This is a preprint of: “Limit cycles for two classes of control piecewise linear differential systems”, Jaume Llibre, Regilene D. S. Oliveira, Camila A. B. Rodrigues, Sao Paulo J. Math. Sci., 1–17, 2020. DOI: [10.1007/s40863-020-00163-7] LIMIT CYCLES FOR TWO CLASSES OF CONTROL PIECEWISE LINEAR DIFFERENTIAL SYSTEMS JAUME LLIBRE1 , REGILENE D. OLIVEIRA2 AND CAMILA AP. B. RODRIGUES3 Abstract. We study the bifurcation of limit cycles from the periodic orbits of 2n– dimensional linear centers when they are perturbed inside classes of continuous and classes of discontinuous piecewise linear differential systems of control theory of the form ẋ = Ax + ϕ(x1 )b, where ϕ is a continuous or discontinuous piecewise linear function respectively. 1. Introduction and statement of the main results In the control theory are relevant the continuous piecewise linear differential systems of the form (1) ẋ = Ax + φ(x1 )b, with A a m × m matrix, x, b ∈ Rm , φ : R → R is the continuous piecewise linear function      −1 if x1 ∈ (−∞, −1),    (2) φ(x1 ) = x1 if x1 ∈ [−1, 1],        1 if x1 ∈ (1, ∞), where x = (x1 , . . . , xm )T , and the dot denotes the derivative with respect to the inde- pendent variable t, the time. Also in control theory are important the discontinuous piecewise linear differential systems of the form (1) where ψ : R → R is the discontinuous piecewise linear function     −1 if x1 ∈ (−∞, 0), (3) ψ(x1 ) =    1 if x1 ∈ (0, ∞). For more details on these continuous and discontinuous piecewise linear differential systems see for instance the books [1, 3, 10, 11, 15, 17]. The analysis of discontinuous piecewise linear differential systems goes back mainly to Andronov and coworkers [2] and nowadays still continues to receive attention by many researchers. In particular, discontinuous piecewise linear differential systems appear in 1 http://www.gsd.uab.cat 2 J. LLIBRE, R.D. OLIVEIRA AND C.A.B. RODRIGUES a natural way in control theory and in the study of electrical circuits, see for instance the book [4] and the references quoted there. These systems can present complicated dynamical phenomena such as those exhibited by general nonlinear differential systems. One of the main ingredients in the qualitative description of the dynamical behavior of a differential system is the number and the distribution of its limit cycles. The goal of this paper is to study the existence of limit cycles for a class of continuous and a class of discontinuous piecewise linear differential of the form (1). More precisely, first we consider the class of continuous piecewise linear differential systems ( ) (4) ẋ = A0 x + ε Ax + φ(x1 )b , with |ε| ̸= 0 a sufficiently small real parameter, where A0 is the 2n × 2n matrix having on its principal diagonal the following 2 × 2 matrices    0 −(2k − 1)    for k = 1, . . . , n,   2k − 1 0 in the order that they are defined and zeros in the complement, A is an arbitrary 2n×2n matrix and b ∈ R2n \ {0}. Note that for ε = 0 system (4) becomes (5) ẋ1 = −x2 , ẋ2 = x1 , ... , ẋ2n−1 = −(2n − 1)x2n , ẋ2n = (2n − 1)x2n−1 . Moreover, the origin of (5) is a global isochronous center in R2n , i.e. all its orbits different from the origin are periodic with period 2π. In a similar way we consider the discontinuous piecewise linear differential systems ( ) (6) ẋ = A0 x + ε Ax + ψ(x1 )b . Our main results on the limit cycles of the continuous and discontinuous piecewise linear differential systems (4) are the following ones. Theorem 1. For |ε| > 0 sufficiently small and using averaging theory of first order at most one limit cycle of the continuous piecewise linear differential system (4) bifurcates from the periodic orbits of system (5). Moreover there are systems (4) with |ε| > 0 sufficiently small having a such limit cycle. Theorem 1 is proved in section 3. Theorem 2. For |ε| > 0 sufficiently small and using averaging theory of first order at most one limit cycle of the discontinuous piecewise linear differential system (6) bifurcates from the periodic orbits of system (5). Moreover there are systems (4) with |ε| > 0 sufficiently small having a such limit cycle. Theorem 2 is proved in section 4. LIMIT CYCLES OF CONTROL PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 3 If instead of the matrix A0 we consider the matrix A1 where A1 is the 2n × 2n matrix having on its principal diagonal the following 2 × 2 matrices    0 −k    for k = 1, . . . , n,   k 0 in the order that they are defined and zeros in the complement, then we have the following results. Theorem 3. For |ε| > 0 sufficiently small the averaging theory of first order does not provide any information about the limit cycles of the continuous piecewise linear differential system ( ) (7) ẋ = A1 x + ε Ax + φ(x1 )b , which can bifurcate from the periodic orbits of system (5). Theorem 4. For |ε| > 0 sufficiently small the averaging theory of first order does not provide any information about the limit cycles of the discontinuous piecewise linear differential system ( ) (8) ẋ = A1 x + ε Ax + ψ(x1 )b , which can bifurcate from the periodic orbits of system (5). Theorems 3 and (4) are proved in section 5. Note the difference between the matrices A0 and A1 , in the matrix A0 the non-zero entries are only the odd numbers 1, 3, . . . , 2n − 1, while in the matrix A1 the non-zero entries are the numbers 1, 2, . . . , n. This difference provides that the continuous and discontinuous piecewise linear differential systems (4) and (6) can have limit cycles detected by the averaging theory, while for the continuous and discontinuous piecewise linear differential systems (7) and (8) the averaging theory cannot detect limit cycles. Really for the control differential systems here studied the limit cycles that we obtain bifurcate from some periodic orbit of the 2n-dimensional linear differential center (5). This technique of finding limit cycles bifurcating from centers has been intensively studied in dimension 2, see for instance the book of Christopher and Li [9] and the hundreds of references quoted therein. Other results different to the ones presented here, but which also study the limit cycles of control systems of the form (1) using averaging theory, can be found in [6, 7, 8, 12]. The main tools for proving the previous theorems are the extensions of the classical averaging theory for computing periodic solutions of C 2 differential systems to contin- uous and discontinuous differential systems. In section 2 we summarize the extensions of the averaging theory that we shall use here for proving our results. 4 J. LLIBRE, R.D. OLIVEIRA AND C.A.B. RODRIGUES 2. First order averaging theory For the classical averaging theory for finding periodic orbits of differential systems of class C 2 see for instance the chapter 11 of the book of Verhulst [16]. In this section we present first the result on the continuous averaging theory that we will use for proving our Theorems 1 and 3. This theory uses the Brouwer degree of a continuous function and its proof can be found in [5]. Theorem 5. We consider the following differential system (9) ẋ = εH(t, x) + ε2 R(t, x, ε), where H : R × D → Rn , R : R × D × (−εf , εf ) → Rn are continuous functions, T -periodic in the first variable, and D is an open bounded subset of Rn . We define h : D → Rn as ∫ T (10) h(z) = H(s, z)ds, 0 and assume that (i) H and R are locally Lipschitz with respect to x; (ii) for p ∈ D with h(p) = 0, there exists a neighborhood V of p such that h(z) ̸= 0 for all z ∈ V̄ \ {p} and the Brouwer degree dB (h, V, 0) ̸= 0. Then, for |ε| ̸= 0 sufficiently small, there exists an isolated T - periodic solution x(t, ε) of system (9)such that x(0, ε) → p as ε → 0. Remark 6. Let h : D → Rn be a C 1 function with h(p) = 0, where D is an open bounded subset of Rn and p ∈ D. If the Jacobian of h at p is not zero, then there exists a neighborhood V of p such that h(z) ̸= 0 for all z ∈ V̄ \ {p}, and the Brouwer degree dB (h, V, p) ∈ {−1, 1}. For a proof of Remark 6 see for instance [14]. For proving Theorems 2 and 4 we shall need the following extension of the averaging theory for computing periodic solutions to discontinuous differential systems done in [13]. Theorem 7. We consider the following discontinuous differential system (11) x′ (t) = εH(t, x) + ε2 R(t, x, ε), with H(t, x) = H1 (t, x) + sign(g(t, x))H2 (t, x), R(t, x, ε) = R1 (t, x, ε) + sign(g(t, x))R2 (t, x, ε), where H1 , H2 : R × D → Rn , R1 , R2 : R × D × (−ε0 , ε0 ) → Rn and g : R × D → R are continuous functions, T –periodic in the variable t and D is an open subset of Rn . We also suppose that g is a C 1 function having 0 as a regular value. LIMIT CYCLES OF CONTROL PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 5 Define the average function h : D → Rn as ∫ T (12) h(x) = H(t, x)dt. 0 We assume the following conditions. (i) H1 , H2 , R1 , R2 are locally Lipschitz with respect to x; (ii) there exists an open bounded subset C ⊂ D such that, for |ε| > 0 sufficiently small, every orbit starting in C reaches the set of discontinuity only at its cross- ing regions. (iii) for a ∈ C with h(a) = 0, there exists a neighbourhood U ⊂ C of a such that h(z) ̸= 0 for all z ∈ U \ {a} and dB (h, U, 0) ̸= 0. Then, for |ε| > 0 sufficiently small, there exists a T –periodic solution x(t, ε) of system (11) such that x(0, ε) → a as ε → 0. 3. Proof of Theorem 1 The main tool for proving Theorem 1 is the averaging theory of first order for con- tinuous differential systems presented in Theorem 5. In order to use this theorem we need to write the differential system (4) in the normal form (9), and for obtaining this we need to some changes of variables. Lemma 8. Doing the change of variables (x1 , x2 , . . . , x2n ) 7→ (θ, r, θ1 , r1 , . . . , θn−1 , rn−1 ) defined by x1 = r cos θ, x2 = r sin θ, x2j−1 = rj−1 cos((2j − 1)θ + θj−1 ), x2j = rj−1 sin((2j − 1)θ + θj−1 ), for j = 2, . . . , n system (4) is transformed into the system dr = εH1 (θ, r, θ1 , r1 , . . . , θn−1 , rn−1 ) + O(ε2 ), dθ (13) drj−1 = εH2(j−1) (θ, r, θ1 , r1 , . . . , θn−1 , rn−1 ) + O(ε2 ), dθ dθj−1 = εH2j−1 (θ, r, θ1 , r1 , . . . , θn−1 , rn−1 ) + O(ε2 ), dθ where n ∑ ( ) H1 = rl−1 F1,l cos θ + F2,l sin θ + φ(r cos θ)(b1 cos θ + b2 sin θ), l=1 6 J. LLIBRE, R.D. OLIVEIRA AND C.A.B. RODRIGUES and for j = 2, 3, . . . , n we have ∑ n ( ) H2(j−1) = rl−1 F2j−1,l cos((2j − 1)θ + θj−1 ) + F2j,l sin((2j − 1)θ + θj−1 ) l=1 [ ] +φ(r cos θ) b2j−1 cos((2j − 1)θ + θj−1 ) + b2j sin((2j − 1)θ + θj−1 ) , ∑n ( ) rl−1 H2j−1 = F2j,l cos((2j − 1)θ + θj−1 ) − F2j−1,l sin((2j − 1)θ + θj−1 ) rj−1 l=1 ( ) ∑n rl−1 +(2j − 1) F1,l sin θ − F2,l cos θ r l=1 ( ) b2j b2j−1 +φ(r cos θ) cos((2j − 1)θ + θj−1 ) − sin((2j − 1)θ + θj−1 ) rj−1 rj−1 ( ) b2 b1 −(2j − 1)φ(r cos θ) cos θ − sin θ , r r with Fi,l = Fi,l (r, θ, θl−1 ) = ai(2l−1) cos((2l − 1)θ + θl−1 ) + ai(2l) sin((2l − 1)θ + θl−1 ). We take ε0 sufficiently small, m arbitrarily large and { ( ) [ ( )]n−1 } 1 1 1 Dm = (r, θ1 , r1 , . . . , θn−1 , rn−1 ) ∈ ,m × S × ,m . m m Then the vector field of system (13) is well defined and continuous on S1 × Dm × (−ε0 , ε0 ). Moreover the system is 2π-periodic with respect to variable θ and locally Lipschitz with respect to variables (r, θ1 , r1 , . . . , θn−1 , rn−1 ). Proof. In the variables (θ, r, θ1 , r1 , . . . , θn−1 , rn−1 ) the differential system (4) becomes [ n ( ) ] ε ∑ θ̇ = 1 + rl−1 F2,l cos θ − F1,l sin θ + φ(r cos θ)(b2 cos θ − b1 sin θ) , r l=1 ṙ = εH1 (θ, r, θ1 , r1 , . . . , θn−1 , rn−1 ), ṙj−1 = εH2(j−1) (θ, r, θ1 , r1 , . . . , θn−1 , rn−1 ), θ̇j−1 = εH2j−1 (θ, r, θ1 , r1 , . . . , θn−1 , rn−1 ), for j = 2, 3, . . . , n. Note that for ε = 0 , θ̇(t) > 0 and hence for |ε| ̸= 0 sufficiently small this property remains valid for each t when (θ, r, θ1 , r1 , . . . , θn−1 , rn−1 ) ∈ S1 × Dm . Now we take θ as the new independent variable. The right-hand side of the new system is well defined and continuous in S1 × Dm × (−ε0 , ε0 ) and it is 2π-periodic with respect to the new variable θ and locally Lipschitz with respect to (r, θ1 , r1 , . . . , θn−1 , rn−1 ). Now system (8) can be obtained doing a Taylor series expansion in the parameter ε around ε = 0.  LIMIT CYCLES OF CONTROL PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 7 The next step is to find the corresponding average function (10) of system (8) that we denoted by h = (h1 , h2 , . . . , h2(n−1) , h2n−1 ) : Dm → Rn−1 and it is defined by ∫ 2π h1 = h1 (r, θ1 , r1 , . . . , θn−1 , rn−1 ) = H1 (r, θ1 , r1 , . . . , θn−1 , rn−1 )dθ, 0 ∫ 2π h2(j−1) = h2(j−1) (r, θ1 , r1 , . . . , θn−1 , rn−1 ) = H2(j−1) (r, θ1 , r1 , . . . , θn−1 , rn−1 )dθ, 0 ∫ 2π h2j−1 = h2j−1 (r, θ1 , r1 , . . . , θn−1 , rn−1 ) = H2j−1 (r, θ1 , r1 , . . . , θn−1 , rn−1 )dθ, 0 for j = 1, 2, . . . , n. To calculate these integrals we will use the following equalities ∫ 2π cos((2j − 1)θ + θj−1 ) sin((2l − 1)θ + θl−1 )dθ = 0 for all integers l, j > 1, ∫0 2π { π if l = j, cos((2j − 1)θ + θj−1 ) cos((2l − 1)θ + θl−1 )dθ = 0 0 if l ̸= j, ∫ 2π { π if l = j, sin((2j − 1)θ + θj−1 ) sin((2l − 1)θ + θl−1 )dθ = 0 0 if l ̸= j, and the next lemma. For r > 0 and j = 1, 2, . . . , n we denote ∫ 2π Ij (r) = φ(r cos θ) cos((2j − 1)θ)dθ, ∫0 2π Jj (r) = φ(r cos θ) sin((2j − 1)θ)dθ, 0 where φ is the piecewise linear function (2). Lemma 9. The integrals Ij and Jj (r) satisfy    πr if j = 1 and 0 < r ≤ 1,  0 if j > 1 and 0 < r ≤ 1, Ij (r) =   K(r) if j = 1 and r > 1,   Lj (r) if j > 1 and r > 1; Jj (r) = 0 for all j = 1, 2, . . . , n and r > 0. where ( √ √ 2 2 cos((2j − 1) arctan Lj (r) = (2j − 1) −1 + r −1 + r2 ) j(2j − 1)2 √ ) − sin((2j − 1) arctan −1 + r ) ,2 2√ 2 √ K(r) = πr + r − 1 − 2r arctan( r2 − 1). r 8 J. LLIBRE, R.D. OLIVEIRA AND C.A.B. RODRIGUES Proof. We consider two cases: 0 < r ≤ 1 and r > 1. Case 1: 0 < r ≤ 1 In this case |r cos θ| ≤ 1 and hence φ(r cos θ) = r cos θ for all θ ∈ [0, 2π]. Then if j = 1 ∫ 2π ∫ 2π φ(r cos θ) cos θdθ = r cos2 θdθ = πr, 0 0 and ∫ 2π ∫ 2π φ(r cos θ) sin θdθ = r cos θ sin θdθ = 0. 0 0 And if j > 1 then ∫ 2π ∫ 2π φ(r cos θ) cos((2j − 1)θ)dθ = r cos θ cos((2j − 1)θ)dθ = 0, 0 0 ∫ 2π ∫ 2π φ(r cos θ) sin((2j − 1)θ)dθ = r cos θ sin((2j − 1)θ)dθ = 0. 0 0 Case 2: r > 1 In this case choose θc ∈ (0, π/2) such that cos θc = 1/r. If j = 1 we have ∫ θc ∫ π−θc ∫ π+θc I1 (r) = cos θdθ + r cos2 θdθ − cos θdθ 0 θc π−θc ∫ 2π−θc ∫ 2π 2 +r cos θdθ + cos θdθ π+θc 2π−θc 2√ 2 √ = πr + r − 1 − 2r arctan( r2 − 1). r The same reasoning can be applied to see that J1 (r) = 0. If j > 1 then ∫ θc ∫ π−θc ∫ π+θc Ij (r) = cos((2j − 1)θ)dθ + r cos θ cos((2j − 1)θ)dθ − cos((2j − 1)θ)dθ 0 θc π−θc ∫ 2π−θc ∫ 2π +r cos θ cos((2j − 1)θ)dθ + cos((2j − 1)θ)dθ π+θc ( 2π−θc 2 √ √ = (2j − 1) −1 + r 2 cos((2j − 1) arctan −1 + r2 ) j(2j − 1)2 √ ) − sin((2j − 1) arctan −1 + r ) , 2 and Jj (r) = 0.  With the results presented previously we are able to prove Theorem 1. Since we can choose m sufficiently large to find the zeroes of the average function h in Dm it is sufficient to look for them in (0, ∞) × [S1 × (0, ∞)]n−1 . To calculate the expression of the average function we consider again two cases. LIMIT CYCLES OF CONTROL PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 9 Case 1: 0 < r ≤ 1. In this case the system whose zeros can provide limit cycles of system (4) is h1 = (a11 + a22 + b1 )πr, h2 = (a33 + a44 )πr1 , h3 = (a43 − a34 + 3(a12 − a21 − b2 ))π, (14) .. . h2(n−1) = (a(2n−1)(2n−1) + a(2n)(2n) ))πrn−1 , h2n−1 = (a(2n)(2n−1) − a(2n−1)(2n) + (2n − 1)(a12 − a21 − b2 )π. Note that the variables θ1 , θ2 , . . . , θn−1 does not appear explicitly into system (14). Hence, if this system has zeros, it has a continuum of zeros. Therefore the assumption (ii) of the averaging theory, presented in Theorem 5, is not satisfied and this theorem does not provide any information about the limit cycles of system (13). Case 2: r > 1. Now the system whose zeros can provide limit cycles of system (13) is h1 = (a11 + a22 )πr + b1 K(r), h2 = (a33 + a44 )πr1 + (b3 cos θ1 + b4 sin θ2 )L2 (r), h3 = (a43 − a34 + 3(a12 − a21 ))π− 3b2 r1 K(r) − r(b4 cos θ1 − b3 sin θ1 )L2 (r) , rr1 .. (15) . h2(n−1) = (a(2n−1)(2n−1) + a(2n)(2n) )πrn−1 + (b2n−1 cos θn−1 + b2n sin θn−1 )Ln (r), h2n−1 = (a(2n)(2n−1) − a(2n−1)(2n) + (2n − 1)(a12 − a21 ))π− (2n − 1)b2 rn−1 K(r) − r(b2n cos θn−1 − b2n−1 sin θn−1 )Ln (r) , rrn−1 10 J. LLIBRE, R.D. OLIVEIRA AND C.A.B. RODRIGUES For each j ∈ {2, 3, . . . , n} we will study the zeros of the system h1 = (a11 + a22 )πr + b1 K(r), h2(j−1) = (a(2j−1)(2j−1) + a(2j)(2j) )πrj−1 + (b2j−1 cos θj−1 + b2j sin θj−1 )Lj (r), h2j−1 = (a(2j)(2j−1) − a(2j−1)(2j) + (2j − 1)(a12 − a21 ))π− (2j − 1)b2 rj−1 K(r) − r(b2j cos θj−1 − b2j−1 sin θj−1 )Lj (r) , rrj−1 Claim: The function K : (1, ∞) → (π, 4) is a diffeomorphism. Indeed note that K is twice differentiable with √ r2 − 1 √ ′ K (r) = π − 2 − 2 arctan r2 − 1, r2 and 4 K ′′ (r) = − √ <0 r r2 − 1 3 which implies that K ′ is a strictly decreasing function. Moreover limr→∞ K ′ (r) = 0 what means that K ′ (r) has a horizontal asymptote given by the axis r and then K ′ (r) ≥ 0. Suppose that there exists an r0 ∈ (1, ∞) such that K ′ (r0 ) = 0. Then for all r > r0 we have K ′ (r) < K ′ (r0 ) = 0, contradiction. Therefore it follows that K ′ (r) ̸= 0 for all r ∈ (1, ∞) and the Inverse Function Theorem guarantees that K is a local diffeomorphism and since that K is a injective function we obtain the global diffeomorphism, ending the proof of this claim. First we note that in order that the equation h1 = 0 has solutions with r > 1 it is necessary that b1 (a11 + a22 ) < 0. Moreover K ′′ (r) < 0 implies that the graph of K is convex. In the plane of the graph of K(r) the graph of (a11 + a22 )πr is a straight line passing through the origin and then both graphs can intersect at most in two points. 4 3 2 1 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Figure 1. The graphic of the function K(r). But if some straight line intercept the graph of K(r) in two points then it cannot pass through the origin, as we can see in Figure 1. Then the equation h1 = 0 has at LIMIT CYCLES OF CONTROL PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 11 most one solution if r > 1, and since that K(r) is a diffeomorphism we can choose the coefficients a11 , a22 and b1 so that this solution exists. We denote this solution by r0 and we substitute it into the equations h2(j−1) = 0 and h2j−1 = 0. Defining Aj = (a(2j−1)(2j−1) + a(2j)(2j) )π, Bj = b2j−1 Lj (r0 ), Cj = b2j Lj (r0 ), 1 Dj = (a(2j)(2j−1) − a(2j−1)(2j) + (2j − 1)(a12 − a21 ))π − (2j − 1)b2 K(r0 ), r0 uj = cos θj−1 , vj = sin θj−1 , the system h2(j−1) = h2j−1 = 0 is equivalent to the system Aj rj−1 + Bj uj + Cj vj = 0, Dj rj−1 + Cj uj − Bj vj = 0, u2j + vj2 − 1 = 0. Using the two first equations we obtain (Aj Bj + Cj Dj )rj−1 (Bj Dj − Aj Cj )rj−1 uj = − , vj = . Bj2 + Cj2 Bj2 + Cj2 Substituting these two expressions in the third equation we get (A2j + Dj2 )rj−1 2 − Bj2 − Cj2 = 0. Therefore at most there is one solution rj−1 > 0, which provide a unique uj and vj . Since we fixed an arbitrarily j to solve this system, the same reasoning can be applied to each pair of equations h2(j−1) = 0 and h2j−1 = 0, concluding that system (15) has at most one solution. Moreover taking conveniently the parameters of the initial system (4) this solution exists and its Jacobian is not zero. Hence at most one limit cycle can bifurcate from the periodic orbits of the center of system (5) when we perturbe it as in system (4), and there are systems for which a such limit cycles exist. This completes the proof of Theorem 1. Now we present an explicit example of a continuous piecewise linear differential system (4) in R4 , and repeating for it the proof of Theorem 1 we will show it has one limit cycle. Consider the following differential system (16) ẋ = A0 x + ε(Ax + φ(x1 )b), 12 J. LLIBRE, R.D. OLIVEIRA AND C.A.B. RODRIGUES where       24π  0 −1 0 0   2 1 0 0   − √       3 3 + 2π         1 0 0 0   0 2 0 0         1     ,b =   A0 =  ,A =    9(3 − 2√3π)  .  0 0 0 −3   0 0 −2 −1             √   2     18π − 3    0 0 3 0 0 0 0 −1 9π Doing the change or variables x1 = r cos θ, x2 = r sin θ, x3 = r1 cos(3θ + θ1 ), x4 = r1 sin(3θ + θ1 ) and taking θ as the new independent variable we obtain the system (17) dr r′ (θ) = = εH1 (θ, r, θ1 , r1 ) + O(ε2 ), dθ dr1 r1′ (θ) = = εH2 (θ, r, θ1 , r1 ) + O(ε2 ), dθ dθ1 θ1′ (θ) = = εH3 (θ, r, θ1 , r1 ) + O(ε2 ), dθ ( ) 24πφ(r cos θ) H1 (θ, r, θ1 , r1 ) = 2r + φ(r cos θ) sin θ + cos θ r sin θ − √ , 3 3 + 2π ( 1 H2 (θ, r, θ1 , r1 ) = − 18πφ(r cos θ) sin(3θ + θ1 ) + 9πr1 sin(2(3θ + θ1 )) 18π √ √ + 3r1 + 81π(2 3π − 3)φ(r cos θ) cos(3θ + θ1 )− ) √ ( 3 − 36π)r1 cos(2(3θ + θ1 )) , sin(2(3θ + θ1 )) H3 (θ, r, θ1 , r1 ) = sin2 (3θ + θ1 ) + 2 sin(2(3θ + θ1 )) − √ + 3 sin2 θ 6 3π 72πφ(r cos θ) sin θ φ(r cos θ) cos(3θ + θ1 ) 3φ(r cos θ) cos θ − √ − − √3 3r + 2πr r1 r 9π 3φ(r cos θ) sin(3θ + θ1 ) 27φ(r cos θ) sin(3θ + θ1 ) + − . r1 2r1 After some computations the average function h = (h1 , h2 , h3 ) defined in (10) is ( √ ) 24π 2 r2 − 1 √ h1 (r, θ1 , r1 ) = 4πr − √ πr + 2 − 2r arctan( r − 1) , √ 3 3 + 2π r √ 3 3 √ √ 3 h2 (r, θ1 , r1 ) = sin θ1 + (2 3π − 3) 3 cos θ1 − r1 , 3 2 9 √ √ ( ) 9 3 sin θ1 9π sin θ1 3 cos θ1 3 √ 2π h3 (r, θ1 , r1 ) = − + − 3+ + 4π. 2r1 r1 3r1 2 3 LIMIT CYCLES OF CONTROL PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 13 In order to solve the system h1 = h2 = h3 = 0 we can use the same reasoning applied in the proof of Theorem 1 obtaining that (r∗ , θ1∗ , r1∗ ) = (2, π/2, 3) is a zero of the average function. Moreover if J = J(r, θ1 , r1 ) is the Jacobian matrix of h, then det J(2, π/2, 3) ̸= 0 which implies that we have a simple zero. By Theorem 5 system (17) and consequently system (16) has one limit cycle for |ε| > 0 sufficiently small. 4. Proof of Theorem 2 This section is devoted to prove Theorem 2. According with Theorem 7 the same kind of arguments used for proving Theorem 1 can be applied to the discontinuous system (6), obtaining that the average function h = (h1 , h2 , . . . , h2(n−1) , h2n−1 ) : Dm → Rn−1 defined in (12) is h1 = (a11 + a22 )πr + b1 Ie1 , h2(j−1) = (a(2j−1)(2j−1) + a(2j)(2j) )πrj−1 + (b2j−1 cos θj−1 + b2j sin θj−1 )Iej , (18) h2j−1 = (a(2j)(2j−1) − a(2j−1)(2j) + (2j − 1)(a12 − a21 ))π− (2j − 1)b2 rj−1 Ie1 − r(b2j cos θj−1 − b2j−1 sin θj−1 )Iej , rrj−1 for j = 2, 3, . . . , n, where  4  − if j is even, Iej = (2j − 1)  4  if j is odd. (2j − 1) In fact if we define ∫ 2π Iej = ψ(r cos θ) cos((2j − 1)θ)dθ, ∫0 2π Jej = ψ(r cos θ) sin((2j − 1)θ)dθ, 0 where ψ is the piecewise linear function given by (3). Then we have that ∫ 2π Iej = ψ(r cos θ) cos((2j − 1)θ)dθ ∫0 π/2 ∫ 3π/2 ∫ 2π = cos((2j − 1)θ)dθ − cos((2j − 1)θ)dθ + cos((2j − 1)θ)dθ 0 π/2 3π/2 4 = − cos(jπ), (2j − 1) 14 J. LLIBRE, R.D. OLIVEIRA AND C.A.B. RODRIGUES and ∫ 2π Jej = ψ(r cos θ) sin((2j − 1)θ)dθ ∫0 π/2 ∫ 3π/2 ∫ 2π = sin((2j − 1)θ)dθ − sin((2j − 1)θ)dθ + sin((2j − 1)θ)dθ 0 π/2 3π/2 4 = − sin(2jπ) cos(jπ) = 0. (2j − 1) Note that Iej is a constant real number different from zero, and hence h1 is a straight line, and then system (18) has at most one positive zero. Moreover if we choose conve- niently the coefficients b1 a11 and a22 we can find a simple positive zero of system (18). This completes the proof of Theorem 2. 5. Proof of Theorems 3 and 4 Doing the change of coordinates x1 = r cos θ, x2 = r sin θ, x2j−1 = rj−1 cos(jθ + θj−1 ), x2j = rj−1 sin(jθ + θj−1 ) j ∈ {2, 3, . . . , n}, for j = 2, 3, . . . , n, to the continuous piecewise linear differential system (7), and work- ing as in the proof of Theorem 1 we obtain that the average function h = (h1 , h2 , . . ., h2n−1 ) defined in (10) now is given by h1 = (a11 + a22 )πr + b1 I1 (r), h2(j−1) = (a(2j−1)(2j−1) + a(2j)(2j) )πrj−1 + (b2j−1 cos θj−1 + b2j sin θj−1 )Ij (r), (19) h2j−1 = (a(2j)(2j−1) − a(2j−1)(2j) + j(a12 − a21 ))π− jb2 rj−1 I1 (r) − r(b2j cos θj−1 − b2j−1 sin θj−1 )Ij (r) , rrj−1 where ∫ 2π Ij (r) = φ(r cos θ) cos(jθ)dθ. 0 Using exactly the same arguments than in the proof of Lemma 9 is possible to prove that   πr if j = 1 and 0 < r ≤ 1, Ij (r) = 0 if j is even and 0 < r ≤ 1,   Lj (r) if j is odd and r > 1, where ( √ √ √ ) 4 2 − 1 cos(j arctan( r 2 − 1)) − sin(j arctan( r 2 − 1)) . Lj (r) = j r j(j 2 − 1) LIMIT CYCLES OF CONTROL PIECEWISE LINEAR DIFFERENTIAL SYSTEMS 15 The simple zeros of system (19) provide the existence of limit cycles for system (7) but since Ij (r) = 0 if j is even and r > 1, the variables θj−1 , for j = 2, 4, 6, ... do not appear in the system h1 = h2 = . . . = h2n−1 = 0, so either this system has no zeros, or if it has zeros, then it has a continuum of zeros, and therefore the assumption (ii) of the averaging Theorem 9 does not hold, and consequently the averaging theory cannot say anything about the limit cycles of system (7). The same occurs for the case 0 < r ≤ 1. So we conclude that, using the averaging theory of first order, we can say nothing about the number of the limit cycles of system (7). This completes the proof of Theorems 3. Now if we consider the discontinuous piecewise linear differential system (8), then its average function h = (h1 , h2 , . . . , h2n−1 ) defined in (12) is h1 = (a11 + a22 )πr + b1 Ie1 , h2(j−1) = (a(2j−1)(2j−1) + a(2j)(2j) )πrj−1 + (b2j−1 cos θj−1 + b2j sin θj−1 )Iej , (20) h2j−1 = (a(2j)(2j−1) − a(2j−1)(2j) + j(a12 − a21 ))π− jb2 rj−1 Ie1 − r(b2j cos θj−1 − b2j−1 sin θj−1 )Iej , rrj−1 where ∫ 2π Iej = ψ(r cos θ) cos(jθ)dθ. 0 Again we have that  ∫ 2π 0 if j is even, Iej = ψ(r cos θ) cos(jθ)dθ = 4 0 ± if j is odd, (2j − 1) and we can see that again either no zeros of the function h, or a continuum of zeros, concluding that the averaging theory of first order given by Theorem 7 does not say anything about the limit cycles of system (8). This completes the proof of Theorems 4. Acknowledgements The first author is partially supported by a MINECO grant number MTM2013- 40998-P, an AGAUR grant number 2014SGR568 and a FP7-PEOPLE-2012-IRSES grant number 318999. 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Novaes and M.A. Teixeira, On the birth of limit cycles for non-smooth dy- namical systems, Bull. Sci. Math. 139 (2015), 229–244. [14] N.G. Lloyd, Degree theory, Cambridge Tracts in Math. vol. 73, Cam- bridge Univ. Press, Cam- bridge, Great Britain, 1978. [15] A.I. Mees, Dynamics of feedback systems, John Wiley & Sons, 1981. [16] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2nd edition, Universitext, Springer 1996. [17] M. Vidyasagar, Nonlinear systems analysis, 2nd ed., Prentice-Hall, 1993. 1 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 2,3 Department of Mathematic, ICMC-Universidade de São Paulo, C.P. 668, 13.560-970, São Carlos, SP, Brazil E-mail address: regilene@icmc.usp.br, camilaap@icmc.usp.br