Journal of Applied Mathematics and Physics
Vol.07 No.12(2019), Article ID:97061,6 pages
10.4236/jamp.2019.712213
On Non Liénard Type with One Limit Cycle
Ali E. M. Saeed1, Abderahim B. Hamid2
1Department of Mathematics, Alzaem Alazhari University, Khartoum North, Sudan
2Department of Mathematics, University of Gezira, Wad Madani, Sudan
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: September 29, 2019; Accepted: December 9, 2019; Published: December 12, 2019
ABSTRACT
In the present paper, we have investigated the non Liénard system. We have shown that limit cycles may bifurcate at the origin. Bendixons theorem has been used in our study to prove non-existence of limit cycles. We have also proved that the system has unique limit cycle through change of the parameters.
Keywords:
Limit Cycle, Non-Liénard Equation, Hopf-Bifucation
1. Introduction
In the present investigation, we revisit the problem of bifurcation of limit cycles. The problem of limit cycle was studied intensively. For Liénard, we can read [1] - [8], and for non Liénard we can read [9] - [19].
We give criterion for the non Liénard system to have or not to have limit cycles with some parameters. We also demonstrate that the system exhibits a Hopf-bifurcation. Now we consider the following Liénard equation
(1)
The above equation may be written in two dimensional autonomous dynamical system
. (2)
Therefore, the above equations can be written in the Liénard plane as
(3)
where .
Theorem 1.1 [11] Suppose that for system (1.1), there exist such that , for , for , for , and , then (1.1) has at most one limit cycle in D, which is simple and stable, if exists.
Theorem 1.2 [11] If, in system (1.1), (or ), and equality holds only for at most a finite number of points, then (1.1) has no closed orbits in closed region .
In Section 2, the main system equations results have been presented, the section has been divided in two cases.
The case I considered the conditions that the system has a limit cycle when is an anti saddle.
Finally, the case of saddle point with limit cycle is presented in theorems and lemmas in Section 4 along with the concluding remakes.
2. The Basic System Equations and Results
The main part of this paper is devoted to explain the existence and uniqueness of limit cycles of the following differential equations system
(4)
the singular points of the system are and .
The Jacobian matrix
has the determinant for the origin O is anti saddle and for the origin O is saddle for more details (see [5] ).
The system (2.2) needs to change to the Liénard system (1.1).
Let so after simplify and substitute so that we have
(5)
After change z to y we can get system (1.1) as follows
(6)
The system (2.4) is considered in two cases.
Case I: The Origin is an anti Saddle
The case under consideration is , in this case and as above the system (2.4) has unique equilibrium point which is an anti saddle.
Lemma 2.1
For system (2.4) has no limit cycle.
Proof
Let , then and . Thus and by using theorem (1.2) there is no limit cycle so we just look for .
In the case of O becomes saddle also for or ( ) O is node in two cases no limit cycles surround O. Thus, in the sequel, we only need to consider .
Consider the polynomial Liénard system of degree n
(7)
Lemma 2.2 [11]
For system (2.5) with , the first three focal values at are
where is positive constant.
By scaling and [where new , and ], then system (2.4) becomes
(8)
Therefore the three focal values of and by using Lemma 2.2 namely are
If then O, is strong focus which is unstable for and stable if , and for , then O is weak focus of order one which is stable.
By using Hopf-bifurcation (by changing of stability), for no limit cycle because no change of stability if , then O is weak focus of order one which is stable. Thus as a decreasing from −1 O becomes unstable and one stable limit cycle appears from Hopf-bifurcation.
Theorem 2.3
For the system (2.4) has a unique stable limit cycle.
Proof:
Now we apply theorem (1.1) consider since So has only one root which is . For
the roots are . The roots of are and has minimum at .
Since we have , then we deduce that for
since then the term and the value of in the interval so for . Finally since so we have .
Case II: The Origin is a saddle
In this case, we discuss system (2.4) when and as above the system has three equilibrium points and where trance the to the Origin by the relation
(9)
Let ,, (2.4) is converted into
(10)
By using Hopf-bifurcation, for no limit cycle because no change of stability if , then O becomes weak focus of order one which is stable.
Thus for fixe b is bifurcate value so as increasing, O becomes unstable and one stable limit cycle appear from Hopf-bifurcation.
Lemma 3.1
equivalent to .
Proof
Assume that since , then we have contradiction. Thus for also we get .
Lemma 3.2 [10]
If there exists a constant such that for , System (2.8) has at most one limit cycle.
We have by putting , and after simplify we have
.
Let so we have
Since , we can delete from upper equation and for suitable a as small enough we have
.
3. Conclusion
A non-Liénard system is studied and analyzed by adapting Hopf-bifurcation theory. It has been proved that the system has unique limit cycle under some change of parameters under two cases. Bendixons theorem is used to prove non-existence of limit cycles.
Acknowledgements
I would like to express my thanks to Prof. V.P. Sing, Depr. of Mathematics Albaha University for his voluble suggestion for improving the paper.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Saeed, A.E.M. and Hamid, A.B. (2019) On Non Liénard Type with One Limit Cycle. Journal of Applied Mathematics and Physics, 7, 3031-3036. https://doi.org/10.4236/jamp.2019.712213
References
- 1. Ali, E. and Luo, D.J. (2006) Limit Cycle Problem for the Quadratic System of Type. (III)m=0’ I*. Journal of Nanjing University (Natural Sciences), 23, 1-10.
- 2. Deconinck, B. (2008) Dynamical System and Chaos: An Introduction. University of Washington USA, Seattle.
- 3. Wang, D.M. and Zheng, Z.M. (2005) Differential Equations with Symbolic Computation. Birkhäuser, Basel. https://doi.org/10.1007/3-7643-7429-2
- 4. Delange, E. (2006) Neuron Models of the Generic Bifurcation Type Network Analysis and Data Modeling. M.Sc. in Electrical Engineering, Technische Universiteit Delft, Delft, Pays-Bas de nationalit nerlandaise.
- 5. Perko, L. (2000) Differential Equations and Dynamical Systems. Third Edition, Springer, New York.
- 6. Gaiko, V.A. (2014) Limit Cycle Bifurcations of a Special Liénard Polynomial System. Advances in Dynamical Systems and Applications, 9, 109-123.
- 7. Jiang, F.F. and Sun, J.T. (2014) On the Uniqueness of Limit Cycles in Discontinuous Liénard-Type Systems. Electronic Journal of Qualitative Theory of Differential Equations, No. 71, 1-12. https://doi.org/10.14232/ejqtde.2014.1.71
- 8. Coppel, W.A. (1991) A New Class of Quadratic Systems. Journal of Differential Equations, 92, 360-372. https://doi.org/10.1016/0022-0396(91)90054-D
- 9. Arnaud (2002) The McKean’s Caricature of the FitzHugh-Nagumo Model. I. The Space Champed System. SIAM Journal on Applied Mathematics, 63, 459-484. https://doi.org/10.1137/S0036139901393500
- 10. Chen, B.L. and Martin, C.F. (2004) FitzHugh-Nagumo Model and Signal Processing the Visual Cortex of Fly. 43rd IEEE Conference on Decision and Control, Nassau, 14-17 December 2004, 14-17. https://doi.org/10.1109/CDC.2004.1428696
- 11. Luo, D.J., Wang, X., Zhu, D.M. and Han, M.A. (1997) Bifurcation Theory and Methods of Dynamical Systems. World Scientific Publ. Co., Singapore. https://doi.org/10.1142/2598
- 12. Hayashi, M. (2011) On Canard Homoclinic of a Lienard Perturbation System. Applied Mathematics, 2, 1221-1224. https://doi.org/10.4236/am.2011.210170
- 13. Ringqvist, M. (2006) On Dynamical Behaviour of FitzHugh-Nagumo Systems. Research Reports in Mathematics Number 5, Stockholm University, Stockholm.
- 14. Buric, N. and Todoravic, D. (2003) Dynamics of FitzHugh-Nagumo Excitable Systems with Delayed Coupling. Physical Review E, 67, Article ID: 066222. https://doi.org/10.1103/PhysRevE.67.066222
- 15. Rabinovitch, A. and Friedman, M. (2009) The Modified FitzHugh-Nagumo System as an Oscillator. Mathematical Methods in the Applied Sciences, 32, 371-378. https://doi.org/10.1002/mma.1048
- 16. Ringkvist, M. and Zhou, Y. (2009) On the Dynamical Behaviour of FitzHugh-Nagumo Systems: Revisited. Nonlinear Analysis, 71, 2667-2687. https://doi.org/10.1016/j.na.2009.01.149
- 17. Franca, R.S., Prendergast, I.E., Eva-Shirley, S., Sanchez, M.A. and Berezovsky, F. (2001) The Role of Time Delay in the Fitzhugh-Nagumo Equations: The Impact of Alcohol on Neuron Firing. Cornell University, Dept. of Biometrics Technical Report, BU-1577-M.
- 18. Sabatini, M. and Villari, G. (2011) On the Uniqueness of Limit Cycles for Liénard Equations.
- 19. Hayashi, M. (2016) Unique Existence and Nonexistence of Limit Cycles for a Classical Lienard System. Advances in Dynamical Systems and Applications, 11, 59-65.