. Hasselblatt,
Introduction to the Modern Theory of Dynamical Systems, volume~54. Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
| [9] |
V. Lakshmikantham, T. G. Bhaskar and J. V. Devi,
Theory of Set Differential Equations in Metric Spaces, volume~2, Cambridge Scientific Publishers Cambridge, 2006. |
| [10] |
J. Lewowicz,
Lyapunov functions and topological stability, Journal of Differential Equations, 38 (1980), 192-209.
doi: 10.1016/0022-0396(80)90004-2. |
| [11] |
J. Lewowicz,
Persistence of semi-trajectories, Journal of Dynamics and Differential Equations, 18 (2006), 1095-1102.
doi: 10.1007/s10884-006-9047-9. |
| [12] |
W. Lohmiller and J.-J. E. Slotine,
On contraction analysis for non-linear systems, Automatica J. IFAC, 34 (1998), 683-686.
doi: 10.1016/S0005-1098(98)00019-3. |
| [13] |
A. M. Lyapunov,
The general problem of the stability of motion, International Journal of Control, 55 (1992), 521-790.
doi: 10.1080/00207179208934253. |
| [14] |
M. Malisoff and F. Mazenc,
Constructions of Strict Lyapunov Functions, Springer Science & Business Media, 2009.
doi: 10.1007/978-1-84882-535-2. |
| [15] |
M. Mazur,
On the relationship between hyperbolic and cone-hyperbolic structures in metric spaces, Annales Polonici Mathematici, 109 (2013), 29-38.
doi: 10.4064/ap109-1-2. |
| [16] |
M. Mazur and Ja. Tabor,
Computational hyperbolicity, Discrete and Continuous Dynamical Systems (DCDS-A), 29 (2011), 1175-1189.
doi: 10.3934/dcds.2011.29.1175. |
| [17] |
C. C. McCluskey,
Using Lyapunov functions to construct Lyapunov functionals for delay differential equations, SIAM Journal on Applied Dynamical Systems, 14 (2015), 1-24.
doi: 10.1137/140971683. |
| [18] |
M. Mrozek, Topological dynamics: Rigorous numerics ia cubical homology, In Advances in Applied and Computational Topology: Proc. Symp. Amer. Math. Soc, volume 70, pages 41-73. American Mathematical Society, Providence, RI, 2012.
doi: 10.1090/psapm/070/588. |
| [19] |
S. Newhouse,
Cone-fields, domination, and hyperbolicity, Modern dynamical systems and applications, (2004), 419-432.
|
| [20] |
Ł. Struski and Ja. Tabor,
Expansivity and cone-fields in metric spaces, Journal of Dynamics and Differential Equations, 26 (2014), 517-527.
doi: 10.1007/s10884-014-9373-2. |
| [21] |
Ł. Struski, Ja. Tabor and T. Kulaga,
Cone-fields without constant orbit core dimension, Discrete and Continuous Dynamical Systems, 32 (2012), 3651-3664.
doi: 10.3934/dcds.2012.32.3651. |
| [22] |
Ja. Tabor,
Differential equations in metric spaces, Proceedings of Equadiff 10, 127 (2002), 353-360.
|
| [23] |
Ja. Tabor, Jó. Tabor and M. Żoldak,
On ω-strongly quasiconvex and ω-strongly quasiconcave sequences, Aequationes mathematicae, 82 (2011), 255-268.
doi: 10.1007/s00010-011-0094-x. |
| [24] |
J. Tolosa,
The method of Lyapunov functions of two variables, Contemporary Mathematics, 440 (2007), 243-271.
doi: 10.1090/conm/440/08489. |
| [25] |
D. Wilczak and P. Zgliczyński,
Computer assisted proof of the existence of homoclinic tangency for the Hénon map and for the forced damped pendulum, SIAM Journal on Applied Dynamical Systems, 8 (2009), 1632-1663.
doi: 10.1137/090759975. |
Figure 2.
All possible situations in Case Ⅲ (see Figure 1(c)) depending on the position of $\tilde{n}$ , where $\text{dom}_e(\alpha)\cap\text{dom}_e(\alpha')=[0,n]_Z$
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