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November  2017, 22(9): 3575-3589. doi: 10.3934/dcdsb.2017180

Expansivity implies existence of Hölder continuous Lyapunov function

Łukasz Struski , and  Jacek Tabor

Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

* Corresponding author: Łukasz Struski

Received  August 2016 Revised  May 2017 Published  July 2017

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.

Keywords: Lyapunov function, expansive map, Hölder continuous, cone-fields.
Mathematics Subject Classification: Primary:37D20.
Citation: Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180
References:
[1]

E. ~Akin, The General Topology of Dynamical Systems, volume 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993. Google Scholar

[2]

D. Angeli, A Lyapunov approach to the incremental stability properties, IEEE Trans. Automat. Control, 47 (2002), 410-421. doi: 10.1109/9.989067. Google Scholar

[3]

J. P. Aubin, Mutational equations in metric spaces, Set-Valued Analysis, 1 (1993), 3-46. doi: 10.1007/BF01039289. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, volume 157, Springer-Verlag, New York, 2004. doi: 10.1007/b97315. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

J. Lewowicz, Persistence of semi-trajectories, Journal of Dynamics and Differential Equations, 18 (2006), 1095-1102. doi: 10.1007/s10884-006-9047-9. Google Scholar

[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. Google Scholar

[13]

A. M. Lyapunov, The general problem of the stability of motion, International Journal of Control, 55 (1992), 521-790. doi: 10.1080/00207179208934253. Google Scholar

[14]

M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer Science & Business Media, 2009. doi: 10.1007/978-1-84882-535-2. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

S. Newhouse, Cone-fields, domination, and hyperbolicity, Modern dynamical systems and applications, (2004), 419-432. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

Ja. Tabor, Differential equations in metric spaces, Proceedings of Equadiff 10, 127 (2002), 353-360. Google Scholar

[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. Google Scholar

[24]

J. Tolosa, The method of Lyapunov functions of two variables, Contemporary Mathematics, 440 (2007), 243-271. doi: 10.1090/conm/440/08489. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

D. Angeli, A Lyapunov approach to the incremental stability properties, IEEE Trans. Automat. Control, 47 (2002), 410-421. doi: 10.1109/9.989067. Google Scholar

[3]

J. P. Aubin, Mutational equations in metric spaces, Set-Valued Analysis, 1 (1993), 3-46. doi: 10.1007/BF01039289. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, volume 157, Springer-Verlag, New York, 2004. doi: 10.1007/b97315. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

J. Lewowicz, Persistence of semi-trajectories, Journal of Dynamics and Differential Equations, 18 (2006), 1095-1102. doi: 10.1007/s10884-006-9047-9. Google Scholar

[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. Google Scholar

[13]

A. M. Lyapunov, The general problem of the stability of motion, International Journal of Control, 55 (1992), 521-790. doi: 10.1080/00207179208934253. Google Scholar

[14]

M. Malisoff and F. Mazenc, Constructions of Strict Lyapunov Functions, Springer Science & Business Media, 2009. doi: 10.1007/978-1-84882-535-2. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

S. Newhouse, Cone-fields, domination, and hyperbolicity, Modern dynamical systems and applications, (2004), 419-432. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

Ja. Tabor, Differential equations in metric spaces, Proceedings of Equadiff 10, 127 (2002), 353-360. Google Scholar

[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. Google Scholar

[24]

J. Tolosa, The method of Lyapunov functions of two variables, Contemporary Mathematics, 440 (2007), 243-271. doi: 10.1090/conm/440/08489. Google Scholar

[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. Google Scholar

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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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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