Volume 1, No. 8, August 2013 Journal of Global Research in Mathematical Archives RESEARCH PAPER Available online at http://www.jgrma.info ANALYTICAL PROCESS FOR DETERMINATION OF HOPF BIFURCATIONS Tarini Kumar Dutta 1, Anil Kr.Jain 2 and Nabajyoti Das 3 1 Professor (Dr.) Tarini Kumar Dutta, Department of Mathematics, Gauhati University, Guwahati, Assam, Pin code- 781014, India. Email: tkdutta2001@yahoo.co.in 1 2 Anil Kumar Jain, Assistant Professor, Department of Mathematics, Barama College, Barama, Assam, Pincode-781346 and India Email: jainanil965@gmail.com 2 3 Nabajyoti Das, Assistant Professor, Department of Mathematics, Jawaharlal Nehru College, Boko 781123: India E-mail: dnabajyoti09@gmail.com 3 Abstract: In this paper we consider a two dimensional model of the type where a and b are tunable parameters. The Hopf bifurcation theorem provides a powerful analytical tool for exploring properties of periodic solutions of ordinary differential equations [2]. We have used this theorem [1, 4, and 12] for the determination of Hopf bifurcations in nonlinear differential equations and finally applied to show the existence of supercritical and subcritical Hopf bifurcations with some snapshots of periodic oscillations in our model. Keywords: Limit Cycles, Periodic oscillation, Hopf bifurcation, Subcritical and Supercritical Hopf bifurcation 1. INTRODUCTION: In the theory of bifurcations, a Hopf bifurcation refers to the local birth and death of a periodic solution as a pair of complex conjugate eigenvalues of the linearization around the fixed point which crosses the imaginary axis of the complex plane as the JGRMA 2012, All Rights Reserved 44
parameter varies. Under reasonably generic assumptions about the dynamical system, we can expect to see a small amplitude limit cycle branching from the fixed point[2-8]. In 1980, the Predator-Prey model (1) was considered by G.M Odell where and are the dimensionless populations of the prey and the predator, and is the control parameter[12]. Here to discuss the Hopf bifurcations, we consider the cousin of the equation (1) as (2) where a and b are tunable parameters. 2 HOPF BIFURCATION THEOREM:[1,4,12] Let us consider the planer system where is a parameter.suppose it has a fixed point, which may depend on. Let the eigenvalues of the linearised system about this fixed point be given by. Suppose further that for certain value of,say, the following conditions are satisfied. 1. where (non-hyperbolicity condition: conjugate pair of imaginary eigenvalues) 2. (transversality condition: the eigenvalues cross the imaginary axis with non-zero speed) 3. With etc. (genericity condition) Then a unique curve of periodic solutions bifurcates from the origin into the region if or if. The origin is a stable fixed point for (resp. ) and an unstable fixed point for (resp. ) if (resp. ) whilst the periodic solutions are stable (resp. unstable) if the origin is unstable (resp. stable) on the side of where the periodic solutions exist. The amplitude of the periodic orbits grows like whilst their periods tend to as. The bifurcation is called supercritical if the bifurcating periodic solutions are stable, and subcritical if they are unstable. If then bifurcation is subcritical meaning that the periodic orbit is unstable. If then bifurcation is supercritical meaning that the periodic orbit is stable. 2.1: OUR MAIN STUDY: Here the concern model is JGRMA 2012, All Rights Reserved 45
where a is the control parameter and b is constant. The fixed points of the system are The jacobian of the system is Eigenvalues at are, which are clearly real values for all a Eigenvalues at are,which are clearly real values for any values of a and b. Eigenvalues at are,which gives complex eigenvalues for some values of a and b. Now the linearised system of above equation at the fixed point is given as Jacobian of the linearised system is Now equilibrium point of linearised system is ( 0, 0 ) Hence eigenvalues at this fixed point are For complex eigenvalues we must have This gives where and And for Hopf bifurcation we must have (3) This gives (4) In the following Table2.1, we have shown different values of the parameter where Hopf bifurcation occurs. JGRMA 2012, All Rights Reserved 46
a b = 2 a D Eigenvalues 0.1 0.2 0.102382 4.29762 True 0.2 0.4 0.20911 4.59089 True 0.3 0.6 0.319649 4.88035 True 0.4 0.8 0.433568 5.16643 True 0.5 1.0 0.55051 5.44949 True 0.6 1.2 0.670178 5.72982 True 0.7 1.4 0.792319 6.00768 True 0.8 1.6 0.916718 6.28328 True 0.9 1.8 1.04319 6.55681 True 1.0 2.0 1.17157 6.82843 True 1.1 2.2 1.30172 7.09828 True 1.2 2.4 1.43352 7.36648 True 1.3 2.6 1.56685 7.63315 True 1.4 2.8 1.701613 7.89839 True 1.5 3.0 1.837722 8.16228 True 1.6 3.2 1.975097 8.4249 True 1.7 3.4 2.113665 8.68634 True 1.8 3.6 2.253360 8.94664 True 1.9 3.8 2.394123 9.20588 True 2.0 4.0 2.535898 9.4641 True 2.1 4.2 2.678637 9.72136 True 2.2 4.4 2.822291 9.97771 True 2.3 4.6 2.966820 10.2332 True 2.4 4.8 3.112182 10.4878 True 2.5 5.0 3.258343 10.7417 True 2.6 5.2 3.405267 10.9947 True 2.7 5.4 3.552923 11.2471 True 2.8 5.6 3.701282 11.4987 True 2.9 5.8 3.850316 11.7497 True 3.0 6.0 4.000000 12.0000 True 3.1 6.2 4.15031 12.2497 True 3.2 6.4 4.30122 12.4988 True 3.3 6.6 4.45271 12.7473 True 3.4 6.8 4.60476 12.9952 True 3.5 7.0 4.75736 13.2426 True 3.6 7.2 4.91048 13.4895 True 3.7 7.4 5.0641 13.7359 True 3.8 7.6 5.21822 13.9818 True 3.9 7.8 5.37281 14.2272 True JGRMA 2012, All Rights Reserved 47
4.0 8.0 5.52786 14.4721 True 4.1 8.2 5.68336 14.7166 True 4.2 8.4 5.8393 14.9607 True 4.3 8.6 5.99565 15.2043 True 4.4 8.8 6.15242 15.4476 True 4.5 9.0 6.30958 15.6904 True 4.6 9.2 6.46714 15.9329 True 4.7. 9.4 6.62507 16.1749 True 4.8 9.6 6.78336 16.4166 True 4.9 9.8 6.94202 16.658 True 5.0 10.0 7.10102 16.899 True Table2.1 : Showing different values of the parameter where Hopf bifurcation occurs In particular if we consider Then (4) implies From (3) we see that Clearly Hopf bifurcation occurs at Now we verify different conditions of Hopf bifurcation theorem. is satisfied. 1. Non hyperbolic condition: Now to determine the sign of let Hence at the fixed point. So we take sign of as positive i.e. 2. Transversality condition: At ie we have Thus JGRMA 2012, All Rights Reserved 48
3. Genericity condition Tarini Kumar Dutta et al, Journal of Global Research in Mathematical Archives, August 2013, 44-52 Again let Now Since Now the origin is (i) stable fixed point if if for (ii) Unstable fixed point if if for Since, i.e. stability coefficient is negative, the limit cycle is stable and the Hopf bifurcation is supercritical.[1] Fig2.1a: Phase portrait at a=0.5 JGRMA 2012, All Rights Reserved 49
Fig2.1b: Phase portrait at a=0.61 Fig2.1c: Phase portrait at a= 1.21 JGRMA 2012, All Rights Reserved 50
Fig2.1d: Phase portrait at a= 1. 0 and b=2.0 REFERENCES: [1] Alecea, M.R., Introduction to Bifurcations and The Hopf Bifurcation Theorem for Planar Systems Dynamics at the Horsetooth, M640, 2011 [2] Das.N, Dutta.T.K, Determination of Supercritical and Subcritical Hopf Bifurcation on Two Dimensional Chaotic Model IJASRT.ISSN2249-9954,Issue2,Vol.1Feb2012. [3] Hopf, E., Abzweigung einer periodischen Losung von einer stationaren Losung eines Differential systems, Ber. Verh. Sachs. Akad. Wiss. Leipsig Math.-Nat. 94(1942), 3-22, Translation to English with commentary by L. Howard and N. Kopell,in[81;163-205] [4] Heijden,V.D., Hopf Bifurcation http://www.ucl.ac.uk/ ucesgvd/hopf.pdf [5] Marsden, J. E. and McCracken, M., The Hopf Bifurcation and Its Applications, Springer-Verlag,New York, 1976 [6] Moiola, J. L. and Chen, G., Hopf Bifurcation Analysis: a frequency domain approach, World Scientific, 1996 [7] Murray, J. D., Mathematical Biology I: An Introduction, Third Edition (2002), Springer [8] Peitgen H.O., Jurgens H. and Saupe D., Chaos and Fractal, New Frontiers of Science, SpringerVerlag, 1992 [9]Roose, D. and Hlavacek, V., A Direct Method for the computation of Hopf bifurcation points, SIAM J. APPL. MATH., Vol. 45, No. 6, December 1985 [10].Roussel,M.R., Introduction to bifurcations Sept16,2005 JGRMA 2012, All Rights Reserved 51
[11]Sarmah, H.K, Paul.R anddutta,n, Hopf Bifurcation and Normal Forms of Nonlinear Oscillators IJRPH,ISSN0976-5840,Vol.4 No.3(2012)PP.309-321. [12] Strogatz, S. H., Nonlinear Dynamics and Chaos. Cambridge MA: Perseus, 1994 JGRMA 2012, All Rights Reserved 52