New approach to study the van der Pol equation for large damping

Transcription

1 Electronic Journal of Qualitative Theor of Differential Equations 2018, No. 8, 1 10; New approach to stud the van der Pol equation for large damping Klaus R. Schneider Weierstrass Institute for Applied Analsis and Stochastics, Mohrenstr. 9, Berlin, German Received 25 September 2017, appeared 12 Februar 2018 Communicated b Gabriele Villari Abstract. We present a new approach to establish the eistence of a unique limit ccle for the van der Pol equation in case of large damping. It is connected with the bifurcation of a stable hperbolic limit ccle from a closed curve composed of two heteroclinic orbits and of two segments of a straight line forming continua of equilibria. The proof is based on a linear time scaling (instead of the nonlinear Liénard transformation in previous approaches), on a Dulac Cherkas function and on the propert of rotating vector fields. Kewords: relaation oscillations, Dulac Cherkas function, rotated vector field Mathematics Subject Classification: 4C05 4C26 4D20. 1 Introduction The scalar autonomous differential equation d 2 dt 2 + λ(2 1) d + = 0, (1.1) dt where λ is a real parameter, has been introduced b Balthasar van der Pol [7] in 1926 to describe self-oscillations in a triod circuit. It is well-known (see e.g. [6]) that for all λ = 0 this equation has a unique isolated periodic solution p λ (t) (limit ccle) whose amplitude stas ver near 2 for all λ ([1,9]), but its maimum velocit grows unboundedl as λ tends to infinit. If we rewrite the second order equation (1.1) as the planar sstem d dt =, d dt = λ(2 1) (1.2) and if we represent p λ (t) as a closed curve Γ λ in the (, )-phase plane, then Γ λ does not sta in an bounded region as λ tends to. schneider@wias-berlin.de

2 2 K. R. Schneider In the paper [8], van der Pol noticed that for large parameter λ the form of the oscillations is characterized b discontinuous jumps arising each time the sstem becomes unstable and that the period of these new oscillations is determined b the duration of a capacitance discharge, which is sometimes named a relaation time. From that reason, van der Pol popularized the name relaation oscillations []. A proof of the conjecture of van der Pol that the limit ccles Γ λ for large λ is located in a small neighborhood of a closed curve representing a discontinuous period solution has been given b A. D. Flanders and J. J. Stoker [2] in 1946 b considering (1.1) in the so-called Liénard plane [5] which is related to the nonlinear Liénard transformation of the phase plane which implies that the corresponding famil {Γ λ } of limit ccles stas for all λ in a uniforml bounded region of the Liénard plane. For defining the Liénard transformation for λ > 0 we introduce a new time τ b t = λτ and a new parameter ε b ε = 1 λ 2. B this wa, equation (1.1) can be rewritten in the form Appling the nonlinear Liénard transformation [ d ε d ] dτ dτ + + = 0. (1.) η =, ξ = ε + = ε d dτ + equation (1.) is equivalent to the sstem dξ dτ = η, ε dη dτ = ξ + η η which represents a singularl perturbed sstem for small ε > 0. The corresponding degenerate sstem (1.4) dξ dτ = η, 0 = ξ + η η is a differential-algebraic sstem. This sstem does not define a dnamical sstem on the curve ξ = η + η / since on that curve there eist two points A(η = 1, ξ = 2/) and C(η = 1, ξ = 2/) which are so-called impass points: no solution of (1.5) can pass these points. Let B (D) be the point where the tangent to the curve ξ = η + η / at the point A (C) intersects the curve ξ = η + η / (see Figure 1.1). We denote b Z 0 the closed curve in the (η, ξ)-liénard plane which consists of two segments of the curve ξ = η + η / bounded b the points D, A and C, B and of two segments of the straight lines connecting the points A, B and D, C, respectivel (see Figure1.1). Using this closed curve Z 0, D. A. Flanders and J. J. Stoker [2] constructed an annulus whose boundaries are crossed transversall b the trajectories of sstem (1.4) which contains the stable limit ccle Γ ε of sstem (1.4) for sufficientl small positive ε. In what follows we present a new approach to establish the eistence of a unique hperbolic limit ccle Γ λ of the van der Pol equation (1.1) for large λ which is based on a linear time scaling instead of appling the mentioned nonlinear Liénard transformation and on using an appropriate Dulac Cherkas function. (1.5)

3 New approach to stud the van der Pol equation for large damping ξ ξ=-η+η / A B η D C Figure 1.1: Closed curve Z 0. 2 Time scaled van der Pol sstem We start with the van der Pol equation (1.1) and appl the time scaling σ = λ t for λ > 0. Using the notation ε = 1/λ 2 we get the equation d 2 dσ 2 + (2 1) d + ε = 0 (2.1) dσ which is equivalent to the sstem d dσ =, d dσ = ε (2 1). (2.2) Setting ε = 0 in (2.2) we obtain the sstem d dσ =, d dσ = (2 1) (2.) which has the -ais as invariant line consisting of equilibria. In the half-planes > 0 and < 0, the trajectories of sstem (2.) are determined b the differential equation and can be eplicitl described b d d = 2 1 = + c, (2.4) where c is an constant. Net we note the following properties of sstem (2.2). Lemma 2.1. Sstem (2.2) is invariant under the transformation,. This lemma implies that the phase portrait of (2.2) is smmetric with respect to the origin. Lemma 2.2. The origin is the unique equilibrium of sstem (2.2), it is unstable for ε > 0.

4 4 K. R. Schneider Lemma 2.. If sstem (2.2) has a limit ccle Γ ε, then Γ ε must intersect the straight lines = ±1 in the lower and upper half-planes. The proof of Lemma 2. is based on the Bendison criterion and the location of the unique equilibrium. For the sequel we denote b X(ε) the vector field defined b (2.2). Lemma 2.4. Ψ(,, ε) ε 1 (2.5) is a Dulac Cherkas function of sstem (2.2) in the phase plane for ε > 0. Proof. For the function Φ defined in (4.2) in the Appendi, we get from (2.5), (2.2) and κ = 2 Φ := (grad Ψ, X(ε)) + κψ div X(ε) 2( 2 1) 2 0 for (,, ε) R 2 R +. (2.6) Since the set N, where Φ vanishes, consists of the two straight lines = ±1 which do not contain a closed curve, we get from Definition 4.1 and Remark 4.2 in the Appendi that Ψ represents a Dulac Cherkas function for sstem (2.2) in the phase plane for ε > 0, q.e.d. For ε > 0, the set W ε, where Ψ(,, ε) vanishes, consists of a unique oval, the ellipse } E ε := {(, ) R 2 : ε = 1. We denote b A ε the open simpl connected region bounded b E ε. Lemma 2.5. For ε > 0, E ε is a curve without contact with the trajectories of sstem (2.1). Proof. B Lemma 4. in the Appendi, an trajector of sstem (2.2) which meets E ε intersects E ε transversall. It is eas to verif that an trajector of sstem (2.2) which crosses E ε will leave the region A ε for increasing σ. Lemma 2.6. Sstem (2.2) has no limit ccle in A ε for ε > 0. Proof. Ψ(,, ε) is negative for (, ) A ε. Thus, b Lemma 4.4 in the Appendi, Ψ(,, ε) 2 is a Dulac function in the simpl connected region A ε for ε > 0 which implies that there is no limit ccle of sstem (2.2) in A ε for ε > 0. Lemma 2.7. Sstem (2.2) has at most one limit ccle for ε > 0. Proof. For ε > 0, the set W ε consists of the unique ellipse E ε. Thus, according to Theorem 4.6 in the Appendi, sstem (2.2) has at most one limit ccle. Lemma 2.8. If sstem (2.2) has a limit ccle Γ ε, then Γ ε is hperbolic and stable. Proof. If sstem (2.2) has a limit ccle Γ ε, then b Lemma 2.6 and b Lemma 2.5, Γ ε must surround the ellipse E ε. Thus, we have Ψ > 0, κ = 2, Φ > 0 on Γ ε. Theorem 4.5 in the Appendi completes the proof. We recall that for λ > 0 the van der Pol equation (1.1) and sstem (2.2) have the same phase portrait. Thus, the results above impl that the van der Pol equation has at most one limit ccle Γ λ for λ = 0. In the following section we construct for sufficientl small ε a closed curve K ε surrounding the ellipse E ε such that an trajector of sstem (2.2) which meets K ε enters for increasing σ the doubl connected region B ε with the boundaries E ε and K ε.

5 New approach to stud the van der Pol equation for large damping 5 Eistence of a unique limit ccle Γ ε of sstem (2.2) for sufficientl small ε If we assume that there eists some positive number ε 0 such that for 0 < ε < ε 0 sstem (2.2) has a famil {Γ ε } of limit ccles which are uniforml bounded, then the limit position of Γ ε as ε tends to zero is a bounded closed curve Γ 0 smmetric with respect to the origin which can be described b means of solutions of sstem (2.), that is, b segments of curves eplicitl given b (2.4) and b segments of the invariant line = 0 consisting of equilibria of sstem (2.). Since Γ 0 is a bounded smooth curve in the half-planes 0 and 0 and is smmetric with respect to the origin, we get for Γ 0 in the upper half-plane the representation = for 2 1, = 0 for 1 2. The part in the half-plane 0 is determined b the mentioned smmetr (see Figure.1) Figure.1: Closed curve Γ 0. We note that Γ 0 consists of two heteroclinic orbits of sstem (2.) connecting the equilibria ( 2, 0) and (1, 0) in the upper half-plane and the equilibria (2, 0) and ( 1, 0) in the lower half-plane, and of the two segments 2 1 and 1 2 of the -ais consisting of equilibria of sstem (2.). In what follows, to given ε > 0 sufficientl small, we construct a closed curve K ε surrounding the ellipse E ε such that an trajector of sstem (2.2) which meets K ε intersects K ε transversall. For the construction of the closed curve K ε we will take into account the following observations. Remark.1. The amplitude of a possible limit ccle {Γ ε } of sstem (2.2) tends to 2 as ε tends to zero. Remark.2. Let ε > 0. In the fourth quadrant ( < 0, > 0), the vector field X(ε) rotates in the positive sense (anti-clockwise) for increasing ε. Remark.. Let ε > 0. In the first quadrant, the vector field X(ε) rotates in the positive sense for decreasing ε. Let P be the point ( 2.2, 0) on the -ais and let R be the point (2.2, 0) smmetric to P with respect to the origin. B Remark.1, there eists a sufficientl small positive number ε 0

6 6 K. R. Schneider such that for ε (0, ε 0 ) an limit ccle Γ ε of sstem (2.2) intersects the -ais in the intervals ( 2.2, 0) and (0, 2.2). We denote b T 0 (P) the trajector of sstem (2.) in the upper half-plane having P as ω-limit point. T 0 (P) intersects the -ais at the point Q 0 = (0, (2.2) / 2.2) (0, 1.49) and the straight line = 2.2 at the point R 0 = (2.2, 2[(2.2) / 2.2]) (2.2, 2.699). T 0 (P) is represented as solid curve in Figure.2. Let T ε/2 (P) be the trajector of sstem (2.2), where ε is replaced b ε/2, in the fourth quadrant which passes the point P. For sufficientl small ε, it cuts the -ais at the point Q ε/2 which is near the point Q 0. T ε/2 (P) is represented as dotted curve in Figure R 0 R2ε ε/2 (P) (P) 1.5 Q Q ε/2 2ε (Q ε/2 ) P R Figure.2: Trajectories T 0 (P), T ε/2 (P) and T 2ε (Q ε/2 ) for ε = Now we denote b T 2ε (Q ε/2 ) the trajector of sstem (2.2), where ε is replaced b 2ε, in the first quadrant which passes the point Q ε/2. For sufficientl small ε, T 2ε (Q ε/2 ) is located near the trajector T 0 (P) and intersects the straight line = 2.2 at the point R 2ε which is located near the point R 0. T 2ε (Q ε/2 ) is represented as dashed curve in Figure.2. According to Remark.2, an trajector of sstem (2.2) (with the parameter ε), which meets the trajector T ε/2 (P) in the fourth quadrant, intersects T ε/2 (P) transversall and enters for increasing σ the region D ε/2 in the fourth quadrant bounded b T ε/2 (P), the -ais and the -ais (see Figure.). B Remark., an trajector of sstem (2.2) (with the parameter ε) which meets the trajector T 2ε (Q ε/2 ) in the first quadrant intersects T 2ε (Q ε/2 ) transversall and enters for increasing σ the region D 2ε in the fist quadrant bounded b T 2ε (Q ε/2 ), the -ais, the -ais and the straight line = 2.2 (see Figure.). 2.5 R 2ε ε/2 (P) Q ε/2 P 1.0 S ε/2 0.5 S 2ε 2ε (Q ε/2 ) Figure.: Trajectories T ε/2 (P) and T 2ε (Q ε/2 ) for ε = R It can be easil verified that the trajectories of sstem (2.2) intersect the segment of the

7 New approach to stud the van der Pol equation for large damping 7 straight line = 2.2 bounded b the points R and R 2ε transversall in the upper half-plane entering D 2ε for increasing σ. Since at the point R we have d/dσ > 0, also the trajector of sstem (2.2) passing R enters the region D 2ε for increasing σ. If we denote b K ε,u the curve composed b the trajectories T ε/2 (P) and T 2ε (Q ε/2 ) and the segment of the straight line = 2.2 bounded b the points R and R 2ε, and introduce the set D ε,u = D ε/2 D 2ε, then we have the following result: For sufficientl small ε, an trajector of sstem (2.2) which meets the curve K ε,u enters for increasing σ the region D ε,u. Eploiting the smmetr described in Lemma 2.1 there eists a curve K ε,l in the lower half-plane such that the closed curve K ε = K ε,u K ε,l bounding the simpl connecting region D ε has the properties of the wanted closed curve K ε. Figure.4 shows the closed curves K ε and E ε bounding the annulus B ε = D ε \ A ε containing the limit ccle Γ ε for ε = Z ε,u P E ε R Z ε,l -1 Γ ε -2 - Figure.4: Closed curves K ε and E ε together with the limit ccle Γ ε for ε = An improvement of the curve K ε and consequentl of the annulus B ε containing the limit ccle Γ ε can be obtained as follows. Let T 2ε (R) be the trajector of sstem (2.2), where ε is replaced b 2ε, in the first quadrant starting at the point R. Since on the interval 0 < < 2.2 of the -ais it holds d/d = ε > 0, the trajector T 2ε (R) will intersect the -ais at the point Q 2ε which is located between the origin and the point Q ε/2. We denote b D 2ε the region in the first quadrant bounded b the trajector T 2ε (R), the -ais and the -ais (see Figure.5). Using Remark., we can conclude as above that an trajector of sstem (2.2) which meets T 2ε (R) enters D 2ε for increasing σ. For the following we denote b I ε the interval on the -ais bounded b the points Q ε/2 and Q 2ε. From (2.2) it follows that an trajector of sstem (2.2) which meets I ε enters for increasing σ the region D ε/2. Thus, we have the following result: There is a sufficientl small ε 0 such that for ε (0, ε 0 ) the trajectories T ε/2 (P) and T 2ε (R) and the interval I ε form a curve K ε,u which bounds in the upper half-plane the bounded region D ε,u = D ε/2 D 2ε such that an trajector of sstem (2.2) which meets K ε,u enters for increasing σ the region D ε,u. Eploiting the smmetr described in Lemma 2.1 there eists a curve K ε,l in the lower half-plane such that K ε = K ε,u K ε,l forms the wanted closed curve K ε. In Figure.6 the closed curves K ε and E ε are represented as solid curves which form the boundaries of the doubl connected region B ε which contains the unique limit ccle Γ ε (dashed curve) for ε = Summarizing our considerations we have the following results.

8 8 K. R. Schneider 2.0 Q ε/2 ε/2 (P) 1.5 I ε S ε/ Q 2ε S 2ε 2ε (R) P R Figure.5: Curve K ε,u = T 2ε (R) I ε T ε/2(p) (R) for ε = Γ ε Eε Z ε P R -1-2 Figure.6: Closed curves K ε and E ε and the limit ccle Γ ε for ε = Theorem.4. For sufficientl small positive ε, the closed curve K ε is a curve without contact for sstem (2.2), that is, an trajector of sstem (2.2) which meets the curve K ε crosses it transversall b entering the doubl connected region B ε for increasing σ. Taking into account Lemma 2.7 and Lemma 2.8 we get the following theorem from Theorem.4. Theorem.5. For sufficientl small positive ε, sstem (2.2) has a unique limit ccle Γ ε. This ccle is hperbolic and stable and located in the region B ε. 4 Appendi Consider the planar autonomous differential sstem d dt = P(,, λ), d dt = Q(,, λ) (4.1) depending on a scalar parameter λ. Let Ω be some region in R 2, let Λ be some interval and let X(λ) be the vector field defined b sstem (4.1). Suppose that P, Q : Ω Λ R are continuous in all variables and continuousl differentiable in the first two variables.

9 New approach to stud the van der Pol equation for large damping 9 Definition 4.1. A function Ψ : Ω Λ R with the same smoothness as P, Q is called a Dulac Cherkas function of sstem (4.1) in Ω for λ Λ if there eists a real number κ = 0 such that Φ := (grad Ψ, X(λ)) + κψ div X(λ) > 0 (< 0) for (,, λ) Ω Λ. (4.2) Remark 4.2. Condition (4.2) can be relaed b assuming that Φ ma vanish in Ω on a set N of measure zero, and that no closed curve of N is a limit ccle of (4.1). In the theor of Dulac Cherkas functions the set W λ := {(, ) Ω : Ψ(,, λ) = 0} (4.) plas an important role. The following results can be found in [4]. Lemma 4.. An trajector of sstem (4.1) which meets W λ intersects W λ transversall. Lemma 4.4. Let Ω 1 Ω, Λ 1 Λ be such that Ψ(,, λ) is different from zero for (,, λ) Ω 1 Λ 1. Then Ψ κ is a Dulac function in Ω 1 for λ Λ. Theorem 4.5. Let Ψ be a Dulac Cherkas function of (4.1) in Ω for λ Λ. Then an limit ccle Γ λ of (4.1) in Ω is hperbolic and its stabilit is determined b the sign of the epression κφψ on Γ λ, it is stable (unstable) if κφψ is negative (positive) on Γ λ Theorem 4.6. Let Ω be a p-connected region, let Ψ be a Dulac Cherkas function of (4.1) in Ω such that the set W λ consists of s ovals in Ω. Then sstem (4.1) has at most p 1 + s limit ccles in Ω. References [1] E. Fisher, The period and amplitude of the Van Der Pol limit ccle, J. Appl. Phs. 25(1954), MR [2] D. A. Flanders, J. J. Stoker, The limit case of relaation oscillations, in: Studies in Nonlinear Vibration Theor, Institute for Mathematics and Mechanics, New York Universit, 1946, pp MR [] J.-M. Ginou, C. Letellier, Van der Pol and the histor of relaation oscillations: toward the emergence of a concept, Chaos 22(2012), 02120, 15 pp. MR8857; / [4] A. A. Grin, K. R. Schneider, On some classes of limit ccles of planar dnamical sstems, Dn. Contin. Discrete Impuls. Sst. Ser. A, Math. Anal. 14(2007), MR25489 [5] A. Liénard, Étude des oscillations entretenues (in French), Revue générale de l électricité 2(1928), [6] L. Perko, Differential equations and dnamical sstems, Tets in Applied Mathematics, Vol. 7, Springer, [7] B. van der Pol, On relaation-oscillations, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Ser. 7 2(1926),

10 10 K. R. Schneider [8] B. van der Pol, J. van der Mark, Le battement du cœur considéré comme oscillation de relaation et un modèle électrique du cœur, L Onde Électrique 7(1928), [9] H. Yanagiwara, Maimum of the amplitude of the periodic solution of van der Pol s equation, J. Sci. Hiroshima Univ. Ser. A-I Math. 25(1961), MR