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


Unpredictable Solutions of Impulsive Quasi-Linear Systems

Discontinuity, Nonlinearity, and Complexity 11(1) (2022) 73--89 | DOI:10.5890/DNC.2022.03.006

Marat Akhmet$^1$ , Madina Tleubergenova$^{2,3}$, Zakhira Nugayeva$^{2,3}$

$^1$ Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey

$^2$ Department of Mathematics, Aktobe Regional State University, 030000, Aktobe, Kazakhstan

$^3$ Institute of Information and Computational Technologies, 050010, Almaty, Kazakhstan

Download Full Text PDF



The impulsive differential equations with unpredictable perturbations are under investigation. The definition of the discontinuous unpredictable function with the discrete unpredictable set of discontinuity moments is introduced. The existence and uniqueness of asymptotically stable discontinuous unpredictable solutions for quasi linear systems are proved. Examples with simulations are given to illustrate the results.


M. Akhmet has been supported by 2247-A National Leading Researchers Program of TUBITAK, Turkey, N 120C138.


  1. [1]  Akhmet, M.U. (2010), Principles of discontinuous dynamical systems, Springer-Verlag, New York.
  2. [2]  Akhmet, M.U. (2020), Almost periodicity, chaos, and asymptotic equivalence, Springer International Publishing, Switzerland.
  3. [3]  Bogolyubov, N.N. and Mitropol'skii, Yu.A. (1961), Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach.
  4. [4]  Lakshmikantham, V., Bainov, D.D., and Simeonov P.S. (1989), Theory of impulsive differential equations, World Scientific, Singapore.
  5. [5]  Samoilenko, A.M. and Perestyuk, N.A. (1995), Impulsive Differential Equations, World Scientific, Singapore.
  6. [6]  Erbe, L.H. and Liu, X. (1991), Existence of periodic solutions of impulsive differential systems, Appl.Math. and Stoch. Anal., 4(2), 137-146.
  7. [7]  Girel, S. and Crauste, F. (2018), Existence and stability of periodic solutions of an impulsive differential equation and application to CD8 T-cell differentiation, J. Math. Biology. 76, 1765-1795.
  8. [8]  Li, X., Bohner, M., and Wang, Ch.-K. (2015), Impulsive differential equations: Periodic solutions and applications, Automatica, 52, 173-178.
  9. [9]  Liz, E. and Nietto, J.J. (1991), Periodic solutions of discontinuous impulsive differential systems, J. Math. Anal. Appl., 161(2), 388-394.
  10. [10]  Niu, Y. and Li, X. (2018), Periodic solutions of sublinear impulsive differential equations, Taiwanese J Math., 22(2), 439-452.
  11. [11]  Niu, Y. and Li X. (2018), Periodic solutions of semilinear Duffing equations with impulsive effects, J. Math. Anal.Appl., 467, 349-370.
  12. [12]  Wang, Sh. (2018), The existence of affine-periodic solutions for nonlinear impulsive differential equations, Bound. Value Probl., 113.
  13. [13]  Zhang, T. and Xiong L. (2020), Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101, 106072.
  14. [14]  Akhmet, M.U. (1989), Quasiperiodic solutions of systems with impulses, in Asymptotic Methods in Problems of Mathematical Physics, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 12-18.
  15. [15]  Halanay, A. and Wexler, D. (1971), The Qualitative theory of impulse systems, (russian), Mir, Moscow.
  16. [16]  Gurgula, S.I. (1982), A study of the stability of solutions of impulse systems by Lyapunov's second method, Ukrainian Math. J., 34, 84-87.
  17. [17]  Mil'man, V.D. and Myshkis, A.D. (1960), Stability of motion in the presence of shocks, (russian), Sib. Mat. Zh., 1(2), 233-237.
  18. [18]  Myshkis, A.D. and Samoilenko A.M. (1967), Systems with impulses at fixed moments of time, (russian), Math. Sb., 74, 202-208.
  19. [19]  Abdelaziz, M. and Ch{e}rif, F. (2020), Piecewise asymptotic almost periodic solutions for impulsive fuzzy Cohen-Grossberg neural networks, Chaos, Solitons and Fractals, 132, 109575.
  20. [20]  Agarwal, R.P., Hristova, S., and O'Regan, D. (2017), Non-Instantaneous Impulses in Differential Equations, Springer International Publishing, Switzerland.
  21. [21]  Akhmet, M.U., Fen, M.O., and Kirane, M. (2016), Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument, Neural Comput. Appl., 27, 2483-2495.
  22. [22]  Cherif, F. (2014), Pseudo almost periodic solutions of impulsive differential equations with delay, Diff. Equat. Dynamical Systems, 22(1), 73-91.
  23. [23]  Fe\v ckan, M. (2000), Existence of almost periodic solutions for jumping discontinuous systems, Acta Math. Hungar., 86(4), 291-303.
  24. [24]  Fe\v ckan, M. and Wang, J.R. (2019), Periodic impulsive fractional differential equations, Adv. Nonlinear Anal., 8, 482-496.
  25. [25]  Frigon, M. and O'Regan, D. (1999), First order impulsive initial and periodic problems with variable moments, J. Math. Anal. Appl., 233(2), 730-739.
  26. [26]  Hakl, R., Pinto, M., Tkachenko, V., and Trofimchuk, S. (2017), Almost periodic evolution systems with impulse action at state-dependent moments, J. Math. Anal. Appl., 446, 1030-1045.
  27. [27]  He, M.X., Chen, F.D., and Li, Z. (2010), Almost periodic solution of an impulsive differential equation model of plankton allelopathy, Nonlinear Anal. RWA., 11, 2296-2301.
  28. [28]  Hern{a}ndez, E. and O'Regan, D. (2013), On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141(5), 1641-1649.
  29. [29]  Karakoc, F., Unal, A., and Bereketoglu, H. (2018), Oscillation of a nonlinear impulsive differential equation system with piecewise constant argument, Advances in Diff. Equat., 99, 11.
  30. [30]  Li, J., Nieto, J.J., and Shen, J. (2007), Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl., 325(1), 226-236.
  31. [31]  Liu, J. and Zhang, Ch. (2012), Existence and stability of almost periodic solutions for impulsive differential equations, Adv. Differ. Equ., 34.
  32. [32]  Luo, A.C.J. and Wang, Y. (2009), Switching dynamics of multiple linear oscillators, Commun Nonlinear Sci Numer Simulat, 14(8), 3472-3485.
  33. [33]  Luo, A.C.J. (2012), Regularity and Complexity in Dynamical Systems, Springer, NY.
  34. [34]  Luo, A.C.J. and Wang, Y. (2010), On Periodic Flows of a 3-D Switching System with Many Subsystems, in Dynamical Systems, (Ed. Albert C J Luo), Springer, NY.
  35. [35]  Ma, X., Shu, X.-B., and Mao, J. (2020), Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay, Stochastics and Dynamics, 20(1), 2050003.
  36. [36]  Rach{{\r u}}nkov{a}, I. and Tvrdy, M. (2004), Existence results for impulsive second-order periodic problems, Nonlinear Anal., 59(2), 133-146.
  37. [37]  Rach{{\r u}}nkov{a}, I. and Tome{c}ek, J. (2018), Equivalence between distributional differential equations and periodic problems with state-dependent impulses, Electron. J. Qual. Theory Differ. Equ., 2(2), 1-22.
  38. [38]  Ronto, N.I. and Tuzson, {A}. (1994), Construction of periodic solutions of differential equations with impulse effect, Publ. Math. Debrecen, 44, 335-357.
  39. [39]  Stamov, G.Tr. and Stamova, I.M. (2007), Almost periodic solutions for impulsive neural networks with delay, Appl. Math. Model, 31(7), 1263-1270.
  40. [40]  Jiang, G., Lu, Q., and Qians, L. (2007), Chaos and its control in an impulsive differential system, Chaos Solit. Fract., 34(4), 1135-1147.
  41. [41]  Akhmet, M.U. (2009), Li-Yorke chaos in the system with impacts, J. Math. Anal. Appl., 351(2), 804-810.
  42. [42]  Li, C., Liao, X., and Zhang, R. (2004), Impulsive synchronization of nonlinear coupled chaotic systems, Phys. Lett. A, 328(1), 47-50.
  43. [43]  Li, C. Liao, X., and Zhang, R. (2005), Impulsive synchronization of chaotic systems, Chaos, 15, 023104.
  44. [44]  Li, P., Li, Z., Halang, W.A., and Chen, G. (2007), Li-Yorke chaos in a spatiotemporal chaotic system, Chaos Solit. Fract., 33(2), 335-341.
  45. [45]  Birkhoff, G.D. (1927), Dynamical Systems, American Mathematical Soc. New York.
  46. [46]  Akhmet, M. and Fen, M.O. (2016), Unpredictable points and chaos, Commun. Nonlinear Sci. Numer. Simulat., 40, 1-5.
  47. [47]  Akhmet, M.U., Fen, M.O., and Alejaily, E.M. (2020), Dynamics with Chaos and Fractals, Springer International Publishing, Switzerland.
  48. [48]  Miller, A. (2019), Unpredictable points and stronger versions of Ruelle-Takens and Auslander-Yorke chaos, STopology and its Appl., 253, 7-16.
  49. [49]  Thakur, R. and Das R. (2019), Strongly Ruelle-Takens, strongly Auslander-Yorke and Poincar{e} chaos on semiflows, Commun. Nonlinear. Sci. Numer. Simulat., 81, 105018.
  50. [50]  Akhmet, M. and Fen, M.O. (2018), Non-autonomous equations with unpredictable solutions, Commun. Nonlinear. Sci. Numer. Siml., 59, 657-670.
  51. [51]  Akhmet, M., Fen, M.O., Tleubergenova, M., and Zhamanshin, A. (2019), Unpredictable solutions of linear differential and discrete equations, Turk. J Math., 43(5), 2377-2389.
  52. [52]  Akhmet, M., Tleubergenova, M. and Zhamanshin, A. (2020), Quasilinear differential equations with strongly unpredictable solutions, Carpatian J Math., 36, 341-349.