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# Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems

## Abstract

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.

## References

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[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.
[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.
[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.
[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
[5] Bohner, M., Riccati matrix difference equations and linear Hamiltonian difference systems. Dynam. Contin. Discrete Impuls. Systems 2(1996), no. 2, 147–159.
[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
[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
[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
[9] Bohner, M/ and Hilscher, R., An eigenvalue problem for linear Hamiltonian dynamic systems. Fasc. Math. 35(2005), 35–49.
[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
[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
[12] Chetaev, N., Stability of motion. Second ed., Pergamon Press, Oxford, 1961.
[13] Dirichlet, G., Uber Die Stabilitat Des Gleichgewitchts. J. Reine Angrew Math. 32(1846), 85–88.
[14] Došlá, Z. and Škrabáková, D., Phases of linear difference equations and symplectic systems. Math. Bohem. 128(2003), no. 3, 293–308.
[15] Elaydi, S. N., An introduction to diòerence equations. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1996.
[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
[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
[18] Lagrange, J., Essai sur le problème des trois corps. Oeuvres complètes, VI, 1772, pp. 229–324.
[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.
[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.
[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
[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
[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
[24] Meyer, K. R., Normal forms for Hamiltonian systems. Celestial Mech. 9(1974), 517–522.http://dx.doi.org/10.1007/BF01329333
[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.
[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
[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
[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
[29] Zheng, Bo., Multiple periodic solutions to nonlinear discrete Hamiltonian systems. Adv. Difference Equ. 2007, Art. ID 41830.
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# Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems

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