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Orbital stability of periodic solutions of an impulsive system with a linear continuous-time part

Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetsky av. 28, Stary Peterhof, 198504, St. Petersburg, Russia

Special Issues: Initial and Boundary Value Problems for Differential Equations

An impulsive system with a linear continuous-time part and a nonlinear discrete-time part is considered. A criterion for exponential orbital stability of its periodic solutions is obtained. The proof is based on linearization by the first approximation of an auxiliary discrete-time system. The formulation of the criterion depends significantly on a number of impulses per period of the solution. The paper provides a mathematical rationale for some results previously examined in mathematical biology by computer simulations.
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Keywords systems with impulses; hybrid systems; periodic solutions; exponential stability; orbital stability

Citation: Alexander N. Churilov. Orbital stability of periodic solutions of an impulsive system with a linear continuous-time part. AIMS Mathematics, 2020, 5(1): 96-110. doi: 10.3934/math.2020007

References

  • 1. A. N. Churilov, A. Medvedev, A. I. Shepeljavyi, Mathematical model of non-basal testosterone regulation in the male by pulse modulated feedback, Automatica, 45 (2009), 78-85.    
  • 2. A. N. Churilov, A. Medvedev, A. I. Shepeljavyi, State observer for continuous oscillating systems under intrinsic pulse-modulated feedback, Automatica, 48 (2012), 1005-1224.    
  • 3. A. N. Churilov, A. Medvedev, An impulse-to-impulse discrete-time mapping for a time-delay impulsive system, Automatica, 50 (2014), 2187-2190.    
  • 4. A. N. Churilov, A. Medvedev, P. Mattsson, Periodical solutions in a pulse-modulated model of endocrine regulation with time-delay, IEEE Trans. Automat. Contr, 59 (2014), 728-733.    
  • 5. Z. T. Zhusubaliyev, E. Mosekilde, A. N. Churilov, et al. Multistability and hidden attractors in an impulsive Goodwin oscillator with time delay, Eur. Phys. J. Special Topics, 224 (2015), 1519-1539.    
  • 6. A. N. Churilov, A. Medvedev, Discrete-time map for an impulsive Goodwin oscillator with a distributed delay, Math. Control Signals Syst, 28 (2016), 1-22.    
  • 7. A. N. Churilov, A. Medvedev, Z. T. Zhusubaliyev, Impulsive Goodwin oscillator with large delay: Periodic oscillations, bistability, and attractors, Nonlin. Anal. Hybrid Syst., 21 (2016), 171-183.    
  • 8. A. Churilov, A. Medvedev, Z. Zhusubaliyev, Discrete-time mapping for an impulsive Goodwin oscillator with three delays, Intern. J. Bifurc. Chaos, 27 (2017), 1750182.
  • 9. A. Medvedev, A. V. Proskurnikov. Z. T. Zhusubaliyev, Mathematical modeling of endocrine regulation subject to circadian rhythm, Ann. Rev. Contr., 46 (2018), 148-164.    
  • 10. H. Taghvafard, A. Medvedev, A. V. Proskurnikov, et al. Impulsive model of endocrine regulation with a local continuous feedback, Math. Biosci., 310 (2019), 128-135.    
  • 11. V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, Singapore: World Scientific, 1989.
  • 12. D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, UK: Longman Harlow, 1993.
  • 13. A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, Singapore: World Scientific, 1995.
  • 14. W. M. Haddad, V. Chellaboina, S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton and Oxford: Princeton University Press, 2006.
  • 15. M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions, New York: Hindawi Publishing Corporation, 2006.
  • 16. I. Stamova, Stability Analysis of Impulsive Functional Differential Equations, Berlin: Walter de Gruyter, 2009.
  • 17. I. Stamova and G. Stamov, Applied Impulsive Mathematical Models, Berlin: Springer, 2016.
  • 18. A. K. Gelig, A. N. Churilov, Stability and Oscillations of Nonlinear Pulse-Modulated Systems, Boston: Birkhäuser, 1998.
  • 19. R. Bellman, Stability Theory of Differential Equations, New York: McGraw-Hill, 1953.
  • 20. A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, New York: Academic Press, 1966.
  • 21. P. S. Simeonov, D. D. Bainov, Orbital stability of the periodic solutions of autonomous systems with impulse effect, Int. J. Syst. Sci., 19 (1988), 2561-2585.    
  • 22. P. S. Simeonov, D. D. Bainov, Orbital stability of the periodic solutions of autonomous systems with impulse effect, Publ. RIMS, Kyoto Univ., 25 (1989), 321-346.    
  • 23. J. W. Grizzle, G. Abba, F. Plestan, Asymptotically stable walking for biped robots: Analysis via systems with impulse effects, IEEE Trans. Autom. Contr., 46 (2001), 51-64.    
  • 24. S. G. Nersesov, V. Chellaboina, W. M. Haddad, A generalization of Poincaré's theorem to hybrid and impulsive dynamical systems, Proc. Amer. Contr. Conf., 2 (2002), 1240-1245.
  • 25. K. G. Dishlieva, A. B. Dishliev, V. I. Radeva, Orbital Hausdorff dependence on impulsive differential equations, Int. J. Diff. Equat. Appl., 13 (2014), 145-163.
  • 26. K. G. Dishlieva, Orbital Hausdorff stability of the solutions of differential equations with variable structure and impulses, Amer. Rev. Math. Statist., 3 (2015), 70-87.
  • 27. K. G. Dishlieva, Orbital Euclidean stability of the solutions of impulsive equations on the impulsive moments, Pure Appl. Math. J., 4 (2015), 1-8.
  • 28. O. Vejvoda, On the existence and stability of the periodic solution of the second kind of a certain mechanical system, Czechoslovak Math. J., 9 (1959), 390-415.
  • 29. G. A. Leonov, D. V. Ponomarenko, V. B. Smirnova, Local instability and localization of attractors. From stochastic generator to Chua's systems, Acta Appl. Math., 40 (1995), 179-243.    
  • 30. A. Halanay, D. Wexler, Teoria Calitativǎ a Sistemelor cu Impulsuri, Bucureşti: Ed. Acad. R. S. România, 1968.
  • 31. A. Halanay, V. Rǎsvan, Stability and Stable Oscillations in Discrete Time Systems, Boca Raton: CRC Press, 2000.
  • 32. A. N. Michel, K. Wang, B. Hu, Qualitative Theory of Dynamical Systems: The Role of Stability Preserving Mappings, 2 Eds., New York: Marcel Dekker, 2001.
  • 33. R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge: Cambridge University Press, 1990.
  • 34. L. Glass, M. C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton: Princeton University Press, 1988.
  • 35. L. Glass, Synchronization and rhythmic processes in physiology, Nature, 410 (2001), 277-284.    
  • 36. A. Goldbeter, Computational approaches to cellular rhythms, Nature, 420 (2002), 238-245.    
  • 37. L. Glass, Dynamical disease: Challenges for nonlinear dynamics and medicine, Chaos, 25 (2015), 097603.
  • 38. J. J. Walker, J. R. Terry, K. Tsaneva-Atanasova, et al. Encoding and decoding mechanisms of pulsatile hormone secretion, J. Neuroendocrinol., 22 (2009), 1226-1238.
  • 39. E. Zavala, K. C. A. Wedgwood, M. Voliotis, et al. Mathematical modelling of endocrine systems, Trends Endocrin. Metab., 30 (2019), 244-257.    
  • 40. N. Bagheri, S. R. Taylor, K. Meeker, et al. Synchrony and entrainment properties of robust circadian oscillators, J. R. Soc. Interface, 5 (2008), S17-S28.
  • 41. J. D. Murray, Mathematical Biology, I: An Introduction, 3 Eds., New York: Springer, 2002.
  • 42. L. S. Farhy, J. D. Veldhuis, Joint pituitary-hypothalamic and intrahypothalamic autofeedback construct of pulsatile growth hormone secretion, Am. J. Physiol. Regul. Integr. Comp. Physiol., 285 (2003), R1240-R1249.
  • 43. Z. T. Zhusubaliyev, A. N. Churilov, A. Medvedev, Bifurcation phenomena in an impulsive model of non-basal testosterone regulation, Chaos, 22 (2012), 013121.

 

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