Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory

. In this paper we study the maximal number of limit cycles for a class of piecewise smooth near-Hamiltonian systems under polynomial pertur- bations. Using the second order averaging method, we obtain the maximal number of limit cycles of two systems respectively. We also present an appli- cation.


1.
Introduction. As we know, one of the main problems, in the qualitative theory of ODEs, is to study the number of limit cycles of planar differential systems, which is related to the Hilbert's 16th problem. Many works have been done on the problem for smooth planar differential systems, see for instance [1,3,5,9,16] and the references therein. In recent years, a lot of papers have appeared to study periodic solutions by qualitative theory for the non-smooth systems, see [2,4,7,12,13,15,18].
In order to obtain the maximal number of periodic solutions or an upper bound of the number of periodic solutions for piecewise smooth differential equations, people have developed various methods. Among them, the Melnikov function method and the averaging method are the most widely used ones. For example, the authors in [11] developed the Melnikov function method for piecewise smooth planar systems, and established a formula for the first order Melnikov function. From [7,18,19], we know that one can consider the number of limit cycles for piecewise polynomial systems by using the method of first order Melnikov function in Hopf and generalized homoclinic bifurcations. Recently the authors of [17] applied the Melnikov function theory to high-dimensional piecewise smooth near-integrable systems and gave a formula for the first order Melnikov vector function. One can use the first and second order averaging methods for studying the periodic solutions of piecewise smooth periodic differential systems, see [4,15,13]. The authors [10] used the higher order averaging theory for studying a class of quartic polynomial differential systems. In [14], the authors obtained the formulas of averaged functions at any order for discontinuous piecewise differential systems with many zones.
It is worth noting that the averaging method is equivalent to the Melnikov function method for studying the number of limit cycles of planar analytic (or C ∞ ) near-Hamiltonian systems, see [6].
X. Liu and M. Han [11] investigated the following piecewise polynomial system: where and proved that system (1) has at most n limit cycles bifurcating from the unperturbed period annulus by using the first order Melnikov function. If further the constant terms a ± 00 , b ± 00 do not appear, the system has the following form Inspired by [6], we study the maximal number of limit cycles for systems (1) and (2) via the second order averaging method. For any given ε 0 > 0 sufficiently small and N > ε 0 sufficiently large, denote by H 1 (n) the maximal number of limit cycles of system (1) bifurcating from the region ε 0 ≤ x 2 + y 2 ≤ N , H 2 (n) the maximal number of limit cycles of system (2) bifurcating from the region x 2 + y 2 ≤ N . Our main result is as follows: Theorem 1.1. For |ε| > 0 sufficiently small, using the second order averaging method we have H 1 (n) ≤ 2n − 1 and H 2 (n) ≤ 2n − 2.
The paper is organized as follows. In section 2, we introduce the first order and the second order averaging methods for piecewise smooth systems. In section 3, we compute the averaged functions associated to systems (1) and (2). Then we study the maximal number of zeros of averaged functions and prove Theorem 1.1. Finally we give an application.
2. Preliminary theorems. In this section we present some known basic results from the averaging theory for discontinuous differential systems, see [4] for more details.
Consider a T -periodic differential equation of the form with T > 0 constant, F being given for 0 ≤ t ≤ T by being an open set containingD j ,D j denoting the closure of the set D j which have the following form Note that F is periodic in t with period T and may not be continuous on the switch lines l 1 , · · · , l k−1 , where The equation (3) is called a k-piecewise C r smooth periodic equation, as called in [4]. Let hj (x) hj−1(x) For x 0 ∈ J, define the solution of equation (3) satisfying The P oincaré map of (3) is given by In [4], the author developed the averaging theory and obtained the following results.
Lemma 2.1. Consider the periodic equation (3). We have (I) For any given closed interval I ⊂ J, there exists ε * > 0 such that the function g k (x 0 , ε, δ) is well defined and of C r in (x 0 , ε, δ) for all x 0 ∈ I, |ε| < ε * and δ ∈ V . (II) If there exists an integer m, 1 ≤ m ≤ r, such that the function f defined in (4) has at most m zeros in x ∈ J for all δ ∈ V , multiplicity taken into account, then for any closed interval I ⊂ J, there exists ε 1 = ε 1 (I) > 0, such that for 0 < |ε| < ε 1 , δ ∈ V the periodic equation (3) has at most m T -periodic solutions with the property that the range of each of them is a subset of I.
Using the second order averaging theory, we can do further study to the maximal number of periodic solutions for the piecewise smooth periodic equations.
In fact, from (5) we know that if f (x, δ) = 0, the P oincaré map of (3) can be written as where εg k (x 0 , ε, δ) =ḡ k (x 0 , ε, δ). Moreover, according to Lemma 2.1 and [8], for any given closed interval I ⊂ J, there exists ε * > 0 such that the functiong k (x 0 , ε, δ) is well defined and of C r−1 in (x 0 , ε, δ) for all x 0 ∈ I, |ε| < ε * and δ ∈ V . By lemma 9 (the fundamental lemma) of [13], we havẽ where f 2 (x 0 , δ) is given by satisfyingF Clearly, f 2 ∈ C r−1 . Similar to the proof of Theorem 1.1 in [4], we can obtain Lemma 2.2. Consider the periodic equation (3). Suppose f (x, δ) = 0. If there exists an integer m, 1 ≤ m ≤ r − 1, such that the function f 2 defined in (7) has at most m zeros in x ∈ J for all δ ∈ V , multiplicity taken into account, then for any closed interval I ⊂ J, there exists ε 1 = ε 1 (I) > 0, such that for 0 < |ε| < ε 1 , δ ∈ V , the periodic equation (3) has at most m T -periodic solutions with the property that the range of each of them is a subset of I. Lemma 2.3. If the condition of Lemma 2.2 is satisfied together with F (t, 0, ε, δ) = 0 and J = (0, +∞), then for any N > 0, there exists ε 1 = ε 1 (N ) > 0 such that for 0 < |ε| < ε 1 , δ ∈ V , (3) has at most m positive periodic solutions whose ranges are subsets of (0, N ]. 3. Proof of main results. In this section, we proceed to prove our main theorem. The proof is divided into two steps. First we study the limit cycle bifurcations of system (1). For the purpose, we introduce polar coordinate transformation.
From the equations (9) and (10) in Lemma 3.1, it yields the following 2π-periodic equation where It is obvious that seeking the limit cycles of system (1) bifurcated from the period annulus is equivalent to searching for the 2π-periodic solutions of (14). Let Then from the formula (4) we have where We can easily see that (16) has at most n isolated positive zeros, as in [11]. If f 1 (r) ≡ 0, then we need to study the function f 2 (r) by using formula (7). Through direct computation we get Then by (15) and (17), we obtain where M ± k , N ± k are constants that depend on the coefficients of system (1) for k = 0, 1, · · · , 2n − 1.
Remark 2. The reason we require ε 0 ≤ x 2 + y 2 is that equation (14) may not be well defined at r = 0.
Second we consider the limit cycle bifurcations of system (2). Note that a ± 00 = b ± 00 = 0. This implies v 0 = V 0 = 0. Using the above results, we can easily see that the function f 1 (r) has at most n − 1 isolated positive zeros. If f 1 (r) ≡ 0, clearly f 2 has at most 2n − 2 isolated positive zeros. Since equation (14) is well defined at the origin and r = 0 is its zero solution, we get the next result from Lemmas 2.2 and 2.3. Theorem 3.3. For |ε| > 0 sufficiently small, system (2) has at most 2n − 2 limit cycles on any given compact set of the plane containing the origin using the second order averaging method.
Combining Theorems 3.2 and 3.3, we complete the proof of Theorem 1.1.
Proof. Applying Theorem 1.1, it is easy to see that system (19) has at most 4 limit cycles by the averaging method of second order. Now we prove 4 limit cycles can appear.
From (16), the function f 1 is given by Noting that r i , i = 1, 2, 3 are linearly independent, we have f 1 (r) ≡ 0 if the condition (H) holds.
In order to use the second order averaging theory, we need to compute the function f 2 . From the formula (7), we obtain , π).
From this example we know that more limit cycles can be produced by using the second order averaging method than using the first order averaging method.
By the second order averaging method, we conjecture that 2n − 1 limit cycles can appear for system (1) and 2n − 2 limit cycles can appear for system (2) for |ε| > 0 sufficiently small. Take system (1) with n = 1, 2 for example.
Then we have It follows that V 0 , V 1 , V 2 can be taken as free parameters. Similarly we can prove system (1) has 3 limit cycles for n = 2.