Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Tikhonov Theorem for Differential Equations with Singular Impulses

Discontinuity, Nonlinearity, and Complexity 7(3) (2018) 291--303 | DOI:10.5890/DNC.2018.09.007

M. Akhmet; S. Çağ

Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

Download Full Text PDF



The paper considers impulsive systems with singularities. The main novelty of the present research is that impulses (impulsive functions) are singular. This is beside singularity of differential equations. The most general Tikhonov theorem for the impulsive systems s obtained. Illustrative examples with simulations are given to support the theoretical results.


  1. [1]  Segel, L.A. and Slemrod, M. (1989), The quasi-steady state assumption: a case study in perturbation, SIAM Review, 31, 446-477.
  2. [2]  Hek, G. (2010), Geometric singular perturbation theory in biological practice, J. Math. Biol., 60, 347-386.
  3. [3]  Owen, M.R. and Lewis, M.A. (2001), How predation can slow, stop, or reverse a prey invasion, Bulletin of Mathematical Biology, 63, 655-684.
  4. [4]  Damiano, E.R. and Rabbitt, R.D. (1996), A singular perturbationmodel of fluid dynamics in the vestibular semicircular canal and ampulla, Journal of Fluid Mechanics, 307, 333-372.
  5. [5]  Kokotovic, P.V. (1984), Applications of singular perturbation techniques to control problems, SIAM Review, 26, 501- 550.
  6. [6]  Gondal, I.A. (1988), On the application of singular perturbation techniques to nuclear engineering control problems, IEEE Transactions on Nuclear Science, 35, 1080-1085.
  7. [7]  Kadalbajoo, M.K. and Patidar, K.C. (2003), Singularly perturbed problems in partial differential equations: a survey, Applied Mathematics and Computation, 134, 371-429.
  8. [8]  Veliov, V. (1997), A generalization of the Tikhonov theorem for singularly perturbed differential inclusions, Journal of Dynamical and Control Systems, 3, 291-319.
  9. [9]  Donchev, T. and Slavov, I. (1995), Tikhonov’s theorem for functional-differential inclusions, Annuaire Univ. Sofia Fac. Math. Inform., Session Dedicated to the Centenary of the Birth of Nikola Obreshkoff(Sofia, 1996), 89, 69-78.
  10. [10]  Chen, W.H., Chen, F., and Lu, X. (2010), Exponential stability of a class of singularly perturbed stochastic time-delay systems with impulse effect, Nonlinear Analysis: Real World Applications, 11, 3463-3478.
  11. [11]  Chen,W.H.,Wei, D., and Lu, X. (2013), Exponential stability of a class of nonlinear singularly perturbed systems with delayed impulses, Journal of the Franklin Institute, 350, 2678-2709.
  12. [12]  Chen, W.H., Yuan, G., and Zheng, W.X. (2013), Robust stability of singularly perturbed impulsive systems under nonlinear perturbation, Automatic Control, IEEE Transactions on, 58, 168-174.
  13. [13]  Simeonov, P.S. and Bainov, D.D. (1988), Stability of the solutions of singularly perturbed systems with impulse effect, Journal of Mathematical Analysis and Applications, 136, 575-588.
  14. [14]  Simeonov, P.S. and Bainov, D.D. (1990), Exponential stability of the solutions of singularly perturbed systems with impulse effect, Journal of Mathematical Analysis and Applications, 151, 462-487.
  15. [15]  Akhmet, M. (2010), Principles of Discontinuous Dynamical Systems, Springer: New York.
  16. [16]  Akhmet, M. (2011), Nonlinear hybrid continuous/discrete-time models, Atlantis Press: Paris.
  17. [17]  Akhmet, M. and Fen, M.O. (2015), Replication of Chaos in Neural Networks, Economics and Physics, Nonlinear Physical Science, Springer: Berlin Heidelberg.
  18. [18]  Tikhonov, A.N., Vasil’eva, A.B., and Sveshnikov, A.G. (1985), Differential Equations, Springer-Verlag: Berlin.
  19. [19]  Bainov, D. and Covachev, V. (1994), Impulsive Differential Equations with a Small Parameter, World Scientific.
  20. [20]  Vasil’eva, A., Butuzov, V., and Kalachev, L. (1995), The Boundary Function Method for Singular Perturbation Problems, Society for Industrial and Applied Mathematics.
  21. [21]  O’Malley, R.E. (1991), Singular Perturbation Methods for Ordinary Differential Equations, Applied Mathematical Sciences, Springer: New York.
  22. [22]  Verhulst, F. (2005), Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Texts in Applied Mathematics, Springer: New York.
  23. [23]  Tikhonov, A.N. (1952), Systems of differential equations containing small parameters in the derivatives, Matematicheskii sbornik, 73, 575-586.