Approximate solutions by truncated Taylor series expansions of nonlinear differential equations and related shadowing property with applications. (English) Zbl 1474.34077

Summary: This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points \(t_i \in [t_0, t_J]\) for \(i = 0,1, \ldots, J\) of the solution. Two examples are provided.


34A45 Theoretical approximation of solutions to ordinary differential equations
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
Full Text: DOI


[1] de la Sen, M., Application of the nonperiodic sampling to the identifiability and model matching problems in dynamic systems, International Journal of Systems Science, 14, 4, 367-383, (1983) · Zbl 0511.93023
[2] de la Sen, M., Stability of switched feedback time-varying dynamic systems based on the properties of the gap metric for operators, Abstract and Applied Analysis, 2012, (2012) · Zbl 1255.93115
[3] de la Sen, M., About the stabilization of a nonlinear perturbed difference equation, Discrete Dynamics in Nature and Society, 2012, (2012) · Zbl 1244.39015
[4] De la Sen, M.; Soto, J. C.; Ibeas, A., Stability and limit oscillations of a control event-based sampling criterion, Journal of Applied Mathematics, 2012, (2012) · Zbl 1244.93131
[5] Gu, R., The average-shadowing property and topological ergodicity, Journal of Computational and Applied Mathematics, 206, 2, 796-800, (2007) · Zbl 1115.37005
[6] Gu, R., On ergodicity of systems with the asymptotic average shadowing property, Computers & Mathematics with Applications, 55, 6, 1137-1141, (2008) · Zbl 1163.37004
[7] Gu, R., Recurrence and the asymptotic pseudo-orbit tracing property, Nonlinear Analysis: Theory, Methods & Applications, 66, 8, 1698-1706, (2007) · Zbl 1139.37005
[8] Lee, K.; Sakai, K., Various shadowing properties and their equivalence, Discrete and Continuous Dynamical Systems A, 13, 2, 533-540, (2005) · Zbl 1078.37015
[9] van Vleck, E. S., Numerical shadowing using componentwise bounds and a sharper fixed point result, SIAM Journal on Scientific Computing, 22, 3, 787-801, (2000) · Zbl 0976.65112
[10] Reich, S., Backward error analysis for numerical integrators, SIAM Journal on Numerical Analysis, 36, 5, 1549-1570, (1999) · Zbl 0935.65142
[11] Apostol, T. M., Mathematical Analysis, (1958), Reading, Mass, USA: Addison-Wesley, Reading, Mass, USA
[12] Zhang, Q. L.; Dai, G. Z.; Lam, J.; Zhang, L. Q.; De La Sen, M., Asymptotic stability and stabilization of descriptor systems, Acta Automatica Sinica, 24, 2, 208-211, (1998)
[13] de la Sen, M., Fundamental properties of linear control systems with after-effect. I. The continuous case, Mathematical and Computer Modelling, 10, 7, 473-489, (1988) · Zbl 0671.93004
[14] Maher, R. A.; Samir, R., Robust stability of a class of unstable systems under mixed uncertainty, Journal of Control Science and Engineering, 2011, (2011) · Zbl 1235.93218
[15] Boutat, D., Extended nonlinear observer normal forms for a class of nonlinear dynamical systems, International Journal of Robust and Nonlinear Control, (2013) · Zbl 1328.93063
[16] Benchohra, M.; Ziane, M., Impulsive evolution inclusions with state-dependent delay and multivalued jumps, Electronic Journal of Qualitative Theory of Differential Equations, 42, 1-21, (2013) · Zbl 1340.34289
[17] Tunç, C.; Ateş, M., Boundedness of solutions to differential equations of fourth order with oscillatory restoring and forcing terms, Discrete Dynamics in Nature and Society, 2013, (2013)
[18] Tunc, C.; Gozen, M., Stability and uniform boundedness in multidelay functional differential equations of third order, Abstract and Applied Analysis, 2013, (2013) · Zbl 1276.34058
[19] Diblík, J.; Fečkan, M.; Pospíšil, M., Representation of a solution of the Cauchy problem for an oscillating system with multiple delays and pairwise permutable matrices, Abstract and Applied Analysis, 2013, (2013) · Zbl 1277.34093
[20] Baculikova, B.; Dzurina, J.; Rogovchenko, Y. V., Oscillation of third order trinomial delay differential equations, Applied Mathematics and Computation, 218, 13, 7023-7033, (2012) · Zbl 1252.34074
[21] Baštinec, J.; Berezansky, L.; Diblík, J.; Šmarda, Z., On the critical case in oscillation for differential equations with a single delay and with several delays, Abstract and Applied Analysis, 2010, (2010) · Zbl 1209.34080
[22] Hasanbulli, M.; Rogovchenko, Y. V., Oscillation criteria for second order nonlinear neutral differential equations, Applied Mathematics and Computation, 215, 12, 4392-4399, (2010) · Zbl 1195.34098
[23] Han, Z.; Zhao, Y.; Sun, Y.; Zhang, C., Oscillation for a class of fractional differential equation, Discrete Dynamics in Nature and Society, 2013, (2013)
[24] Soto, J. C.; Delasen, M., Nonlinear oscillations in nonperiodic sampling systems, Electronics Letters, 20, 20, 816-818, (1984)
[25] de la Sen, M., Oscillatory behavior in linear difference equations under unmodeled dynamics and parametrical errors, Mathematical Problems in Engineering, 2007, (2007) · Zbl 1295.70010
[26] Soto, J. C.; Delasen, M., On the derivation and analysis of a non-linear model for describing a class of adaptive sampling laws, International Journal of Control, 42, 6, 1347-1368, (1985) · Zbl 0614.93037
[27] Chacón, J.; Sánchez, J.; Visioli, A.; Yebra, L.; Dormido, S., Characterization of limit cycles for self-regulating and integral processes with PI control and send-on-delta sampling, Journal of Process Control, 23, 6, 826-838, (2013)
[28] Memarbashi, R.; Rasuli, H., Notes on the dynamics of nonautonomous discrete dynamical systems, Journal of Advanced Research in Dynamical and Control Systems, 6, 2, 8-17, (2014)
[29] Mazenc, F.; Malisoff, M.; de Querioz, M., Tracking control and robustness analysis for a nonlinear model of human heart rate during exercise, Automatica, 47, 5, 968-974, (2011) · Zbl 1233.93084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.