Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T04:35:27.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems

Published online by Cambridge University Press:  20 November 2018

Cristian Carcamo  and
Claudio Vidal
Cristian Carcamo
Affiliation:
Depto. de Mat., Fac. de Ciencias, Universidad del Bíio-Bío, Casilla 5-C, Concepción, VIII-región, Chile. e-mail: [email protected], [email protected]
Claudio Vidal
Affiliation:
Depto. de Mat., Fac. de Ciencias, Universidad del Bíio-Bío, Casilla 5-C, Concepción, VIII-región, Chile. e-mail: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we study the stability in the Lyapunov sense of the equilibrium solutions of discrete or difference Hamiltonian systems in the plane. First, we perform a detailed study of linear Hamiltonian systems as a function of the parameters. In particular we analyze the regular and the degenerate cases. Next, we give a detailed study of the normal form associated with the linear Hamiltonian system. At the same time we obtain the conditions under which we can get stability (in linear approximation) of the equilibrium solution, classifying all the possible phase diagrams as a function of the parameters. After that, we study the stability of the equilibrium solutions of the first order difference system in the plane associated with mechanical Hamiltonian systems and Hamiltonian systems defined by cubic polynomials. Finally, we point out important differences with the continuous case.

Keywords

34D2034E10difference equationsHamiltonian systemsstability in the Lyapunov sense
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 6 , 01 December 2015 , pp. 1270 - 1289
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Agarwal, R. P., Difference equations and inequalities: theory, methods, and applications. Second ed., Monographs in Pure and Applied Mathematics, 228, Marcel Dekker, New York, 2000.Google Scholar
[2] Agarwal, R., Ahlbrandt, C., Bohner, M., and Peterson, A., Discrete linear Hamiltonian systems: a survey. Dynam. Systems Appl. 8(1999), no. 3-4, 307–333.Google Scholar
[3] Ahlbrandt, C. and Peterson, A., Discrete Hamiltonian systems: Difference equations, continued fractions, and Riccati equations. Kluwer Texts in Mathematical Sciences, 16, Kluwer Academic Publishers, Dordrecht, 1996.Google Scholar
[4] Ahlbrandt, C. D., Bohner, M. ,J and Ridenhour, , Hamiltonian system on time scales. J. Math. Anal. Appl. 250(2000), no. 2, 561–578. http://dx.doi.org/10.1006/jmaa.2000.6992 Google Scholar
[5] Bohner, M., Riccati matrix difference equations and linear Hamiltonian difference systems. Dynam. Contin. Discrete Impuls. Systems 2(1996), no. 2, 147–159.Google Scholar
[6] Bohner, M., Discrete linear Hamiltonian eigenvalue problems. Advances in difference equations. II. Comput. Math. Appl. 36(1998), no. 10-12, 179–192.http://dx.doi.org/10.1016/S0898-1221(98)80019-9 Google Scholar
[7] Bohner, M., Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions. J. Math. Anal. Appl. 199(1996), no. 3, 804–826.http://dx.doi.org/10.1006/jmaa.1996.0177 Google Scholar
[8] Bohner, M., Doslý, O., and Hilscher, R., Linear Hamiltonian dynamic systems on time scales: Sturmian property of the principal solution. In: Proceedings of the Third World Congress of Nonlinear Analysts, PArt 2 (Catania, 2000). Nonlinear Anal. 47(2001),no.2, 849–860.http://dx.doi.org/10.1016/S0362-546X(01)00228-0 Google Scholar
[9] Bohner, M/ and Hilscher, R., An eigenvalue problem for linear Hamiltonian dynamic systems. Fasc. Math. 35(2005), 35–49.Google Scholar
[10] Cárcamo, C. and Vidal, C., The Chetaev theorem for ordinary difference equations. Proyecciones 31(2012), no. 4, 391–402.http://dx.doi.org/10.4067/S0716-09172012000400007 Google Scholar
[11] Chen, W., Yang, M., and Ding, Y., Homoclinic orbits of first order discrete Hamiltonian systems with super linear terms. Sci. China Math. 54(2011), no. 12, 2583–2596.http://dx.doi.org/10.1007/s11425-011-4276-8 Google Scholar
[12] Chetaev, N., Stability of motion. Second ed., Pergamon Press, Oxford, 1961.Google Scholar
[13] Dirichlet, G., Uber Die Stabilitat Des Gleichgewitchts. J. Reine Angrew Math. 32(1846), 85–88.Google Scholar
[14] Došlá, Z. and Škrabáková, D., Phases of linear difference equations and symplectic systems. Math. Bohem. 128(2003), no. 3, 293–308.Google Scholar
[15] Elaydi, S. N., An introduction to diòerence equations. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1996.Google Scholar
[16] Erbe, L. and Yan, P., Disconjugacy for linear Hamiltonian difference systems. J. Math. Anal. Appl. 167(1992), 355–367.http://dx.doi.org/10.1016/0022-247X(92)90212-V Google Scholar
[17] Erbe, L. and Yan, P., Qualitative properties of Hamiltonian difference systems. J. Math. Anal. Appl. 171(1992),no. 2, 334–345.http://dx.doi.org/10.1016/0022-247X(92)90347-G Google Scholar
[18] Lagrange, J., Essai sur le problème des trois corps. Oeuvres complètes, VI, 1772, pp. 229–324.Google Scholar
[19] Lakshmikantham, V. and Trigiante, D., Theory of difference equations: Numerical methods and applications. Mathematics in Science and Engineering, 181, Academic Press INc., Boston, MA, 1988.Google Scholar
[20] LaSalle, J. P., Stability theory for difference equations. In: Studies in ordinary differential equations, Stud. in Math., 14, Math. Assoc. of America, Washington, DC, 1977, pp. 1–31.Google Scholar
[21] Laub, A. and Meyer, K., Canonical forms for symplectic and Hamiltonian matrices. Celestial Mech. 9(1974), 213–238.http://dx.doi.org/10.1007/BF01260514 Google Scholar
[22] Long, Y. and Dong, D., Normal forms of symplectic matrices. Acta Math. Sin. (Engl. Ser.) 16(2000), no. 2, 237–260.http://dx.doi.org/10.1007/s101140000048 Google Scholar
[23] Mert, R. and Zafer, A., On disconjugacy and stability criteria for discrete Hamiltonian systems. Comput. Math. Appl. 62(2011), no. 8, 3015–3026.http://dx.doi.org/10.1016/j.camwa.2011.08.013 Google Scholar
[24] Meyer, K. R., Normal forms for Hamiltonian systems. Celestial Mech. 9(1974), 517–522.http://dx.doi.org/10.1007/BF01329333 Google Scholar
[25] Meyer, K. R., Hall, H., and Offin, D., Introduction to Hamiltonian dynamical systems and the N-body problem. Applied Mathematics Sciences, 90, Springer, New York, 2009.Google Scholar
[26] Zhang, Q.-M. and Tang, X. H., Lyapunov inequalities and stability for discrete linear Hamiltonian systems. Appl. Math. Comput. 218(2011), no. z, 574–582.http://dx.doi.org/10.1016/j.amc.2011.05.101 Google Scholar
[27] Zhang, Q.-M. and Tang, X. H., Lyapunov inequalities and stability for discrete linear Hamiltonian systems. J. Difference Equ. Appl. 18(2012), no. 9, 1467–1484.http://dx.doi.org/10.1080/10236198.2011.572071 Google Scholar
[28] Zhang, Q.-M., Jiang, J., and Tang, X., Stability for planar linear discrete Hamiltonian systems with perturbations. Appl. Anal. 92(2012), no. 8, 1704–1716.http://dx.doi.org/10.1080/00036811.2012.698269 Google Scholar
[29] Zheng, Bo., Multiple periodic solutions to nonlinear discrete Hamiltonian systems. Adv. Difference Equ. 2007, Art. ID 41830.Google Scholar