The Hopf Bifurcation Theorem: Abstract. Introduction. Transversality condition; the eigenvalues cross the imginary axis with non-zero speed

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1 Supercritical and Subcritical Hopf Bifurcations in Non Linear Maps Tarini Kumar Dutta, Department of Mathematics, Gauhati Universit Pramila Kumari Prajapati, Department of Mathematics, Gauhati Universit Sankar Haloi, Department of Mathematics, Gauhati Universit Abstract We consider a modified form of the Brusselator model as described below x 1 ( b 1) x ax bx ax where a, b > 0 are parameters and x, 0 are dimensionless concentrations We develop a general theor for evaluating Supercritical and Subcritical Hopf Bifurcation for Brusselator model and obtain the Hopf bifurcation points We also highlight some dnamical properties and time series analsis of the bifurcation points A few open problems are posed at the end Ke words: Limit ccles/ Stabilit/ Supercritical /Subcritical Hopf Bifurcation/ Time series analsis 010 Subject Classification: 37 G 15(P), 37 G 35(S) Introduction The term Hopf Bifurcation(also known as Poincare Andronov-Hopf Bifurcation),named after Henri Poincare, Eberhard Hopf and Aleksander Andronov, refers to the local birth and death of a periodic solution from an equilibrium as a parameter crosses a critical value It is the simplest bifurcation not just involving equilibria and therefore belongs to what is sometimes termed as dnamic bifurcation theor Hopf bifurcation occurs when a complex conjugate pair of eigenvalues linearised about the fixed point crosses the imaginar axis of the complex plane There are two tpes of Hopf bifurcation, supercritical and Subcritical Hopf bifurcation A supercritical hopf bifurcation occurs when stable limit ccles are created about an unstable critical point whereas subcritical hopf bifurcation occurs when an unstable limit ccle is created about stable critical point In this paper we have studied the nullclines and have shown its direction field Next we have shown that our model satisfies Hopf bifurcation theorem and have viewed the time evaluation and phase space graphwe have also studied the behavior of the real and imaginar eigenvalues The Hopf Bifurcation Theorem: We consider the planar sstem dx d f ( x, ; ); g( x, ; ) dt dt where μ is an adjustable parameter Suppose it has a fixed point, which ma depend on μ Let the eigenvalues of the linearised sstem about the fixed point be _ given b ( ), ( ) i Suppose further that for a certain value of, sa the following conditions are satisfied: I Non-hperbolicit condition ; conjugate pair of imaginar eigenvalues ( 0 ) 0, ( 0 ) 0 where g sign( ) sign[ ( x 0, 0 )] x II III with Transversalit condition; the eigenvalues cross the imginar axis with non-zero speed d ( ) d 0 Genericit Condition: f x d 0 f ( ) ( x0, 0 ) x 0 Then a unique curve of periodic solutions bifurcates from the origin into the region If A d<0 INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY & SCIENCE VOLUME 4, NUMBER, MARCH

2 or μ < μ0 if Ad > 0 The origin is a stable fixed point for μ > μ0 (resp μ < μ0) and an unstable fixed point for μ < μ0 (resp μ > μ0) if d < 0 (resp d > 0) whilst the periodic solutions are stable (resp unstable) if the origin is unstable (resp stable) on the side of μ = μ0 where the periodic solutions exist The amplitude of the periodic orbits grows like whilst their periods tend to as μ tends to μ0 The bifurcation is called supercritical if the bifurcating periodic solutions are stable, and subcritical if the are unstable If > 0 the periodic orbit is unstable ie, the bifurcation is subcritical If < 0 the periodic orbit is stable ie, the bifurcation is supercritical (a) 1 The main Stud of our model We have considered the Brusselator model x 1 ( b 1) x ax bx ax where, are controlled parameter and are dimensionless concentration 1 The Nullclines and vector field : Our model contains two variables,so it is useful to resort to the nullclines representation These curves are defined b on one hand and on the other hand Thus, the intersection of these curves corresponds to the stead state, for which we have simultaneousl and SUPERCRITICAL AND SUBCRITICAL HOPF BIFURCATIONS These nullclines are sketched along with the some representative vectors The delimit the region in the phase space where the vector field has a particular directionin the different regions delimited b these nullclines, it is possible to determine the direction of the evolution of the sstem b studing the sign of and of (b) Figure1: (a) shows nullclines and (b) direction field; where the horizontal axis shows the x-axis and the vertical axis shows the -axis Our Main Result : After finding the nullclines and vector field we proceed towards linear stabilit analsis of our model The fixed point of the model is obtained as The Jacobian of the sstem is At the stead state we have 15

3 Eigenvalues at are which gives complex root for some values of and Now linearising the sstem about the fixed point we get True True True True True True True True True True True True True Jacobian about the origin Hence eigenvalues at the fixed point is For complex eigenvalues we must have This gives This implies Table showing different values of Hopf bifurcation with the increasing value of the parameter and with the condition Since and,let us assume the value of as Consequentl the value of becomes Now when and, we find this implies ) where For Hopf Bifurcation to occur we must have This gives In the following table we have shown the different values of the parameters and where Hopf Bifurcation occurs b appling the Mathematica Software [4,5] 1 A b p Eigenvalues True True True True True True True True True True True True True Clearl Hopf Bifurcation occurs at and Now we verif the different conditions of the Hopf Bifurcation theorem [5] i Non-hperbolicit Condition: INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY & SCIENCE VOLUME 4, NUMBER, MARCH 016 and Now we determine the sign of Let At the fixed point Hence the sign of is negative ie, ii Transversalit condition: (sa) 16

4 iii Genericit condition : Let (d) (e) Fig: (c) and (d) shows the phase portraits when the value of a=11 and b=18, whereas in (e) the value of a= and b=3 The horizontal line shows x-axis while the vertical line shows -axis Since ie, stabilit coefficient is negative,the limit ccle is stable and hence our concerned model undergoes Supercritical Bifurcation 1 Linear Stabilit Analsis The Brusselator model is x 1 ( b 1) x ax bx ax where, are controlled parameter and are dimensionless concentration The stead state is given as (c) The trace Tr and the determinant are: of the Jacobian Matrix SUPERCRITICAL AND SUBCRITICAL HOPF BIFURCATIONS 17

5 We stud the sign of = We find that the determinant the trace is positive if is alwas positive whereas otherwise negative is positive if The following table shows the different behavior as a function of the parameter and [3,4] b tpes stable sta- unsta unstable of node ble ble node stead fo- fo- state cus cus When parameter b increases, the stead state turns from a stable node to a stable focus, then, it loses its stabilit (the sstem then evolves towards a limit ccle) and the stead state turns from an unstable focus to an unstable node These time evaluation and phase portrait is shown for different values of the parameter and where parameter is increased keeping constant at 3 INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY & SCIENCE VOLUME 4, NUMBER, MARCH

6 We obtain the graph for the following Fig: 4 In the above plot we keep constant at 3 and var within the range of to and we obtain a straight line parallel to the axis Fig: 5 Different kind of behavior obtained for the Brusselator model with parameter a=3 and 05<b<14 Left panels: time evolution Right panels: phase space Time is plot along the abscissa and x, is plot along the ordinate From the following figures we can stud the behavior of our model We have fixed at 3 and have observed the different behavior of the time evaluation and phase portraits as the parameter increases As we increase keeping constant at 3, we find that the sstem evolves towards a stead state (stable node) and then evolves towards a stable focus There is a formation of limit ccle at which is quite distinct at Then the sstem leaves its stead state (unstable focus) to reach a limit ccle at and an unstable node occurs at Hence this transition from an asmptoticall stable equilibrium point to an unstable equilibrium point enclosed b an attracting limit ccle is supercritical Hopf Bifurcation[,7] Hence from the equation we have obtained the following Fig: 5 In Fig: 5, we obtain a curve which shows that when constant at 3 and complex conjugate pairs is we obtain Fig: 6 SUPERCRITICAL AND SUBCRITICAL HOPF BIFURCATIONS 19

7 The following plot is for real part of In the Fig: 6 we find that the real of eigenvalues remains negative when lies between to 39 At it the real part becomes zero and then it gives the positive value of the real part [7] Roussel, MR, Introduction To Bifurcation Sept 16, 005 [8] S Lnch, Dnamical Sstem with Application Using Mathematica, Birkhauser, 007 [9] Strogatz S H, "Nonlinear Dnamics and Chaos" Cambridge MA: Perseus, 1994 [10] Sarma, H K, Paul R and Dutta, N, "Hopf Bifurcation and Normal Forms of Nonlinear Oscillators", IJRPH,ISSN ,Vol4 No3(01)PP Biographies Fig:7 The above plot is for imaginar part of In the Fig:7 we find that the imaginar part of the eigenvalues appears onl when lies between at constant Open Problems: 1 Can we find the Hopf bifurcation for higher dimensional equations b appling the same technique developed in this paper Can we establish a suitable relation between Hopf Bifurcation and Period Doubling Bifurcation 3 Can we develop a sophisticated theor in order to find Hopf Bifurcation in case of nonlinear maps 4Can we appl the theor of Hopf Bifurcation in order to control chaos in nonlinear sstem References [1] Alecea, MR, "Introduction to Bifurcation and The Hopf Bifurcation Theorem for Planar Sstems", Dnamics at the Horsetooth,M640,011 [] Brian G Higgins, Nonlinear Analsis: Case Stud, 010 [3] D Gonze, M Kaufman, "Theor of non-linear dnamical sstems", Januar 6, 014 [4] Dutta TK, Das N, Determination of Supercritical and Subcritical Hopf Bifurcation on Two Dimensional Chaotic Model,IJASRT,ISSN ,Issue,Vol1Feb01 [5] Dutta TK, Jain AK, Das N," Analtical Process For Determination Of Hopf Bifurcations" JGRMA, ISSN 30-58,Vol 1,No8,Aug 013 [6] Marsden, JE and McCracken, M, The Hopf bifurcation and Its Application Springer-Verlag, New York, 1976 TARINI KUMAR DUTTA a senior professor of the Mathematics department of Gauhati universit He obtained MSc degree with First Class First position from Gauhati universit in 1974, PhD from Edinburgh Universit (Scotland) in 1987, Post Doct from Edinburgh Universit and ETH, Switzerland in under Commonwealth Scholarship and Fellowship schemes respectivel After working for a few ears in Ara Vidapeeth college, Handique Girls college and Dibrugarh universit in Assam state, he has been working in Gauhati universit since 1978 till date he visited UK, France, German, Ital, Switzerland, USA and Bangladesh for different academic programs, has published about 60 research papers, and has produced 11 PhD students His research fields are Dnamical Sstems, Functional Analsis and Abstract algebras Professor Tarini Kumar Dutta ma be reached at tkdutta001@ahoocoin Pramila Prajapati is a research scholar at the department of Mathematics of Gauhati Universit, She obtained her M Sc degree with First Class from Gauhati Universit in 011 Her research field are Nonlinearit and Chaos She ma be reached at pramilaprajapati1987@gmailcom Sankar Haloi is an Assistant professor in the Mathematics department of Cotton college in Assam State He obtained his M Sc Degree with First Class from Gauhati universit in 1995, working as an Assistant professor in Haflong Govt college during , Jorhat Engineering college during and is at present in Cotton college, Guwahati His research field is Dnamical Sstems He ma be reached at sankarcottongh@rediffmailcom INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH IN TECHNOLOGY & SCIENCE VOLUME 4, NUMBER, MARCH 016 0