# Analysis of 2-Term Fractional-Order Delay Differential Equations

• Sachin Bhalekar
Conference paper
Part of the Trends in Mathematics book series (TM)

## Abstract

The value of a state variable at past time usually affects its rate of change at the present time. So, it is very natural to consider the delay while modeling the real-life systems. Further, the nonlocal fractional derivative operator is also useful in modeling memory in the system. Hence, the models involving delay as well as fractional derivative are very important. In this chapter, we review the basic results regarding the dynamical systems, fractional calculus, and delay differential equations. Further, we analyze 2-term nonlinear fractional-order delay differential equation $$D^\alpha x + c D^\beta x = f\left( x,x_\tau \right)$$, with constant delay $$\tau >0$$ and fractional orders $$0<\alpha<\beta <1$$. We present a numerical method for solving such equations and present an example exhibiting chaotic oscillations.

## Notes

### Acknowledgements

The author acknowledges the Council of Scientific and Industrial Research, New Delhi, India for funding through Research Project [25(0245)/15/EMR-II] and the Science and Engineering Research Board (SERB), New Delhi, India for the Research Grant [Ref. MTR/2017/000068] under Mathematical Research Impact Centric Support (MATRICS) Scheme.

## References

1. 1.
Agrawal, O.P.: Generalized variational problems and Euler-Lagrange equations. Comput. Math. Appl. 59, 1852–1864 (2010)
2. 2.
Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58(9), 1838–1843 (2009)
3. 3.
Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems. Springer, New York (2008)Google Scholar
4. 4.
Babakhani, A., Daftardar-Gejji, V.: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 278(2), 434–442 (2003)
5. 5.
Babakhani, A., Daftardar-Gejji, V.: Existence of positive solutions for N-term non-autonomous fractional differential equations. Positivity 9, 193–206 (2005)
6. 6.
Baleanu, D., Magin, R., Bhalekar, S., Daftardar-Gejji, V.: Chaos in the fractional order nonlinear Bloch equation with delay. Commun. Nonlinear Sci. Numer. Simul. 25(1), 41–49 (2015)
7. 7.
Bhalekar, S.: Dynamical analysis of fractional order Ucar prototype delayed system. Signals Image Video Process. 6(3), 513–519 (2012)
8. 8.
Bhalekar, S.: Stability analysis of a class of fractional delay differential equations. Pramana 81(2), 215–224 (2013)
9. 9.
Bhalekar, S.: Stability and bifurcation analysis of a generalized scalar delay differential equation. Chaos 26(8), 084306 (2016)
10. 10.
Bhalekar, S., Daftardar-Gejji, V.: Fractional ordered Liu system with time-delay. Commun. Nonlinear Sci. Numer. Simul. 15(8), 2178–2191 (2010)
11. 11.
Bhalekar, S., Daftardar-Gejji, V.: A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. J. Fract. Calc. Appl. 1(5), 1–9 (2011)
12. 12.
Bhalekar, S., Daftardar-Gejji, V.: Solving fractional-order logistic equation using a new iterative method. Int. J. Differ. Equ. 2012, Article number 975829 (2012)Google Scholar
13. 13.
Bhalekar, S., Daftardar-Gejji, V.: Existence and uniqueness theorems for fractional differential equations: A new approach. In: Daftardar-Gejji, V., (ed.) Fractional Calculus: Theory and Applications. Narosa Publishing House, New Delhi (2013). ISBN 978-81-8487-333-7Google Scholar
14. 14.
Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., Magin, R.: Fractional Bloch equation with delay. Comput. Math. Appl. 61(5), 1355–1365 (2011)
15. 15.
Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., Magin, R.: Generalized fractional order Bloch equation with extended delay. Int. J. Bifurc. Chaos 22(4), 1250071 (2012)
16. 16.
Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., Magin, R.: Transient chaos in fractional Bloch equations. Comput. Math. Appl. 64(10), 3367–3376 (2012)
17. 17.
Chen, Y., Moore, K.L.: Analytical stability bound for a class of delayed fractional-order dynamic systems. Nonlinear Dyn. 29(1), 191–200 (2002)
18. 18.
Choudhari, S., Daftardar-Gejji, V.: Existence uniqueness theorems for multi-term fractional delay differential equations. Fract. Calc. Appl. Anal. 5(18), 1113–1127 (2015)
19. 19.
Daftardar-Gejji, V.: Positive solutions of a system of non-autonomous fractional differential equations. J. Math. Anal. Appl. 302(1), 56–64 (2005)
20. 20.
Daftardar-Gejji, V., Babakhani, A.: Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 293(2), 511–522 (2004)
21. 21.
Daftardar-Gejji, V., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301(2), 508–518 (2005)
22. 22.
Daftardar-Gejji, V., Bhalekar, S., Gade, P.: Dynamics of fractional ordered Chen system with delay. Pramana-J. Phys. 79(1), 61–69 (2012)
23. 23.
Daftardar-Gejji, V., Sukale, Y., Bhalekar, S.: A new predictorcorrector method for fractional differential equations. Appl. Math. Comput. 244, 158–182 (2014)
24. 24.
Daftardar-Gejji, V., Sukale, Y., Bhalekar, S.: Solving fractional delay differential equations: A new approach. Fract. Calc. Appl. Anal. 18(2), 400–418 (2015)
25. 25.
Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996)
26. 26.
Deng, W., Li, C., Lü, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)
27. 27.
Diethelm, K., Ford, N.J., Freed, A.D.: A predictorcorrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)
28. 28.
Feliu, V., Rivas, R., Castillo, F.J.: Fractional robust control to delay changes in main irrigation canals. In: Proceedings of the 16th International Federation of Automatic Control World Congress. Czech Republic, Prague (2005)Google Scholar
29. 29.
Feliu, V., Rivas, R., Castillo, F.: Fractional order controller robust to time delay variations for water distribution in an irrigation main canal pool. Comput. Electron. Agric. 69(2), 185–197 (2009)
30. 30.
Hotzel, R.: Summary: some stability conditions for fractional delay systems. J. Math. Syst. Estim. Control 8, 499–502 (1998)
31. 31.
Hwang, C., Cheng, Y.C.: A numerical algorithm for stability testing of fractional delay systems. Automatica 42, 825–831 (2006)
32. 32.
Jafari, H., Daftardar-Gejji, V.: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Appl. Math. Comput. 180(2), 700–706 (2006)
33. 33.
Kiryakova, V.: A brief story about the operators of the generalized fractional calculus. Fract. Calc. Appl. Anal. 11(2), 203–220 (2008)
34. 34.
Lazarevic, M.P., Debeljkovic, D.L.: Finite time stability analysis of linear autonomous fractional order systems with delayed state. Asian J. Control 7(4), 440–447 (2005)
35. 35.
Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems and Application multiconference, vol. 2, pp. 963–968, IMACS, IEEE-SMC Proceedings. Lille, France (1996)Google Scholar
36. 36.
37. 37.
Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D.Y., Feliu, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, London (2010)
38. 38.
Moornani, K., Haeri, M.: On robust stability of LTI fractional-order delay systems of retarded and neutral type. Automatica 46, 362–368 (2010)
39. 39.
40. 40.
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)Google Scholar
41. 41.
Si-Ammour, A., Djennoune, S., Bettayeb, M.: A sliding mode control for linear fractional systems with input and state delays. Commun. Nonlinear Sci. Numer. Simul. 14, 2310–2318 (2009)
42. 42.
Srivastava, V., Rai, K.N.: A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math. Comput. Model. 51, 616–624 (2010)
43. 43.
Tavazoei, M.S., Haeri, M.: Regular oscillations or chaos in a fractional order system with any effective dimension. Nonlinear Dyn. 54(3), 213–222 (2008)
44. 44.
Tavazoei, M.S., Haeri, M.: Chaotic attractors in incommensurate fractional order systems. Phys. D 237, 2628–2637 (2008)