American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks
  • DCDS-B Home
  • This Issue
  • Next Article
    Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth
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 $\text{dom}_\varepsilon(\alpha)\cap\text{dom}_\varepsilon(\alpha')=[0,n]_Z$, $n\in{\bar{\mathbb{N}}}$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  All possible situations in Case Ⅲ (see Figure 1(c)) depending on the position of $\tilde{n}$, where $\text{dom}_e(\alpha)\cap\text{dom}_e(\alpha')=[0,n]_Z$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Łukasz Struski, Jacek Tabor, Tomasz Kułaga. Cone-fields without constant orbit core dimension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3651-3664. doi: 10.3934/dcds.2012.32.3651
[2]

Boris Muha. A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. Networks & Heterogeneous Media, 2014, 9 (1) : 191-196. doi: 10.3934/nhm.2014.9.191
[3]

Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197
[4]

Andrey Kochergin. A Besicovitch cylindrical transformation with Hölder function. Electronic Research Announcements, 2015, 22: 87-91. doi: 10.3934/era.2015.22.87
[5]

Walter Allegretto, Yanping Lin, Shuqing Ma. Hölder continuous solutions of an obstacle thermistor problem. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 983-997. doi: 10.3934/dcdsb.2004.4.983
[6]

Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059
[7]

Jianhai Bao, Xing Huang, Chenggui Yuan. New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts. Communications on Pure & Applied Analysis, 2019, 18 (1) : 341-360. doi: 10.3934/cpaa.2019018
[8]

Charles Pugh, Michael Shub, Amie Wilkinson. Hölder foliations, revisited. Journal of Modern Dynamics, 2012, 6 (1) : 79-120. doi: 10.3934/jmd.2012.6.79
[9]

Jinpeng An. Hölder stability of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 315-329. doi: 10.3934/dcds.2009.24.315
[10]

Jordi-Lluís Figueras, Thomas Ohlson Timoudas. Sharp $ \frac12 $-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4519-4531. doi: 10.3934/dcds.2020189
[11]

Markus Dick, Martin Gugat, Günter Leugering. A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 225-244. doi: 10.3934/naco.2011.1.225
[12]

Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015
[13]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391
[14]

Yong Chen, Hongjun Gao, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 79-98. doi: 10.3934/dcds.2014.34.79
[15]

Luis Barreira, Claudia Valls. Hölder Grobman-Hartman linearization. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 187-197. doi: 10.3934/dcds.2007.18.187
[16]

Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157
[17]

Vincent Lynch. Decay of correlations for non-Hölder observables. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 19-46. doi: 10.3934/dcds.2006.16.19
[18]

Luca Lorenzi. Optimal Hölder regularity for nonautonomous Kolmogorov equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 169-191. doi: 10.3934/dcdss.2011.4.169
[19]

Slobodan N. Simić. Hölder forms and integrability of invariant distributions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 669-685. doi: 10.3934/dcds.2009.25.669
[20]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (79)
  • HTML views (59)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences