Rocky Mountain Journal of Mathematics

Asymptotic behavior of a planar dynamic system

Gro Hovhannisyan

Full-text: Open access


We investigate the asymptotic solutions of the planar dynamic systems and the second order equations on a time scale by using a new version of Levinson's asymptotic theorem. In this version the error estimate is given in terms of the characteristic (Riccati) functions which are constructed from the phase functions of an asymptotic solution. It means that the improvement of the approximation depends essentially on the asymptotic behavior of the Riccati functions. We describe many different approximations using the flexibility of this approach. As an application we derive the analogue of D'Alembert's formula for the one dimensional wave equation in a discrete time.

Article information

Rocky Mountain J. Math., Volume 44, Number 4 (2014), 1203-1242.

First available in Project Euclid: 31 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34E10: Perturbations, asymptotics 39A10: Difference equations, additive

Dynamic systems on a time scale asymptotic solutions characteristic function Riccati equation perturbation method


Hovhannisyan, Gro. Asymptotic behavior of a planar dynamic system. Rocky Mountain J. Math. 44 (2014), no. 4, 1203--1242. doi:10.1216/RMJ-2014-44-4-1203.

Export citation


  • R. Aulbach and S. Hilger, Linear dynamic processes with inhomogeneous time scale, in Non-linear dynamics and quantum dynamical systems, Acad. Verlag, Berlin, 1990.
  • Z. Benzaid and D.A. Lutz, Asymptotic representation of solutions of perturbed systems of linear difference equations, Stud. Appl. Math. 77 (1987), 195–221.
  • G.D. Birkhoff, General theory of linear difference equations, Trans. Amer. Soc. 12 (1911), 243–284.
  • S. Bodine, M. Bohner and D.A. Lutz, Asymptotic behavior of solutions of dynamic equations, J. Math. Sci. 124 (2004), 5110–5118.
  • M. Bohner and D. Lutz, Asymptotic behavior of dynamic equations on time scales, J. Diff. Equat. Appl. 7 (2001), 21–50.
  • ––––, Asymptotic expansion and analytic time scales, ZAMM-Z. Math. Mech. 86 (2006), 37–45.
  • M. Bohner and A. Peterson, Dynamic equations on time scales: An introduction with applications, Birkhauser, Boston, 2001.
  • ––––, Advances in dynamic equations on time scales, Birkhauser, Boston, 2002.
  • M. Bohner and S. Stevic, Asymptotic behavior of second-order dynamic equations, Appl. Math. Comp. 188 (2007), 1503–1512.
  • L. Cesary, Asymptotic behavior and stability problems in ordinary differential equations, 3rd edition, Springer Verlag, Berlin, 1970.
  • C.V. Coffman, Asymptotic behavior of solutions of ordinary difference equations, Trans. Amer. Math. Soc. 110 (1964), 22–51.
  • M.S.P. Eastham, The asymptotic solution of linear differential systems. Applications of the Levinson theorem, Clarendon Press, Oxford, 1989.
  • T. Gard and J. Hoffacker, Asymptotic Behavior of natural growth on time scales, Dyn. Syst. Appl. 12 (2003), 131–148.
  • W.A. Harris and D.A. Lutz, On the asymptotic integration of linear differential systems, J. Math. Anal. Appl. 48 (1974), 1–16.
  • ––––, A unified theory of asymptotic integration, J. Math. Anal. Appl. 57 (1977), 571–586.
  • P. Hartman, Ordinary differential equations, John Wiley and Sons, New York, 1973.
  • S. Hilger, Analysis on measure chains–a unified approach to continuous and discrete calculus, Results Math. 18 (1990), 18–56.
  • G. Hovhannisyan, Levinson theorem for $2\times2$ system and applications to the asymptotic stability and Schrodinger equation, Inter. J. Evol. Equat. 3 (2008), 181-—203.
  • ––––, Asymptotic stability for dynamic equations on time scales, Adv. Diff. Equat. Article ID 18157 (2006), 17 pages.
  • ––––, Asymptotic stability for $2 \times 2$ linear dynamic systems on time scales, Inter. J. Diff. Equat. 2 (2007), 105–121.
  • ––––, On oscillations of solutions of $n$-th order dynamic equation, ISRN Math. Anal. 2013 (2013), Article ID 946453, 11 pages.
  • G. Hovhannisyan and W. Liu, On non-autonomous Dirac equation, J. Math. Phys. 50 (2009), 123507, 24 pages.
  • N. Levinson, The asymptotic nature of solutions of linear systems of differential equations, Duke Math. J. 15 (1948), 111–126.
  • C. Potzsche, S. Siegmund and F. Wirth, A spectral characterization of exponential stability for linear time-unvariant systems on time scales, Discr. Cont. Dyn. Syst. 9 (2003), 1223–1241.
  • G. Ren, Y. Shi and Y. Wang, Asymptotic behavior of solutions of perturbed linear difference systems, Linear Alg. Appl. 395 (2005), 283–302.
  • V. Volterra, Sui fondamenti della teoria delle equazioni differenziali lineari, Mem. Soc. Ital. Sci. 6 (1887), 1–104.