September  2013, 33(9): 4187-4205. doi: 10.3934/dcds.2013.33.4187

Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory

1. 

Chebyshev laboratory, Saint Petersburg State University, 14th line of Vasiljevsky Island, 29B, Saint-Petersburg, 199178, Russian Federation

Received  February 2012 Revised  February 2013 Published  March 2013

We generalize two classical results of Maizel and Pliss that describe relations between hyperbolicity properties of linear system of difference equations and its ability to have a bounded solution for every bounded inhomogeneity. We also apply one of this generalizations in shadowing theory of diffeomorphisms to prove that some sort of limit shadowing is equivalent to structural stability.
Citation: Dmitry Todorov. Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4187-4205. doi: 10.3934/dcds.2013.33.4187
References:
[1]

A. G. Baskakov, On the invertibility and the Fredholm property of difference operators, Mat. Zametki, 67 (2000), 816-827. doi: 10.1007/BF02675622.

[2]

_______, Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations, Izvestiya: Mathematics, 73 (2009), 215-278. doi: 10.1070/IM2009v073n02ABEH002445.

[3]

M. S. Bichegkuev, On conditions for invertibility of difference and differential operators in weight spaces, Izvestiya: Mathematics, 75 (2011), 665-680. doi: 10.1070/IM2011v075n04ABEH002548.

[4]

W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978.

[5]

Yu. L. Dalecki and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Spaces," Moscow, 1970.

[6]

A. Fakhari, K. Lee and K. Tajbakhsh, Diffeomorphisms with $L^p$-shadowing property, Acta Math. Sin. (Engl. Ser.), 27 (2011), 19-28. doi: 10.1007/s10114-011-0050-7.

[7]

Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, Journal of Operator Theory, 58 (2007), 387-414.

[8]

Yu. Latushkin and Yu. Tomilov, Fredholm properties of evolution semigroups, Illinois Journal of Mathematics, 48 (2004), 999-1020.

[9]

R. Mañé, Characterizations of as diffeomorphisms, in "Geometry and Topology" (eds. Jacob Palis and Manfredo do Carmo), Lecture Notes in Mathematics, Vol. 597, Springer Berlin, (1977), 389-394.

[10]

A. D. Maĭzel, On stability of solutions of systems of differential equations, (Russian) Ural. Politehn. Inst. Trudy, 51 (1954), 20-50.

[11]

M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst., 9 (2003), 383-397.

[12]

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

[13]

O. Perron, Die stabilittsfrage bei differentialgleichungen, (German) Mathematische Zeitschrift, 32 (1930), 703-728. doi: 10.1007/BF01194662.

[14]

S. Pilyugin, G. Vol'fson and D. Todorov, Dynamical systems with Lipschitz inverse shadowing properties, Vestnik St. Petersburg University: Mathematics, 44 (2011), 208-213. doi: 10.3103/S106345411103006X.

[15]

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

[16]

_______, Generalizations of the notion of hyperbolicity, Journal of Difference Equations and Applications, 12 (2006), 271-282. doi: 10.1080/10236190500489350.

[17]

_______, Sets of dynamical systems with various limit shadowing properties, Journal of Dynamics and Differential Equations, 19 (2007), 747-775. doi: 10.1007/s10884-007-9073-2.

[18]

S. Yu Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009.

[19]

V. A. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations, in "Problems of Asymptotic Theory of Nonlinear Oscillations" (Russian), Naukova Dumka, Kiev, (1977), 168-173.

[20]

K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.

[21]

A. L. Sasu and B. Sasu, Translation invariant spaces and asymptotic properties of variational equations, Abstract and Applied Analysis, 2011 (2011). doi: 10.1155/2011/539026.

[22]

B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478. doi: 10.1016/j.jmaa.2005.12.002.

[23]

B. Sasu and A. L. Sasu, Exponential dichotomy and $(l^p,l^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047.

[24]

Sergey Tikhomirov, Hölder shadowing and structural stability, preprint, arXiv:1106.4053v1.

show all references

References:
[1]

A. G. Baskakov, On the invertibility and the Fredholm property of difference operators, Mat. Zametki, 67 (2000), 816-827. doi: 10.1007/BF02675622.

[2]

_______, Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations, Izvestiya: Mathematics, 73 (2009), 215-278. doi: 10.1070/IM2009v073n02ABEH002445.

[3]

M. S. Bichegkuev, On conditions for invertibility of difference and differential operators in weight spaces, Izvestiya: Mathematics, 75 (2011), 665-680. doi: 10.1070/IM2011v075n04ABEH002548.

[4]

W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978.

[5]

Yu. L. Dalecki and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Spaces," Moscow, 1970.

[6]

A. Fakhari, K. Lee and K. Tajbakhsh, Diffeomorphisms with $L^p$-shadowing property, Acta Math. Sin. (Engl. Ser.), 27 (2011), 19-28. doi: 10.1007/s10114-011-0050-7.

[7]

Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, Journal of Operator Theory, 58 (2007), 387-414.

[8]

Yu. Latushkin and Yu. Tomilov, Fredholm properties of evolution semigroups, Illinois Journal of Mathematics, 48 (2004), 999-1020.

[9]

R. Mañé, Characterizations of as diffeomorphisms, in "Geometry and Topology" (eds. Jacob Palis and Manfredo do Carmo), Lecture Notes in Mathematics, Vol. 597, Springer Berlin, (1977), 389-394.

[10]

A. D. Maĭzel, On stability of solutions of systems of differential equations, (Russian) Ural. Politehn. Inst. Trudy, 51 (1954), 20-50.

[11]

M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst., 9 (2003), 383-397.

[12]

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

[13]

O. Perron, Die stabilittsfrage bei differentialgleichungen, (German) Mathematische Zeitschrift, 32 (1930), 703-728. doi: 10.1007/BF01194662.

[14]

S. Pilyugin, G. Vol'fson and D. Todorov, Dynamical systems with Lipschitz inverse shadowing properties, Vestnik St. Petersburg University: Mathematics, 44 (2011), 208-213. doi: 10.3103/S106345411103006X.

[15]

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

[16]

_______, Generalizations of the notion of hyperbolicity, Journal of Difference Equations and Applications, 12 (2006), 271-282. doi: 10.1080/10236190500489350.

[17]

_______, Sets of dynamical systems with various limit shadowing properties, Journal of Dynamics and Differential Equations, 19 (2007), 747-775. doi: 10.1007/s10884-007-9073-2.

[18]

S. Yu Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009.

[19]

V. A. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations, in "Problems of Asymptotic Theory of Nonlinear Oscillations" (Russian), Naukova Dumka, Kiev, (1977), 168-173.

[20]

K. Sakai, Pseudo-orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.

[21]

A. L. Sasu and B. Sasu, Translation invariant spaces and asymptotic properties of variational equations, Abstract and Applied Analysis, 2011 (2011). doi: 10.1155/2011/539026.

[22]

B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478. doi: 10.1016/j.jmaa.2005.12.002.

[23]

B. Sasu and A. L. Sasu, Exponential dichotomy and $(l^p,l^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047.

[24]

Sergey Tikhomirov, Hölder shadowing and structural stability, preprint, arXiv:1106.4053v1.

[1]

Davor Dragičević. Admissibility, a general type of Lipschitz shadowing and structural stability. Communications on Pure and Applied Analysis, 2015, 14 (3) : 861-880. doi: 10.3934/cpaa.2015.14.861

[2]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901

[3]

Augusto Visintin. Weak structural stability of pseudo-monotone equations. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2763-2796. doi: 10.3934/dcds.2015.35.2763

[4]

Jonathan Meddaugh. Shadowing as a structural property of the space of dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2439-2451. doi: 10.3934/dcds.2021197

[5]

Andrejs Reinfelds, Klara Janglajew. Reduction principle in the theory of stability of difference equations. Conference Publications, 2007, 2007 (Special) : 864-874. doi: 10.3934/proc.2007.2007.864

[6]

Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123

[7]

Jean Lerbet, Noël Challamel, François Nicot, Félix Darve. Kinematical structural stability. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 529-536. doi: 10.3934/dcdss.2016010

[8]

Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963

[9]

Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527

[10]

M'hamed Kesri. Structural stability of optimal control problems. Communications on Pure and Applied Analysis, 2005, 4 (4) : 743-756. doi: 10.3934/cpaa.2005.4.743

[11]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577

[12]

Cónall Kelly, Alexandra Rodkina. Constrained stability and instability of polynomial difference equations with state-dependent noise. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 913-933. doi: 10.3934/dcdsb.2009.11.913

[13]

Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315

[14]

José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873

[15]

M. Zuhair Nashed, Alexandru Tamasan. Structural stability in a minimization problem and applications to conductivity imaging. Inverse Problems and Imaging, 2011, 5 (1) : 219-236. doi: 10.3934/ipi.2011.5.219

[16]

Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257

[17]

Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, Javier Gómez-Serrano. Structural stability for the splash singularities of the water waves problem. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 4997-5043. doi: 10.3934/dcds.2014.34.4997

[18]

Zayd Hajjej, Mohammad Al-Gharabli, Salim Messaoudi. Stability of a suspension bridge with a localized structural damping. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1165-1181. doi: 10.3934/dcdss.2021089

[19]

Jihoon Lee, Ngocthach Nguyen. Flows with the weak two-sided limit shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4375-4395. doi: 10.3934/dcds.2021040

[20]

Gregory Berkolaiko, Cónall Kelly, Alexandra Rodkina. Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations. Conference Publications, 2011, 2011 (Special) : 163-173. doi: 10.3934/proc.2011.2011.163

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (93)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]