This is a preprint of: “Non-existence, existence, and uniqueness of limit cycles for a generalization of the Van der Pol-Duffing and the Rayleigh-Duffing oscilators”, Murilo R. Cândido, Jaume Llibre, Clàudia Valls, Phys. D, vol. 407, 132458:1–4, 2020. DOI: [10.1016/j.physd.2020.132458] NON-EXISTENCE, EXISTENCE, AND UNIQUENESS OF LIMIT CYCLES FOR A GENERALIZATION OF THE VAN DER POL–DUFFING AND THE RAYLEIGH–DUFFING OSCILLATORS MURILO R. CÂNDIDO1 , JAUME LLIBRE2 AND CLAUDIA VALLS3 Abstract. We study the limit cycles of some cubic family of dif- ferential equations, containing the well-known van der Pol-Duffing and Rayleigh-Duffing oscillators. In particular, we characterize for the class of differential systems here studied the non-existence, ex- istence and uniqueness of limit cycles. Moreover we provide their global phase portraits in the Poincaré disc. 1. Introduction and statement of the main results In this paper we consider the planar system (1) x0 = y, y 0 = −a1 x − a2 x3 + µ(a3 + a4 x2 + a5 y 2 )y, where a1 , a2 > 0, a3 , a4 , a5 ∈ R, a3 6= 0, µ > 0 is a sufficiently small parameter, a5 < 0 and the prime denotes derivative with respect to the time t. The Van der Pol oscillator was discovered by engineer and physicist Balthasar Van der Pol while working at Philips company. Van der Pol [26] found in circuits that employ vacuum valves stable oscillations, which are now known as limit cycles. Van der Pol also found that at certain frequencies, some irregular noise appears near the coupling frequencies. It will be one of the first experimental discoveries of Chaos Theory [13, 27]. The Van der Pol equation has a long history not only in physics but also in biology. Thus in biology Fitzhugh [8] and Nagumo [22] applied the equation to a two-dimensional field in the Fitzhugh- Nagumo model, as it is known now, to describe the potential action of neurons. It can also be used in seismology to model the behavior of 2010 Mathematics Subject Classification. Primary 34A05. Secondary 34C05, 37C10. Key words and phrases. limit cycles, van der Pol-Duffing oscillator, Rayleigh- Duffing oscillator. 1 2 M.R. CÂNDIDO, J. LLIBRE AND C. VALLS plates in a failure [1]. We note that the Rayleigh equation introduced by Rayleigh in 1875 in his published book The Theory of Sound [24], where he shown the first physical phenomenon modeled by a limit cycle. In fact the Rayleigh equation is more general than the Van der Pol equation. The Duffing equation introduced in [3] essentially adds a term x3 to the Rayleigh equation. The differential systems (1) contain as particular cases the so-called Van der Pol-Duffing oscillator and Rayleigh-Duffing oscillator both with positive linear damping and sufficiently small stiffness. Both mod- els have been studied intensively because they are two essential oscilla- tors in the nonlinear dynamical systems, see for instance the books of [22, 23] and the references cited therein. Many researchers have inves- tigated the existence of limit cycles for autonomous nonlinear systems depending on parameters and in special for system (1) due to the fact that they can be a mechanism for the creation of chaos, see [11, 19]. This problem is also related with the well-known 16th Hilbert prob- lem which asks for the number of limit cycles in polynomial differential equations in function of their degree, see for instance [9, 10, 15]. The main objective of this paper is to characterize the dynamics of the differential system (3) in an easy way using the qualitative theory of differential equations, and in particular we provide the non-existence, existence and uniqueness of limit cycles for the differnetial equation (3). Consequently we provide a new unified proof in the study of the limit cycles of the Van der Pol-Duffing equation and of the Rayleigh-Duffing equation. The differential equation (1) when a5 = 0 becomes a subclass of the classical polynomial Liénard differential equation (2) ẍ + f (x)ẋ + g(x) = 0, introduced in [17], when we write it as the differential system of first order ẋ = y, ẏ = −g(x) − f (x)y. Here the dot denotes differentiation with respect to the time t, and f (x) and g(x) are polynomials in the variable x of degrees n and m respectively. This differential equation has been studied by many au- thors. We denote by H(m, n) the maximum number of limit cycles that the differential equation (2) with m and n fixed can have. Now we describe briefly some of the main results on the limit cicles on the Liénard differential equation (2) LIMIT CYCLES OF SOME LIÉNARD SYSTEMS 3 Rx (i) In 1928 Liénard [17] showed if m = 1 and F (x) = 0 f (s)ds is a continuous odd function, which has a unique root at x = a and is monotone increasing for x ≥ a, then equation (2) has a unique limit cycle. Rx (ii) In 1973 Rychkov [25] proved that if m = 1 and F (x) = 0 f (s)ds is an odd polynomial of degree five, then equation (2) has at most two limit cycles. (iii) In 1977 Lins, de Melo and Pugh [18] proved that H(1, 1) = 0 and H(1, 2) = 1. (iv) In 1998 Coppel [2] proved that H(2, 1) = 1. (v) Dumortier, Li and Rousseau in [7] and [4] proved that H(3, 1) = 1. (vi) In 1997 Dumortier and Chengzhi [5] proved that H(2, 2) = 1. For the differential system (1), with µ = 1, a1 = −1, a2 = 1, a4 = 0 and a5 = −1, the authors in [12] studied numerically the creation and annihilation of limit cycles depending on the negative parameter a3 (that is for negative linear damping and negative linear stiffness). In particular in [12] nothing is studied when a1 is positive and for other the values of the parameters ak for k = 2, ..., 5. Note that since a1 , a2 > 0, by the change of coordinates and a reparametrization of time of the form r a1 a1 1 x= X, y = √ Y, t = √ τ, a2 a2 a1 we can write system (1) as ẋ = y (3) ẏ = −x − x3 − ε(by + x2 y + ay 3 ), √ where ε = −a4 a1 µ/a2 , b = a2 a3 /(a4 a1 ) and a = a5 a1 /a4 . The dot denotes derivative with respect to the new variable τ and we have renamed the new variables (X, Y ) as (x, y). System (3) is invariant by the change (x, y, ε, t) 7→ (−x, y, −ε, −t), and so from now on and without loss of generality we can assume that ε > 0, i.e. a4 ≤ 0. Since a5 < 0 we have that a > 0. So in the rest of this paper and without loss of generality we assume that ε > 0 and a > 0 in the differential system (3). Note that system (3) with ε = 0 has a first integral of the form y 2 x2 x4 (4) H= + + . 2 2 4 4 M.R. CÂNDIDO, J. LLIBRE AND C. VALLS To state the main result of this paper we recall that a limit cycle is a periodic orbit which is isolated in the set of all periodic orbits of the system. In general the limit cycles are very difficult to detect. The main result of our paper is the following. Theorem 1. The polynomial differential system (3) has no limit cycles when b ≥ 0. Theorem 1 is proved in section 3. Theorem 2. The differential system (3) has a unique stable limit cycle when b < 0. See Chapter 5 of [6] for the definitions of Poincaré disc and Poincaré compactification of a polynomial differential system, and Chapter 1 of [6] for the definition of topological equivalence. The second main result of the paper is the following. Figure 1 Figure 2 Theorem 3. The global phase portrait of the differential system (3) in the Poincaré disc is topologically equivalent to the one of Figure 1 if b ≥ 0 and to the one of Figure 2 if b < 0. Theorems 2 and 3 are proved in section 4. 2. Preliminary results and definitions If a planar differential system has some orbit such that its ω–limit escapes at infinity is called an unbounded system, otherwise it is called a bounded system. Note that system (3) is bounded. LIMIT CYCLES OF SOME LIÉNARD SYSTEMS 5 Suppose that as α varies in (a, b), the equililbrium points of the vector fields (P (x, y, α), Q(x, y, α)) remain unchanged, and for any given point p = (x, y) and any parameters α1 < α2 in (a, b), we have P (x, y, α1 ) Q(x, y, α1 ) (5) ≥ 0 (or ≤ 0), P (x, y, α2 ) Q(x, y, α2 ) where equality cannot hold on an entire periodic orbit of the vec- tor field (P (x, y, αi ), Q(x, y, αi )), i = 1, 2. Then the vector fields (P (x, y, α), Q(x, y, α)) are called generalized rotated vector fields. Here, the interval (a, b) can be either bounded or unbounded. The generalized rotated vector fields have the following two proper- ties. (i) The stable (respectively unstable) periodic orbits which are run in clockwise sense increase (respectively decrease) their size as the parameter increases. (ii) The periodic orbits of the system for different values of the parameter cannot intersect. For the proofs of these properties see for instance [6, 29]. 3. Proof of Theorem 1 We recall that a center p of a differential system defined in R2 is a equilibrium point of the system for which it exists a neighborhood U such that U \ {p} is filled of periodic orbits. Moreover the center p is global is R2 \ {p} is filled of periodic orbits. Lemma 4. The differential system (3) with ε = 0 has a global center at the origin of coordinates. Proof. The differential system (3) with ε = 0 is a Hamiltonian system with Hamiltonian H given in (4). Since all the level curves of H = h with h > 0 are closed curves surrounding the origin of coordinates, which leaves at H = 0, the origin is a global center.  Proof of Theorem 1. Let (x(t), y(t)) be an arbitrary solution of the polynomial differential system (3) different from the equilibrium point (0, 0). Since the function H = H(x, y) given in (4) evaluated on the solutions of the differential system (3) satisfies dH ∂H ∂H (x(t), y(t)) = ẋ(t) + ẏ(t) = −εy(t)2 (b + x(t)2 + ay(t)2 ) ≤ 0, dt ∂x ∂y 6 M.R. CÂNDIDO, J. LLIBRE AND C. VALLS because a > 0, ε > 0, and by assumption b ≥ 0 is also positive, it follows that on orbit (x(t), y(t)) of the differential system (3) the value of the function H decreases when the time increases. Therefore the orbit (x(t), y(t)) tends to the origin of coordinates, the equilibrium point of the Hamiltonian system (3) with ε = 0, hence the origin is a global attractor for system (3). In other words the function H is a Lyapunov function for system (3) in the whole R2 , and consequently the origin is a global attractor, for more details see Theorem 1.35 of [6]. Since the origin is a global attractor system (3) with b ≥ 0 has no periodic orbits, and so no limit cycles. This completes the proof of Theorem 1.  4. Proofs of Theorems 2 and 3 Proof of Theorem 2. We first study the finite equilibrium points of sys- tem (3). Note that the unique finite equilibrium point is the origin. The Jacobian matrix of system (3) at the origin is   0 1 M= . −1 −bε whose eigenvalues are p −bε ± (bε)2 − 4 . 2 Hence, if b = 0 the origin is a weak focus because the first non–zero Liapunov constant is −ε(3a + 1)π/4. Since ε > 0 it is stable weak focus. Moreover, if b 6= 0 then if |bε| ≥ 2 the origin is a node, and if |bε| < 2 it is a focus. It is stable if b > 0, and unstable if b < 0. Now we study the infinite equilibrium points. On the local chart U1 system (3) becomes u̇ = −1 − v 2 − u2 v 2 − εu(1 + au2 + bv 2 ), v̇ = −uv 3 . Note that this system has the infinite equilibrium point √ √ √ √ 2/3 ! 2 3 3aε2 − 3 2 3a3/2 ε2 27a + 4ε2 + 9a2 ε2 √ √ ,0 . 62/3 aε( 3a3/2 ε2 27a + 4ε2 + 9a2 ε2 )1/3 LIMIT CYCLES OF SOME LIÉNARD SYSTEMS 7 The eigenvalues of the Jacobian matrix at that point are 0 and √ 6 22/3 3 3aε3 ε− √ √ 2/3 a3/2 ε2 81a + 12ε2 + 9 a √ √ √ 2/3 3 232/3 a3/2 ε2 81a + 12ε2 + 9 a − . aε So it is semi-hyperbolic,and applying Theorem 2.19 in [6] we get that it is a semi-hyperbolic saddle. On the local chart U2 system (3) becomes u̇ = v 2 + u4 + u2 v 2 + εu(a + u2 + bv 2 ), v̇ = v(u3 + uv 2 + ε(a + u2 + bv 2 )). So the origin of U2 is an equilibrium point whose eigenvalues of the Jacobian matrix at the origin are both equal to εa. Hence, the origin of the local chart U2 is an unstable node because εa > 0. Now we shall prove that for b < 0 the differential system (3) has a unique limit cycle. Since the origin is the unique finite equilibrium point which is un- stable for b < 0 and the system is bounded, by the Poincaré-Bendixson Theorem (see for instance Theorem 1.25 of [6]) it must exists at least one stable limit cycle. This limit cycle borns in a Hopf bifurcation at b = 0 because for b ∈ (−2, 0]) the origin is an unstable focus, and for b ∈ (0, 2) the origin is a stable focus, for more details on the Hopf bifurcation see [14]. The vector fields associated to our system (3) are generalized rotated vector fields with respect to the parameter b in the whole R, because the determinant (5) for this system is ε(b1 − b2 )y 2 , and the system has the origin as the unique equilibrium point which does not depend on the parameter b. Suppose that system (3) has more than one limit cycle. The closest limit cycle to the origin, γ1 , must be stable because the origin for b < 0 is unstable. The limit cycle γ2 nearest to γ1 must be unstable in the bounded region limited by it. In view of the properties (i) and (ii) given in section 2 we get a contradiction, because increasing the parameter b the limit cycle γ1 increases and the limit cycle γ2 decreases, so for some value of b they would intersect.  Proof of Theorem 3. We recall that when b ≥ 0, by Theorem 1, the differential system (3) has no limit cycles, and that by Theorem 2 8 M.R. CÂNDIDO, J. LLIBRE AND C. VALLS system (3) has a unique stable limit cycle when b < 0. Moreover in the proof of Theorem 2 we have seen: (i) The differential system (3) is bounded for all values of the pa- rameter b. (ii) The endpoints of the x-axis at infinity are saddles, and the endpoints of the y-axis at infinity are unstable nodes. (iii) The unique finite equilibrium point, the origin, is stable when b ≥ 0 and unstable when b < 0. Taking into account all this information on the local phase portraits at the finite and infinite equilibrium points, and the non-existence, existence and uniqueness of the limit cycles, the phase portraits in the Poincaré disc of system (3) are topologically equivalent to the one of Figure 1 if b ≥ 0, or to the one of Figure 2 if b < 0.  Acknowledgements M. Cândido is partially supported by FAPESP 2018/07344-0 and 2019/05657-4. J. Llibre is supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. C. Valls is partially supported by FCT/Portugal through UID/MAT/04459/2013. References [1] J. Cartwright, V. Eguiluz, E. Hernández-Garcia and O. Piro, Dy- namics of elastic excitable mitjan, Internat. J. Bifur. Chaos 9 (1999), 2197– 2202. [2] W. A. 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CÂNDIDO, J. LLIBRE AND C. VALLS 1 Departamento de Matemática, Universidade Estadual de Camp- inas, Rua Sérgio Baruque de Holanda, 651, Cidade Universitária Ze- ferino Vaz, 13083–859, Campinas, SP, Brazil Email address: candidomr@ime.unicamp.br 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 3 Departamento de Matemática, Instituto Superior Técnico, Uni- versidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal Email address: cvalls@math.ist.utl.pt