Research Article Bifurcations in Van der Pol-Like Systems

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1 Mathematical Problems in Engineering Volume 3, Article ID 3843, 8 pages Research Article Bifurcations in Van der Pol-Like Sstems Orhan Ozgur Abar,, Ilknur Kusbezi Abar, 3 and Avadis Simon acinlian,4,5 Department of Mathematics, Gebze Institute of Technolog, Kocaeli, 44 Gebze, Turke Department of Information Sstems and Technologies, Yeditepe Universit, Atasehir, Istanbul, Turke 3 Department of Computer Education and Instructional Technolog, Yeditepe Universit, Atasehir, Istanbul, Turke 4 Department of Phsics, Yeditepe Universit, Atasehir, Istanbul, Turke 5 Department of Phsics, Bogazici Universit, Bebek, 3434 Istanbul, Turke Correspondence should be addressed to Orhan Ozgur Abar; oabar@editepe.edu.tr Received October 3; Accepted 5 November 3 Academic Editor: Panos Liatsis Copright 3 Orhan Ozgur Abar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. Generalizations of the Van der Pol sstem with polnomial interactions involving additional parameters are studied in order to understand qualitative properties such as stabilit and additional bifurcations in these generalized Van der Pol sstems. The generalizations include the one known as the Duffing-Van der Pol sstem which has properties similar to those of the Mawell- Bloch sstem.. Introduction Dnamical sstems underling man oscillator real life problems that arise in applied sciences can be considered as generalizations of the Van der Pol (henceforth referred to as VP) sstem []. Due to its wide area of applications in a number of scientific areas such as electronics and geolog, the VP sstem stands as an important eample of dnamical sstems. It was first developed b the Dutch phsicist Balthasar Van der Pol ( ) to model power amplifiers in vacuum tube radios ehibiting negative resistance []. Methods referred to b names such as averaging b Guckenheimer and olmes [] and two-variable epansion b Rand and Armbruster [3] have been applied to thissstem. Related tpes of averaging methods have also been applied b Strogatz [4], Verhulst [5, 6], Sanders et al. [7], Kuznetsov [8], Jordan and Smith [9], and ale and Koçak []tosimilar oscillator problems. We use standard averaging in the rest of this paper [, 5, 7]. Oscillator solutions to dnamical sstems using the center manifold and normal form methods, and bifurcation and stabilit analsis have been etensivel studiedbcaoandxiao[ 3]. This paper aims to look at possible generalizations of VP and Duffing-like sstems and to stud their bifurcation schemes in order to clarif our understanding. The methodolog, some of the observed bifurcation scenarios, and attractor structures resemble those reported b Cao and Xiao in other sstems [ 3]. Section discusses quadratic generalizations of the VP sstem including the one known as the Duffing-Van der Pol (henceforth referred to as DVP) sstem and compares the averaging results with the averaged versions of some commonl known similar dnamical sstems. Section 3 introduces a new generalization for the DVP sstem and investigates results obtained b methods discussed in previous sections. In Section 4, generalizations that introduce higher nonlinearities are studied in the same manner. Overall results are compared and discussed in Section 5. The VP equation is the second-order ordinar differential equation μ( ) += () modeling the VP oscillator []. The part of the VP equation including together with its coefficient implies that the friction or the resistance represented b the nonlinear-damping term changes sign and becomes negative as the amplitude

2 Mathematical Problems in Engineering increases. This model can be derived from the Raleigh equation [4]andcanbereducedtosimpleharmonicmotionfor μ=with the solution (t) = c cos(t) + c sin(t) (c,c =). The generalized models under stud can be stated as vector equations given in sstem () bthetransforma- tion (, ) (,) in order to use dnamical sstems approaches for their stud [5, ]. The generalized VP equations [] under stud are special cases of the following sstem: =a, () =P() +Q(), where P()andQ() are polnomials in the variable and we can alwas set a=b aand P() ap().sstem ()isalsoknownastheliénard sstem [5]. Additional parameters in the nonlinearities can be introduced to generate and analze further bifurcation varieties as follows: =, (3) = α+β( ), so that the generalized VP equation is now +α β( ) =. (4) We carr out the stabilit analsis b looking into the eigenvalues of the linearized matri of coefficients. In this manner, the sstem has a trivial equilibrium point at the origin with the eigenvalues {(/)(β± β 4α)}. Scaling the time b the undamped angular frequenc α reveals that the onl nontrivial parameter is β (/α) (α =),sothatβ itself affects directl the stabilit of the sstem. Bifurcation analsis is the ke to understanding the qualitativebehaviorofadnamicalsstem.qualitativechanges, if an, in the sstem when an of the parameters are varied, represent bifurcations and help us understand the behavior of the sstem better. Using the software package MAT- CONT, bifurcations of the sstems of concern are studied numericall [8]. Choosing β as the bifurcation parameter, the obvious scenario for the VP sstem is obtained such as a famil of limit ccles growing endlessl from the opf bifurcation point at the trivial equilibrium point indicated b a nonnegative first Lapunov coefficient []. Considering that in a phsical sstem all parameters are nonnegative, it canbesaidthattheequilibriumpointofsstem(3)isalwas unstable. Details of other bifurcation properties for the simple VP sstem as given in sstem (3) including transcritical bifurcation are omitted here for brevit [9,, 6]. Before rescaling, the equation has two nontrivial coefficients, one determining the undamped frequenc and the other determining the damping term. The latter changes sign as the amplitude is increased. A generalization of the VPequationshouldinvolveeitherorbothoftheseterms. Ourobjectofstudwillbegeneralizationswherenonlinear functions on will be used instead of the term α in equation. Nonlinear functions depending on can be used as generalizations [7].. The Generalized Van der Pol Sstem including Quadratic Self-Nonlinearit Replacing the linear term in the equation of sstem (3) b the quadratic combination( ) modifies the sstem to =, = α( ) +β( ). In addition to the trivial equilibrium point at the origin that alwas occurs, a second equilibrium point at (, ) is observed in this case. The eigenvalues (i.e., the eigenvalues of the coefficients matri of the linearized sstem) for the trivial equilibrium point are the same as those of the original one {(/)(β ± β 4α)}andtheeigenvaluesforthesecond equilibrium point are {± α}. Forβ < 4α, the origin has pure imaginar eigenvalues that indicate opf bifurcation. We want to analze opf bifurcation in this sstem b using the following theorem. Theorem (opf bifurcation, [8]). Aopfbifurcationoccurs if a planar sstem = f(, η), R, η R,hasthefollowing eigenvalues at origin for sufficientl small η taken as bifurcation parameter: (5) λ, (η) = ξ (η) ± iγ (η), (6) where real part ξ(η) of λ, (η) is zero for η=and imaginar part γ(η) of λ, (η) is γ =. (.) The first Lapunov coefficient defined as l (η ) = η= w (ig g +w g ) (7) is not equal to zero. (.) Consider dξ(η)/dη η= = (transversalit condition). B using Theorem, we require that the determinant of the Jacobian matri be greater than zero and trace of the Jacobian matri be zero at the related equilibrium point. According to criterion (.) in Theorem, opfpointalso obes the transversalit condition if the rate of change of real part of eigenvalues at the related equilibrium (codimensionone opf) point is greater than zero which means that eigenvalues of the linearized flow cross the imaginar ais with nonzero derivative when bifurcation parameter η is zero [8, 9]. We use the first Lapunov coefficient to epress opf bifurcation criteria for bifurcation parameter η. When the sstem has pure imaginar eigenvalues at the related equilibrium (opf) point, l (η) =. In the neighborhood of the opf point, if the first Lapunov coefficient l (η) is less than zero, supercritical opf bifurcation can be observed; if l (η) is greater than zero, subcritical opf bifurcation can be observed [8, 9]. When these conditions are satisfied, the sstem has codimension one opf bifurcation [8]. If l (η) vanishes and becomes zero, this tpe is called degenerate (generalized) opf Bifurcation [8].

3 Mathematical Problems in Engineering 3 We see that trace and determinant of the Jacobian matri at originare β and α, respectivel. B appling the conditions, we obtain pure imaginar roots ±i α depending on β=, α>; the transversalit condition gives (d/dβ)(β/) β= = / > at the origin. Under the assumption α > and sufficientl small bifurcation parameter β [8], we have the following conditions at the origin: if α>, β>,subcritical opf bifurcation is observed with negative first Lapunov coefficient and if α>, β<, supercritical opf bifurcation is observed with positive first Lapunov coefficient [8]. The famous limit ccles of the VP equation are observed to bifurcate from the opf bifurcation point (indicated b ) of the sstem (5)asshowninFigure.Whenβ=,thefirst Lapunov coefficient is zero. The determinant and trace at the other equilibrium point (, ) are α and zero, respectivel. B using the above conditions, α islessthanzeroandrateofchangeinrealpartof eigenvalues which is epected to be nonzero vanishes. At this equilibrium point we thus do not observe opf bifurcation as shown in Figure according to Theorem. Definition (normal form epansion). Let the sstem of differential equation =f(,)and = g(, ) be given where, R and f, g R R. To consider the sstem near its equilibrium point = and = such that f(, )=and g(, )=,thispointismovedtothe origin b = and =. Talor epansion near =and =gives =a +b +f (,)+, (8) =a +b +g (,)+, where a, a, b, b are coefficients and f (, ) and g (, ) contain terms of degree two and higher. Diagonalizing or bringing the linear part to the Jordan canonical form b a linear transformation produces ( )=(J J )( )+( f (, ) ), (9) J J g (, ) where J ij, i, j {, } denote Jordan canonical form terms of the linear part in sstem (8). Then a sequence of near identit transformations of the form =] +F (],θ) +, () =θ+g (],θ) + is applied to simplif the sstem (8), where F (],θ) and G (],θ)contain terms of degree two and higher. The normal form epansion is given as follows: ] =a ] +b θ+ F (],θ) +, θ=a ] +b θ+ G (],θ) +, () where F (],θ) and G (],θ) contain degree two and higher order terms. We use normal form epansion around the trivial equilibrium point (origin) to ield opf bifurcation as defined Figure:Phaseplanesofsstem(5) withα =, β variable. The degenerate opf bifurcation point (indicated b ) is observed at the trivial equilibrium point (, ). in Definition. Intherestofthiswork,] and θ will refer to the variables in the near identit transformation [9]. For eample, sstem (5) has the following normal form epansion up to the fifth order which leads to degenerate opf bifurcation at origin for β= ] = i] α 3/ ( α + 5α 3 θ= iθ α 3/ (α 5α 3 (]θ) (]θ) ), (]θ) (]θ) ), () which is also consistent with the opf bifurcation observed at origin in Figure. Figure gives the bifurcation curves with lines indicating pitchfork bifurcation points (indicated b BP). The bifurcation analsis while relaing the bifurcation parameter α shows a pitchfork and a degenerate limit point bifurcation (indicated b LP) point with limit point coefficient equal to zero as shown in Figure.Forα=, =,the ais contains equilibrium points with corresponding eigenvalues λ, = {, β} that satisf the limit point (saddle-node) bifurcation criteria [5, 8]... Averaging of the Generalized Van der Pol Sstem including Quadratic Self-Nonlinearit. The method of averaging is a powerful perturbation method which gives a normal form thatcanbecomparedtomorecommonlknowndnamical sstems. Numericall obtained results concerning stabilit and bifurcation can also be analticall confirmed using the method of averaging. In order to understand the behavior of the sstem, we investigate the averaged version of sstem (5) and observe its similarit to averaged versions of some better known sstems. The averaged equations of sstem (3) and(5) arecalculated b the usual method discussed in [, 5, 7]. We introduce a small parameter ε, <ε and replace α and β b +εα, εβ, respectivel, to analze this sstem for small parameter

4 4 Mathematical Problems in Engineering.6.5 BP LP BP α.4 Figure : Bifurcation graph of sstem (5) with α variable where a pitchfork bifurcation point (indicated b BP) is observed for α= that forks a new curve with a limit point bifurcation point (indicated b LP). values while averaging is carried out over a period in t. We use to obtain (t) =u(t) cos t+v (t) sin t, (t) = u(t) sin t+v (t) cos t (3) u (t)= ε βu (t) (βu (t) +αv (t) (u(t) + V(t) ))+O (ε ), 4 V (t)= ε βv (t) ( αu (t) +βv (t) (u(t) + V(t) ))+O (ε ). 4 (4) According to the theorem given b Verhulst [5, 6], the origin isepectedtobetheonlequilibriumpoint.wecalculate uu+v V and uv Vu and introduce r= u + V and θ= arctan(v/u) to get εβ r r( 4 ), θ= εα (5) in the standard averaging sense [, 5, 7]. This result shows a tpical opf bifurcation form [4, 7, 8]. The stead state of r=leads either to the trivial case r=or to the limit ccle condition (β/)( (r /4)) = where β=or r =4> impling the eistence of a limit ccle of radius.numerical and averaged solutions are given in Figure 3 that also gives unstable (stable) equilibrium point for β>(β<).for small ε, asα is increased, the averaged solution does not tend to the numerical solution of the original equation as shown in Figure 4. The determinant of the Jacobian matri at the related equilibrium point (ε β /4) + ((εα ) /4) is greater than zero for all β, α, ε R. The trace becomes zero if β Averaged solutions Numerical solutions Figure 3: Numerical and averaged solutions of sstem (5) when α=, β=and ε =. near the origin where opf bifurcation is observed Averaged solutions Numerical solutions Figure 4: Numerical and averaged solutions of sstem (5)when α= 3, β =, and ε =. near the origin where opf bifurcation is observed. is zero. With the aid of Theorem, subcritical (supercritical) case is observed for sufficientl small β > (β < ), < ε, α>with the first Lapunov coefficient being less (greater) than zero. Sstem (5) is a special case of the following averaged version of the general form of generalized VP sstems: =, = α(α +α +α )+β(β +β +β 4 4 ), (6)

5 Mathematical Problems in Engineering 5 where we introduce parameters α i,β j R, {i =,, } and {j =,, 4} as coefficients of the generalization terms in parentheses. The averaged form of sstem (6) calculated b the usual method discussed in [5, 7]becomes εβ r(β + β 4 r + β 8 r4 ), θ= α (+εα) 3 8 α r ( εα) (7) b introducing ε, <ε and b replacing α, β b +εα, εβ in that given order. A number of other sstems with linear periodic parts also give similar results upon averaging. For eample, the cubic generalization of the Lotka-Volterra sstem in [9] =a( ) b k, (8) = c(+ )+d k upon averaging (replacing a, b, c, d b εa, εb, εc, εd) bthe usual method discussed in [5, 7]gives r=εr( a c 3 8 (a+c) r ), (9) where k=. θ=for k=values and 3. The Duffing-Van der Pol Sstem θ = + (εr /8)(b + d) for A further generalization of the VP sstem that introduces the cubic propert of the Duffing sstem in the potential function is the DVP sstem given b =, = α(α +α )+β( ), () where parameters α and α are introduced in the potential function [4]. Depending on the sign of the parameter α, single or double well potential cases eist. There are three equilibrium points, that is, the trivial equilibrium point at the origin and two equilibrium points at (±i α /α,).the eigenvalues at the equilibrium points are { (β ± β 4αα )}, { α (β (α +α )± 8αα α +β (α +α ) )} () respectivel. When a sufficientl small β is chosen as bifurcation parameter, trace of the Jacobian β is zero and determinant of the Jacobian matri is αα >at the origin. B using Theorem, eigenvalues at the origin are {±i αα } that give opf bifurcation at the origin as shown in Figure 5.Forβ> and α=α =α >, the first Lapunov coefficient is less than zero that gives supercritical opf bifurcation and, for β< and α=α =α >, the first Lapunov coefficient is greater than zero that gives subcritical opf bifurcation. Figure 5: Phase plane of the DVP sstem ()whenα=α =α = and β is varied where opf bifurcation is observed at the origin for β=. The Jacobian matri of sstem ()at(±i α /α,)is ( αα β(+ α ), () ) α where eigenvalues of the Jacobian matri are purel imaginar if the determinant of the Jacobian matri is αα > andthetraceofthejacobianmatriisβ( + (α /α )) =. These two conditions are used to show opf bifurcation. We see that the trace becomes zero when β=or α /α =. B assuming the latter one, the second and third equilibrium points become (±, ). Another necessar condition is that therateofchangeofrealpartofeigenvalueswithrespect to bifurcation parameter should be greater than zero where β/α >. Combining these results, we see that opf bifurcation occurs if β >, α >, α >, α < or β<, α<, α <, α >giveninfigures6 and 7 for the bifurcation parameter chosen as α. As an eample, we set the determinant of the Jacobian matriattheoriginto and obtain α = /α.sincethe pair of eigenvalues cannot have real part according to the opf bifurcation condition, we can assume that β = where β (, ) gives pure imaginar eigenvalues {±i} as shown in Figure 5. The second and third equilibrium points do not give pure imaginar eigenvalues; hence, we do not observe opf bifurcation at these equilibrium points. When α = /α, the second and third equilibrium points possess positive determinants and b using +(α /α )=and β=, sstem has pure imaginar eigenvalues {±i } as shown in Figures 6 and 7. Thisbehaviorobservedforthesinglewell potential case of the DVP sstem with parameter β free is similartothatofthelorenzssteminwhichanunstable equilibrium point is between two stable equilibrium points. Two families of limit ccles bifurcating from the two unstable opf points intersect on the bifurcation curve (line) of the sstem. The pitchfork bifurcation point forks a quadratic curvewithanegativeslopewhichpassesthroughthetwo

6 6 Mathematical Problems in Engineering BP Figure 6: Phase plane of the DVP sstem () whenβ = α = α =and α is varied where a pair of subcritical opf bifurcation points with families of limit ccles and a pitchfork bifurcation point (indicated b BP) are observed Averaged solutions β> Averaged solutions β< Numerical solutions β> Numerical solutions β< Figure 8: Numerical and averaged solutions of sstem () when α=6, α =, α =, β= and α= 6, α =, α =, β= for ε =. near the origin where opf bifurcation is observed BP b introducing a small parameter ε, <ε and replacing α, β b εα, εβ,respectivel. Numerical and averaged solutions are given in Figure 8 where the origin is a stable (unstable) equilibrium point for β<, α>, α >, α >(β>, α<,α <, α <). 3.. The Mawell-Bloch Sstem. It has been of considerable interest to relate models for laser dnamics to VP and DVP sstems. DVP model is known to give oscillator solutions which includes several opf bifurcations []. The Mawell- Blochequations(tobereferredtoasMBhenceforth)are Figure 7: Bifurcation graph of the DVP sstem () in as α is varied. subcritical opf bifurcation points so that the projection of the Lorenz attractor created would seem to intersect at the pitchfork bifurcation point BP in the two-dimensional (, ) graph. In this case, the equilibrium points of the sstem are again (, ), (, ), and(, ) and the eigenvalues are {±i }, {(± 5)/}, and{±i }, respectivel. The limit ccles go around two subcritical opf bifurcation points indicated b two positive first Lapunov coefficients. The bifurcation analsis when α is varied shows a pitchfork bifurcation curve with numerous limit point bifurcation points on it. To understand the bifurcation properties in sstem (), we calculate the averaged equations b the usual methods [, 5, 7] εβ r r( 4 ), θ= εαα 3 8 (εαα r ) (3) E = ke + gp, P= γ P + geδ, Δ= γ (Δ Δ ) 4gPE (4) that contain one unstable equilibrium point at E=P=, Δ = Δ with one positive and two negative eigenvalues and pair of equilibrium points at either side with eigenvalues {α ± βi, μ}. Whenk=σ, γ =g /k=, g (Δ /k) = r, and γ =b,mbsstem(4) can be transformed into the Lorenz sstem about the equilibrium point Δ=Δ b setting =E, =gp/k, z=δ Δ[]. It was reported that operating point shows chaotic behavior that is similar to that of the Lorenz sstem []. The behavior of the MB sstem around the point (, ) iscomparedtothebehaviorofthedvpsstem around each of the equilibrium points (, ), (, ),and(, ) in Figure 9. For both sstems, Figure 9 shows trajectories involving similar structures.

7 Mathematical Problems in Engineering DVP sstem at (, ) DVP sstem at (, ) DVP sstem at (, ) MB sstem at (, ) Figure 9: ais projection of the Mawell-Bloch (MB) sstem (k =.75, g = 5.8, γ =.66, γ =.75, Δ = 8) and Duffing- Van der Pol sstem (α=, α =, β=, α = ). The averaged version of the following further generalized sstem: is =κ, = α(+α +α 4 4 +α 6 6 +α 8 8 ) +β(+β +β 4 4 +β 6 6 +β 8 8 ), κ r(+ β 4 r + β 8 r4 + 5β 3 64 r6 + 7β 4 8 r8 ), (7) θ= a+b +b( 3α 8 r + 5α 6 r4 + 35α 3 8 r6 + 63α 4 56 r8 ), (8) where α i,β i R, {i=,...,4}, i Z +, and scenarios depend on the characteristic of the numerous roots of r=. 4. The Generalized Van der Pol Sstem including Quartic Self Nonlinearit Now we introduce a further generalized sstem including a quartic term in the parentheses. The generalization with this additional quartic term is =κ, = α(α +α +α 4 4 )+β( ). The averaged version of sstem (5)is β r r( 4 ), θ= κ+αα ε αr 8 (3α + 5α 4 r ), (5) (6) b introducing a small parameter ε, < ε and b replacing α, β b εα, εβ in that given order. According to the sign of Δ=α 4β, the sstem ehibits different bifurcation scenarios where some of them are of interest and others are trivial. A bifurcation analsis selecting κ as the bifurcation parameter gives a pitchfork bifurcation point which forks a curve with numerous limit point bifurcation points as in the case for the DVP sstem. Selecting α as the bifurcation parameter, a pitchfork bifurcation is observed, and selecting β as the bifurcation parameter, the usual subcritical opf bifurcation point scenario is observed. Other parameters do not give an bifurcation in this generalization. 5. Conclusion The subcritical opf bifurcation point of the VP sstem at the origin is inherited through all polnomial generalizations in the VP sstem. Generall a pitchfork bifurcation is observed in all bifurcation graphs which forks into higher bifurcations. Man limit point bifurcations are observed for all generalizations. Pitchfork and limit point bifurcations are observed for generalizations of VP sstem including quadratic selfcoupling. opf bifurcation has also been studied using the normal form method as given in Definition, forthe eigenvalues of the linearized sstem at equilibrium points. For generalized VP sstems considered in this stud, averaging over the period of the linear part, a limit ccle propert is observed for all possible cases. To this approimation, a change in the nonlinearit of the sstem does not change the occurrence of limit ccle phenomena. owever θ in polar coordinates does not remain constant. Also the case wherethebifurcationcurvesccleandoscillatearounda pitchfork bifurcation point between two opf points onl occurs for the original DVP sstem which is nonintegrable. A relation between the DVP and MB sstems where both ehibit Lorenz-like behavior has been shown. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments The authors would also like to thank the editor and referees for their valuable and constructive comments. The authors wouldliketothankprofessordr.ferdinandverhulstforhis valuable suggestions and comments that helped to improve the paper throughout the process.

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