Abstract
Time-continuous and time-discrete dynamic systems as a whole (hybrid systems) are of undoubted interest for applications. The mathematical analysis developed on time scales allows to consider the real-world phenomena in a more accurate description.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Atici F.M., Biles D.C., Lebedinsky A. An application of time scales to economics. Math. and Comput. Modelling 43 (2006) 718–726.
Atici F.M. and McMahan C.S. A comparison in the theory of calculus of variations on time scale with an application to the Ramsey model. Nonlinear Dynamics and Systems Theory 9 (1) (2009) 1–10.
Babenko S.V. Revisiting the theory of stability on time scales of the class of linear systems with structural perturbations. Int. Appl. Mech. 47 (1) (2011) 86–96.
Babenko, S.V., Martynyuk, A.A. Stability of a dynamic graph on time scales. Nonlinear Dynamics and Systems Theory 14 (1) (2014) 30–43.
Bartosiewicz Z., and Pawluszewicz E. Realizations of linear control systems on time scales. Control & Cybernetics 35 (4) (2006) 769–786.
Bohner M., Fan M., and Zhang J. Periodicity of scalar dynamic equations and applications to population models J. Math. Anal. Appl. 330 (2007) 1–9.
Bohner M., Martynyuk A.A. Elements of stability theory of A.M. Lyapunov for dynamic equations on time scales. Int. Appl. Mech. 43 (9) (2007) 949–970.
Bohner M., Peterson A. Dynamic Equations on Time Scales: An Introduction with Applications. Boston: Birkhäuser, 2001.
Bohner M., Peterson A. Advances in Dynamic Equations on Time Scales. Boston: Birkhäuser, 2003.
Bohner M., Rao V.S.H., Sanyal S. Global stability of complex-valued neural networks on time scales. Diff Eqns and Dyn. Syst. 19 (1&2) (2011) 3–11.
Ding X., Hao J. and Liu Ch. Multiple periodic solutions for a delayed predator-prey system on time scale. World Academy of Science, Engineering and Technology 60 (2011) 1532–1537.
Gard T., Hoffacker J. Asymptotic behavior of natural growth on time scales. Dynamic Systems and Appl. 12 (1–2) (2003) 131–148.
Hang J. Global stability analysis in Hopfield neural networks. Appl. Math. Let. 16 (2003) 925–931.
Hopfield J.J. Neurons with graded response have collective computational properties like those of two state neurons. Proc. Nat. Acad. Sci. USA 81 (1984) 3088–3092.
Lila D.M. Stability of motion of quasiperiodic systems in critical cases. Int. Appl. Mech. bf46 (2) (2010) 229–240.
Luk’yanova T.A. and Martynyuk A.A. Integral inequalities and stability of an equilibrium state on a time scale. Ukrainian Mathematical Journal 62 (11) (2011) 1490–1499.
Luk’yanova T.A. and Martynyuk A.A. Sufficient conditions of connective stability of motion on time scale Int. Appl. Mech. 49 (2) (2013) 232–244.
Marks II R.J., Gravagne I., Davis J.M., and DaCunha J.J. Nonregressivity in switched linear circuits and mechanical systems. Mathematical and Computer Modeling 43 (2006) 1383–1392.
Martynyuk A.A. On the exponential stability of dynamic systems on time scales Dokl. Akad. Nauk 421 (2008) 312–317.
Martynyuk A.A. Stability Theory of Solutions of Dynamic Equations on Time Scales. Kiev: Phoeniks, 2012.
Martynyuk A.A. Elements of stability theory of hybrid systems. Int. Appl. Mech. 51 (3) (2015) 243–302.
Nitta T. ​​Complex-Valued Neural Networks: Utilizing Hight-Dimensional Parameters. Hershey, PA: Information Science Reference, 2009.
Šiljak D.D. On stability of large-scale systems under structural perturbations. IEEE. Transactions on Systems, Man and Cybernetics 3 (1973) 415–417.
Šiljak D.D. Dynamic graphs. Nonlinear Analysis: Hybrid Systems 2 (2008) 544–567.
Wang L., Zou X. Exponential stability of Cohen-Grossberg neural networks. Neural networks 16 (2002) 415–422.
Yao Z. Existence and Global exponential stability of an almost periodic solution for a host-macroparasite equation on time scales. Advances in Difference Equations (2015) 2015:41 DOI 10.1186/s13662-015-0383-0.
Zhang J. Global stability analysis in Hopfield neural networks. Appl. Math. Let. 16 (2003) 925–931.
Zhuang K. and Wen Zh. Periodic solutions for a delayed population model on time scales. Int. J. Comput. Math. Sci. 4 (3)(2010) 166–168.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Martynyuk, A.A. (2016). Applications. In: Stability Theory for Dynamic Equations on Time Scales. Systems & Control: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-42213-8_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-42213-8_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-42212-1
Online ISBN: 978-3-319-42213-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)