Limit cycles in the equation of whirling pendulum with piecewise smooth perturbations ∗

This paper deals with the problem of limit cycles for the whirling pendulum equation ˙ x = y, ˙ y = sin x (cos x − r ) under piecewise smooth perturbations of polynomials of cos x , sin x and y of degree n with the switching line x = 0. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained by using the Picard-Fuchs equations which the generating functions of the associated ﬁrst order Melnikov functions satisfy. Further, the exact bound of a special case is given by using the Chebyshev system.


Introduction and main results
The piecewise smooth differential system has been attracted many researchers to study its limit cycles. One important reason is that the sudden behavior after slow change is common in natural and artificial systems, which is usually described by piecewise smooth mathematical model [25,31]. Another interesting reason is that this problem can be seen as an extension of the Hilbert's 16th problem provided by Hilbert [6] in 1902, to the piecewise smooth world. We recall that the Hilbert's 16th problem asks for the maximum number of limit cycles of planar polynomial vector fields of degree n, n ∈ N + and their relative distributions on the plane. Later, this problem was posed again by Smale [30] in 1998 for the 21st Century. Although there are plenty of excellent articles corresponding to it, see for instance [8,11,22] and the references quoted there, this problem is still open.
Consider a piecewise smooth differential system with the forṁ with the functions H ± , f ± , g ± being C ∞ smooth.

Assumption (II)
The equation H + (x, y) = h, x ≥ 0, defines an orbital arc L + h starting from A 1 (h) and ending at A 2 (h); the equation H − (x, y) = H − (A 2 (h)), x ≤ 0, defines an orbital arc L − h starting from A 2 (h) and ending at A 1 (h), such that (1.1)| ε=0 has a family clockwise oriented periodic orbits L h = L + h ∪ L − h , h ∈ Σ. In order to study the problem of limit cycle bifurcations, the authors of [9,17,21] obtained the first order Melnikov function formula M (h) of system (1.1) and the beautiful relationship between limit cycles and zeros of M (h) as in the smooth case. We review these results here for the convenience of the reader.

Theorem A. Under the assumptions (I) and (II).
(i) The first order Melnikov function of system (1.1) has the following form (ii) If M (h) has k zeros in h on the interval Σ with each having an odd multiplicity, then system (1.1) has at least k limit cycles bifurcating from the period annulus for |ε| small.
(iii) If the function M (h) has at most k zeros in h on the interval Σ, taking into multiplicities account, then there exist at most k limit cycles of system (1.1) bifurcating from the period annulus.
This theorem has many applications to Hopf bifurcation, homoclinic and heteroclinic bifurcations when f ± (x, y) and g ± (x, y) are polynomials of x and y, see [7,18,[33][34][35][36] for instance. However, when f ± (x, y) and g ± (x, y) are non-polynomials (e.g. trigonometric functions), there are few papers, as far as we know. For the smooth case, the analysis of pendulum-like equations appears in some literatures. Examples include the perturbed whirling penduluṁ x = y,ẏ = sin x(cos x − r) + εy(cos x + α), (1.2) where α is a real parameter, which was considered by Lichardová in [19]. Via the Melnikov method and Li and Zhang criterion [16], the author proved that for ε > 0 small enough this system has a unique limit cycle in a certain region of a two-dimensional space of parameters. In [20], Lichardová proved that the period function of system (1.2)| ε=0 is either monotone or has exactly one critical point by using Picard-Fuchs equation. Another related problem is the study of the periodic solutions of the simple penduluṁ The perturbed of this equation was studied by several authors, see [2,4,10,12,13,27,29]. Gasull, Geyer and Mañosas [4] considered the pendulum-like equatioṅ where Q n,s (x) are trigonometric polynomials of degree at most n and ε > 0 is a small parameter. They gave upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. We refer the reader to the classical monograph [1,26] for a very complete survey on this problem. In this paper, motivated by the above references, we will study the number of limit cycles bifurcating from the period annuluses of the whirling pendulum when it is perturbed inside any polynomials of cos x, sin x and y of degree n with the switching line x = 0. The whirling pendulum is shown in Fig. 1. It consists of a rigid frame that freely rotates about a vertical axis with constant rotation rate ω, to which a planar pendulum with length l and mass m is attached, the pivot being on the vertical axis. If the angle deviation is denoted by x, then the centrifugal moment is mw 2 l 2 sin x cos x, the gravity moment is mgl sin x, and the moment of inertia is ml 2 . Therefore, the motion of the whirling pendulum can be described by the equation (see [15] p. 272) where g is the gravity constant and the dot stands for the derivative with respect to the time t.
Obviously, if there are forces to counteract gravity, then equation (1.3) is reduced to Introducing a new variable y =ẋ and then changing the variables y → ωy, t → t/ω converts (1.3) to an equivalent planar system of first-order equationṡ where r = g lω 2 ≥ 0 (when there are forces to counteract gravity, r = 0). This system is hamiltonian with the energy Depending on r, one can obtain three qualitatively different dynamics of system (1.4) and for all r ≥ 0, the points (±π, 0) in the phase plane are saddles (see Figs. [2][3][4]. Case (I) For r ≥ 1 (i.e. for small rotation rate), dynamics is the same as that of a planar pendulum: it has a center (0, 0), two saddles (±π, 0), and two types of periodic orbits. For h ∈ (0, 2r), the levels L 0 h = {(x, y)|H(x, y) = h} are ovals surrounding the origin. It corresponds to oscillations about the stable equilibrium (0.0). While for h ∈ (2r, +∞), the corresponding levels have two connected components which are again ovals, one of them contained in the region y > 0 denoted by L + h , corresponding to clockwise rotations of the pendulum, and the other one contained in the region y < 0 denoted by L − h corresponding to counterclockwise rotations, see Fig. 2. Case (II) For 0 < r < 1, i.e. if ω passes through the critical value √ g/l, then (0, 0) is a saddle point with two homoclinic loops (symmetric with respect to the y-axis). Inside each loop, there is a family of periodic solutions (deviated oscillations) , which surround centers (arccos r, 0) and (− arccos r, 0), respectively, see Fig. 3.
Case (III) For r = 0, that is to say, there are forces to counteract gravity. (0, 0) is also a saddle point with two heteroclinic loops (symmetric with respect to the y-axis). Inside each loop, there is a family of periodic solutions L * h, , which surround centers ( π 2 , 0) and (− π 2 , 0), respectively, see Fig. 4.
In the sequel, we will take into consideration only the families L 0 h , L + h and L * h,+ , since, due to symmetry, the results for L − h and L * h,− are analogous. The superscripts 0, + and * will denote which L h -family is being used; for instance, M + (h) denotes a function M (h) restricted to L + h . Inspired by the non-smooth perturbation of the whirling pendulum, we will study the following In order to simplify the notations, we first fix some statements. It is well-known that any polynomial of sin x, cos x and y of degree n can be written as From now on we denote by P k (u) and Q k (u) the polynomials of degree at most k and by [x] the integer part for any real number x.
Without loss of generality, we will consider only the case 0 ≤ r < 1 corresponding the phase portraits in Fig. 3 and Fig. 4, since, the case r ≥ 1 is analogous and more digestible. For the convenience of our statements in this paper, we first give the first order Melnikov functions of system (1.6). By Poincaré-Pontryagin Theorem [3,28] and Theorem A, one knows that they can be written as follows and the number of isolated zeros of them, counting multiplicities, provides the upper bounds for the number of ovals of H(x, y) that generate limit cycles of system (1.6) for ε close to zero if they are not identically zeros. Our main results are the following three theorems.

The same result holds for
) .
The bounds given in Theorem 1.1 and Theorem 1.2 are not optimal. In the following theorems, we give optimal bounds for some particular smooth perturbations in the rotary region for r = 0. In this case, there are forces to counteract gravity.
where n, l ∈ N and let M + (h) be its first order Melnikov function. Assume also that M + (h) is not identically zero. Then it has at most 2n + 1 zeros in (0, +∞), counting multiplicity. This bound is optimal.
The techniques of the proofs of Theorems 1.1 and 1.2 we use mainly include the Melnikov function, Picard-Fuchs equation, Chebyshev criterion and Gram determinant. We first obtain the algebraic structure of the first order Melnikov functions (see Lemma 2.3 and Lemma 2.5), which are more complicated than the Melnikov functions corresponding to the smooth case. Then we find that the corresponding generating functions of them satisfy some Picard-Fuchs equations (see Lemma 2.4). Finally we give the upper bounds of the number of the zeros of Melnikov functions by Riccati equations and derivation-division algorithm. For a special case, we get the exact bound by using Chebyshev criterion and Gram determinant which is in fact similar to the ones in the proofs of [5]. It is worth noting that Picard-Fuchs equation method can be applied to other situations of the investigation of limit cycles for differential systems under piecewise smooth non-polynomial perturbations.
The rest of the paper is organized as follows: In Section 2, we will give detailed expressions of the first order Melnikov functions which can be expressed by some generating functions. The Picard-Fuchs equations of these generating functions are also derived. In Section 3 we prove Theorem 1.1, while Sections 4 and 5 address the proofs of Theorems 1.2 and 1.3, respectively.

Algebraic structure of the first order Melnikov function
For i, j ∈ N, we denote By a straightforward calculation, one has can be written as and i,j and µ * i,j are arbitrary constants which can be expressed by the coefficients of f ± (x, y) and g ± (x, y).
Proof. Without loss of generality, we only prove (2.3). The proofs of (2.4) and (2.5) follow in the same way. To see this, we need to add an auxiliary line BC which is perpendicular to x-axis at the saddle point C(π, 0) and intersects L + h,+ at B, see Fig. 3. Let Ω be the interior of In a similar way, we have ∫ (2.8) Therefore, by (1.7), (2.1) and (2.6)-(2.8), one has (2.10) If the subscripts of a ± i,j , b ± i,j , c ± i,j and d ± i,j in (2.10) satisfy i < 0 or j < 0, then they vanish.
This ends the proof. ♢ To understand the algebraic structure of the first order Melnikov functions M 0 (h) and M * (h) we need first to show that for any i, j ∈ N Indeed, this is a direct consequence of the symmetry with respect to the x-axis of the integral paths and Green's Theorem.
(1) It follows from Multiplying both sides of (2.18) by y j cos i−3 x sin xdx and integrating along L + h,+ , in view of (2.6), one gets Analogously, multiplying both sides of (2.17) by y j−2 cos i xdx and integrating along L + h,+ yields We want the sum of subscripts of elements on the right side of equation ( ] .
To show the statement (2.12), in view of (2.7), we proceed by multiplying (2.18) y j cos i−1 xdx and integrating along L + h,+ which gives Multiplying both sides of (2.17) by y j−2 cos i x sin xdx implies ] . (2.27) Then, the first equality in (2.12) follows by induction using the above two equalities. For the sake of brevity we only give details of the second equality of (2.12). In fact, a simple computation shows that which gives the desired result for i + 2j = n.
] . (2.29) The proofs of (2.13) and (2.14) are conducted along the lines of the proofs of (2.11) and (2.12) using (2.28) and (2.29). The proof of statment (3) is much easier and follows by using the same arguments, so we omit for the sake of brevity and readability. The proof of Lemma 2.2 is completed. ♢ and Proof. According to (1.5) one has Differentiating the above equation with respect to h gives ∂y ∂h = 1 y , which implies Hence, Multiplying both sides of (2.36) by h, one gets which together with (2.21) yields (2.34) for i = 1. Similarly, we can prove (2.34) for i = 2, 3. This completes the proof. ♢ By using the above lemma, the forms of first order Melnikov functions M + (h) and M * (h) for r = 0 are simpler, which are given in the following lemma.
where P + k (h) and P * k (h) are polynomials of degree at most k.
(2) Noting that r = 0 and (2.34) one gets Thus, by the above two equations, one can obtain the statement (2.42) for i = 1. The proof of the case i = 2 follows by using the same arguments. The proof is completed. ♢

Proof of Theorem 1.1
In the sequel we will use the notation #{h ∈ (ρ 1 , ρ 2 ) | ϕ(h) = 0} to indicate the number of zeros of the function ϕ(h) in the interval (ρ 1 , ρ 2 ) taking into account their multiplicities.
This ends the proof. ♢ Proof of Theorem 1.3. It is well known that the first order Melnikov function of system (1.10) is and direct computations give that for a certain polynomial P n (h) of degree n. Therefore, it suffices to show that the family ( 1, h, · · · , h n , I 0,l (h), I 2,l (h), · · · , I 2n,l (h) ) is an ECT-system. To this end, consider where c i and d i are constants. Then and Proposition 5.2 implies that ϕ (n+1) (h) has at most k zeros counting multiplicity. By Rolle's Theorem one obtains that ϕ(h) has at most k + n + 1 zeros counting multiplicity. The proof of Theorem 1.3 is completed.