ARBITRARY NUMBER OF LIMIT CYCLES FOR PLANAR DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH TWO ZONES

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1 Electronic Journal of Differential Equations, Vol (2015), No. 228, pp ISSN: URL: or ftp eje.math.txstate.eu ARBITRARY NUMBER OF LIMIT CYCLES FOR PLANAR DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH TWO ZONES DENIS DE CARVALHO BRAGA, LUIS FERNANDO MELLO Abstract. For any given positive integer n we show the existence of a class of iscontinuous piecewise linear ifferential systems with two zones in the plane having exactly n hyperbolic limit cycles. Moreover, all the points on the separation bounary between the two zones are of sewing type, except the origin which is the only equilibrium point. 1. Introuction an statement of main results One of the most challenging problems in the qualitative theory of planar orinary ifferential equations is the secon part of the classical 16th Hilbert problem: the etermination of an upper boun for the number of limit cycles (an their relative positions) for the class of polynomial vector fiels of egree n. This problem remains unsolve if n 2. The case n = 1, that is for the class of planar linear vector fiels the problem has a trivial answer. However, this problem presents a surprising richness when aapte to the class of the planar piecewise linear systems. Planar piecewise linear ifferential systems are wiely stuie nowaays because of their applicability in several branches of science. A lanmark of such stuy was the work of Anronov et al. [1]. There is an expectation that these systems can present all the ynamical behaviors of the classical nonlinear ifferential systems. In this article, we stuy the existence, number, stability an istribution of limit cycles for a class of piecewise linear ifferential systems in the plane. These issues must be stuie taking into account the following aspects: the number an stability of equilibrium points as well as their locations with respect to the separation bounary L (which efines the number of zones) an the behavior of the linear vector fiels on L. Usually, the points of iscontinuity on the separation bounary L are classifie as sewing, sliing or tangency points. Here the term sliing is use in a broa sense meaning sliing or escaping points. See [9] an the references therein for more etails. Without being exhaustive we present below some results relative to the stuy of limit cycles in piecewise linear ifferential systems in the plane. For the simplest 2010 Mathematics Subject Classification. 34C05, 34C25, 37G10. Key wors an phrases. Piecewise linear ifferential system; limit cycle; non-smooth ifferential system. c 2015 Texas State University - San Marcos. Submitte April 28, Publishe September 2,

2 2 D. D. C. BRAGA, L. F. MELLO EJDE-2015/228 case, that is for planar piecewise linear ifferential systems with two zones separate by a straight line L an with only one equilibrium point p L, we have the following results: (i) With the aitional hypothesis of the continuity on L\{p}, there is at most one limit cycle [4]; (ii) With the aitional hypothesis that the points on L\{p} are of sewing type, there is at most one limit cycle [10]; (iii) With the aitional hypothesis of the existence of a sliing segment on L\{p}, there are examples with two limit cycles surrouning the sliing segment [5]. The answer to the existence an the number of limit cycles in piecewise linear systems with more than two zones in the plane was given in [6]. In that article the authors provie an example of a planar piecewise linear Liénar system with an arbitrary number of limit cycles all of them hyperbolic. More precisely, given a positive integer n the authors showe, using the averaging theory, that the system uner consieration can have at least n limit cycles in the strip x 2n + 2 for a parameter ɛ R sufficiently small. Llibre, Ponce an Zhang [8] prove a conjecture of [6] in which the upper boun for the number of limit cycles can be achieve for a fixe n. Recent stuies suggest that three is the maximum number of limit cycles for planar iscontinuous piecewise linear ifferential systems with two zones separate by a straight line L an only one equilibrium point p / L. Numerical examples which support this statement can be foun in [2] an [7]. The separation bounary L between the two zones plays an important role in planar iscontinuous piecewise linear ifferential systems with only one equilibrium point p / L. The article [3] exhibits an example of such a system with seven limit cycles having L as a polygonal curve. Here we show that a planar iscontinuous piecewise linear ifferential system with two zones can have an arbitrary number of limit cycles, that is, the main iea in [8] still remains for only two zones. So, we stuy the following class of iscontinuous piecewise linear ifferential with two zones in the plane { X G X, H(X) < 0, = G + (1.1) X, H(X) 0, the prime enotes erivative with respect to the inepenent variable t, calle here the time, X = (x, y) R 2 an ( G ± g ± = 11 g 12 ± ) g 21 ± g 22 ±, (1.2) are matrices with real entries satisfying the following assumptions: (A1) g ± 12 < 0; (A2) G has complex eigenvalues with negative real parts, λ 1,2 = γ ±iω, while G + has complex eigenvalues with positive real parts, λ + 1,2 = γ+ ±iω +, γ ±, ω ± R an ω ± > 0. In aition, (A3) the function H is at least continuous an the set H 1 (0) ivies the plane in two unboune components, that is the function H implicitly efines

3 EJDE-2015/228 ARBITRARY NUMBER OF LIMIT CYCLES 3 a simple planar curve homeomorphic to the real line an whose trace is unboune. A member of the class (1.1) will be enote by (G, G +, H). Note that the hypothesis (A3) ensures the existence of only two zones whose separation (bounary) set is efine by L H = {X R 2 : H(X) = 0}. Our goal is to buil a suitable function Ψ satisfying the assumptions on H an to choose matrices G an G + in the Joran normal forms J ± such that, for any positive integer n, the system (J, J +, Ψ) has exactly n hyperbolic limit cycles. Furthermore, 0 = (0, 0) H 1 (0) an L H \{0} is a sewing set. As far as we know, it is the first example of such systems. We prove the following main theorem. Theorem 1.1. Given any positive integer n there is a planar piecewise linear ifferential system with two zones (J, J +, Ψ) having n hyperbolic limit cycles. Novaes an Ponce [11] obtaine examples of planar piecewise linear ifferential systems with two zones having n limit cycles, for every positive integer n. However, the limit cycles obtaine can be non-hyperbolic. This article is structure as follows. In Section 2 we present the proof of Theorem 1.1 an we give an example of such a system with ten limit cycles. In the last section we make some concluing remarks. 2. Proof of main results We consier the case in which the matrices G + an G are in the Joran normal forms; that is, ( ) J ± γ = ± ω ± ω ± γ ±. (2.1) The solutions of X = J X will be enote by (t, X 0 ) X (t, X 0 ) = (x (t, X 0 ), y (t, X 0 )), while the solutions of X = J + X will be enote by an (t, X 0 ) X + (t, X 0 ) = (x + (t, X 0 ), y + (t, X 0 )), x ± (t, X 0 ) = e γ± t ( cos(ω ± t)x 0 sin(ω ± t)y 0 ), y ± (t, X 0 ) = e γ± t ( sin(ω ± t)x 0 + cos(ω ± t)y 0 ). (2.2) In Lemma 2.1 we present a result associate with the system (J, J +, Φ), with ρ > 0, s R v(s) = s u(s) an is the unit step function. X Φ(X) = x φ(y) y φ(y) = ρv(y), (2.3) s R u(s) = { 0, s < 0, 1, s 0

4 4 D. D. C. BRAGA, L. F. MELLO EJDE-2015/228 Figure 1. Displacement function (y 0, ρ) δ φ (y 0, ρ). Lemma 2.1. Let η < 0 an η + > 0 be numbers satisfying η < η + < 3η, η ± = γ ± /ω ±, an efine ρ c = tan (π η+ + η ) η + η. Then, the origin of the system (J, J +, Φ) is: (a) An unstable focus, if 0 < ρ < ρ c, (b) A stable focus, if ρ > ρ c, (c) A center, if ρ = ρ c. Proof. Let X 0 = (x 0, y 0 ) = (φ(y 0 ), y 0 ) L Φ be any initial conition with y 0 > 0. The isplacement function is efine by (y 0, ρ) δ φ (y 0, ρ) = y + ( τ +, X 0 ) y (τ, X 0 ), (2.4) τ > 0 is the smallest time such that X (τ, X 0 ) L Φ, an τ + > 0 is the smallest time such that X + ( τ +, X 0 ) L Φ. See Figure 1. From (2.2) an (2.3), the time τ is the solution of the equation x (τ, X 0 ) = 0 an is given by τ = τ (y 0, ρ) = 1 ( ω arctan ( x 0 ) ) + π, (2.5) y 0 an τ + is the solution of x + ( τ +, X 0 ) = 0, an is given by τ + = τ + (y 0, ρ) = 1 ( ω + arctan ( x 0 ) ) π. (2.6) y 0 Thus, y (τ, X 0 ) = e γ τ x y2 0, y + ( τ +, X 0 ) = e γ+ τ + x y2 0, (2.7)

5 EJDE-2015/228 ARBITRARY NUMBER OF LIMIT CYCLES 5 an, therefore, (y 0, ρ) φ (y 0, ρ) = From (2.5) an (2.6), we have (y 0, ρ) δ φ (y 0, ρ) = φ (y 0, ρ) φ(y 0 ) 2 + y0 2, ( e γ τ e γ+ τ +) = e γ+ τ +( ) e γ τ +γ + τ + 1. (2.8) ρ a(ρ) = γ τ + γ + τ + = π(η + + η ) (η + η ) arctan(ρ) (2.9) an a(ρ) = 0 implies ρ = ρ c = tan (π η+ + η ) η + η. If η < η + < 3η, then ρ c > 0. So, for any y 0 > 0 such that x 0 = φ(y 0 ) = ρ c y 0 u(y 0 ) = ρ c y 0, δ φ (y 0, ρ c ) 0. This proves item (c). Items (a) an (b) follow from x 0 = ρv(y 0 ) an (2.9) since s arctan(s) is a monotonically increasing function. Figure 2. The graphs of the functions y φ(y) an y ψ(y) are illustrate by continuous black an orange lines, respectively. Given an integer number n 1, consier the finite sequences {u l } l N an {v l } l N, u l = (2l 3)ru(l 2), v l = 1 ( ul + ( 1) l εu(l 2) ) (2.10), ρ c for l N = {1, 2,..., n+2}, r, ε R satisfying 0 < ε < r. So v 1 < v 2 < < v n+2. Consier X Ψ(X) = x ψ(y), n+1 y ψ(y) = u 1 + α k (v(y v k ) β k v(y v k+1 )), (2.11)

6 6 D. D. C. BRAGA, L. F. MELLO EJDE-2015/228 an the real numbers α k an β k are given by α k = u k+1 u k, k = 1, 2,..., n + 1, v k+1 v k { 1, k = 1,..., n, β k = 0, k = n + 1. The sets L Φ an L Ψ have intersections at the points p i = (x i, y i ) R 2, with for i = 1,..., n. In fact, y g(y) = ψ(y) φ(y) x i = 2ri, y i = 1 ρ c x i = 2r ρ c i, n+1 = u 1 + α k (v(y v k ) β k v(y v k+1 )) ρ c v(y) is a continuous function on the close interval [v j, v j+1 ] satisfying g(v 1 ) = 0 an n+1 j 1 g(v j ) = F (j, k) = F (j, k) + F (j, j) + for j = 2,..., n + 1, n+1 k=j+1 F (j, k) = α k (v(v j v k ) β k v(v j v k+1 )). F (j, k) ρ c v(v j ), Since {v l } l N is a monotone increasing finite sequence, it follows that j 1 g(v j ) = F (j, k) ρv(v j ) j 1 = α k (v k+1 v k ) ρ c v j j 1 = (u k+1 u k ) ρ c v j = (1 + 2(j 2))r ρ c v j = (2j 3)r ( (2j 3)r + ( 1) j ε ) = ( 1) j+1 ε, (2.12) for j = 2,..., n + 1. Thus g(v i+1 )g(v i+2 ) = ε 2 < 0 for i = 1,..., n an by Bolzano s Theorem the function y g(y) has at least n zeros. Since the function y ψ(y) is piecewise linear an the union of straight lines of the form y [v l, v l+1 ] L l (y) = α l (y v l ) + u l, (2.13) for l = 1,..., n + 1, there exist exactly n zeros, that is there exists a unique y i [v i+1, v i+2 ] such that g(y i ) = 0. Moreover, it is easy to see that y i = v i+1 + v i+2 2 = 1 u i+1 + u i+2, ρ c 2 is such as in (2.12) for i = 1,..., n. The n points p i = (x i, y i ) R 2 given by (2.12) are zeros of the isplacement function y 0 δ ψ (y 0 ) associate with the system (J, J +, Ψ).

7 EJDE-2015/228 ARBITRARY NUMBER OF LIMIT CYCLES 7 Here we stuie only the case in which η an η + are such as in Lemma 2.1 an in aition η + < 1/η although other choices are possible. The slope of the straight line y L 1 (y) = α 1 y is α 1 = 1 ( 1 + ε ). ρ c r Thus, we take ρ c such that v 2 v 1 = 1 (1 + ε ) < 2 < 1 u 2 u 1 ρ c r ρ c η an v 3 v 1 = 1 ( 1 ε ) > 2 > η + u 3 u 1 ρ c 3r 3ρ c for 0 < ε < r. Therefore, 1 3η + < η < 0, 2η < tan (π η+ + η ) η + η < 2 3η +. (2.14) Figure 3. The set R is illustrate by the blue region. The continuous black, re an brown lines represent the graphs of the sets C 1, C 2 an C 3, respectively. From the inequalities 1 3η + < η < 0, η < η +, η < η + < 3η, η + < 1/η, 2η < tan (π η+ + η ) η + η < 2 3η +,

8 8 D. D. C. BRAGA, L. F. MELLO EJDE-2015/228 we obtain the following main functions an the set s h 1 (s) = π arctan(2s) π + arctan(2s) s, s h 2 (s) = π arctan ( ) 2 3s π + arctan ( 2 )s, 3s s h 3 (s) = 1 3s R = { (η, η + ) R 2 : η + > h 1 (η ), η < h 2 (η + ) }. The set R is illustrate in Figure 3. The points q 1 an q 2 (isplaye only with six ecimals) are given by q 1 = (η1, η+ 1 ) = (0, 0) C 1 C 2, q 2 = (η2, η+ 2 ) = ( , ) C 1 C 2 C 3, C 1 = {(η, η + ) R 2 : η + = h 1 (η )}, C 2 = {(η, η + ) R 2 : η = h 2 (η + )}, C 3 = {(η, η + ) R 2 : η + = h 3 (η )}. Figure 4. The ashe re lines are the nullclines associate with X = J X an the ashe blue ones are the nullclines associate with X = J + X. The orange an black continuous lines are the graph of the sets Ψ 1 (0) an Φ 1 (0), respectively.

9 EJDE-2015/228 ARBITRARY NUMBER OF LIMIT CYCLES 9 It follows that the vector fiels J X an J + X are both transversal to the set L Ψ if (η, η + ) R. This means that the set L Ψ \{0} is a sewing set accoring to Figure 4. Therefore if X 0 = (x 0, y 0 ) = (ψ(y 0 ), y 0 ) L Ψ an employing the same previous notation, there exists the smallest time τ > 0 such that X (τ, X 0 ) L Ψ or more precisely x (τ, X 0 ) = 0. In the same way there exists the smallest time τ + > 0 such that X + ( τ +, X 0 ) L Ψ or x + ( τ +, X 0 ) = 0. Moreover, the times τ an τ + are the same as given in (2.5) an (2.6); that is, τ = τ (y 0 ) = 1 ( ω arctan ( x 0 ) ) + π, (2.15) y 0 τ + = τ + (y 0 ) = 1 ω + ( arctan ( x 0 y 0 ) π ). (2.16) Thus, the following function is well efine y 0 δ ψ (y 0 ) = y + ( τ +, X 0 ) y (τ, X 0 ), (2.17) y (τ, X 0 ) an y (τ, X 0 ) are such as in (2.7). The function y 0 δ ψ (y 0 ) can be rewritten as y 0 δ ψ (y 0 ) = ψ (y 0 ) ψ(y 0 ) 2 + y0 2, y 0 ψ (y 0 ) = ( e γ τ e γ+ τ +) = e γ+ τ +( ) e γ τ +γ + τ + 1. (2.18) From (2.12), (2.18) an Lemma 2.1, it follows that the system (J, J +, Ψ) has n limit cycles since δ ψ (y i ) = 0, for i = 1,..., n. To show that the n limit cycles are all hyperbolic we take an open interval I i [v i+1, v i+2 ] such that y i I i for i = 1,..., n, since the function y ψ(y) is not ifferentiable at all the points in its omain. Thus, for y 0 I i the erivative of y 0 δ ψ (y 0 ) with respect to y 0 is δ ψ (y 0 ) = ψ (y 0 ) y 0 y 0 y 0 From (2.13) it results that ψ(y 0 ) 2 + y ψ(y 0 ) y 0 ψ (y 0 ) = ( e γ τ e γ+ τ +) y 0 y 0 ψ(y 0 ) 2 + y 2 0, (2.19) = γ e γ τ τ (y 0 ) + γ + e γ + τ + τ + (y 0 ). y 0 y 0 x 0 = L i+1 (y 0 ) = α i+1 (y 0 v i+1 ) + u i+1 an the erivatives of (2.15) an (2.16) with respect y 0 are τ (y 0 ) = α i+1v i+1 u i+1 y 0 ω (L i+1 (y 0 ) 2 + y0 2), τ + (y 0 ) = α i+1v i+1 u i+1 y 0 ω + (L i+1 (y 0 ) 2 + y0 2). Therefore, for y 0 = y i I i an from (2.10) an (2.12), y 0 τ (y i ) = 1 ω ( 1)i εs(ρ c, r, ε, i),

10 10 D. D. C. BRAGA, L. F. MELLO EJDE-2015/228 y 0 τ + (y i ) = 1 ω + ( 1)i εs(ρ c, r, ε, i), S(ρ c, r, ε, i) = for i = 1,..., n an 0 < ε < r. So ψ (y i ) = γ e γ τ y 0 = ( 1) i ε ρ 2 c 2ri(r + ( 1) i ε)(1 + ρ 2 c) > 0 (2.20) y 0 τ (y i ) + γ + e γ + τ + τ + (y i ) y ( 0 η + e γ+ τ + η e γ τ ) S(ρ c, r, ε, i) an taking into account the result (2.20) an that η + e γ+ τ + η e γ τ > 0, 0 < ε < r an ψ (y i ) = 0 it follows from (2.19) that δ ψ (y i ) = ψ (y i ) x 2 i y 0 y + y2 i (2.21) 0 is ifferent from zero. This implies that the n limit cycles are all hyperbolic. Furthermore if i {1,..., n} is o (or even) then the associate limit cycle is stable (or unstable). Note that with the choice (2.10) the first limit cycle is always an attractor limit cycle. Also note that the perio of each limit cycle is constant an given by τ + τ + (see (2.15) an (2.16)) an can be mae equal to 2π through a ifferent time rescaling in each zone which becomes ω = ω + = 1. Theorem 1.1 is prove. Figure 5. The set R is illustrate by the blue region. The black ot represents the pair (η, η + ) = ( 0.3, 0.5) R. Now we present an example with ten limit cycles.

11 EJDE-2015/228 ARBITRARY NUMBER OF LIMIT CYCLES 11 Figure 6. Phase portrait of the system (J, J +, Ψ) with ten limit cycles for (η, η + ) = ( 0.3, 0.5) R, r = 1, ε = 0.5 an n = 10. The stable (unstable) limit cycles are illustrate by blue (re) lines. Figure 7. The continuous black line is the graph of the isplacement function y 0 δ ψ (y 0 ) of (J, J +, Ψ) for (η, η + ) = ( 0.3, 0.5) R, r = 1, ε = 0.5 an n = 10. The stable limit cycles are illustrate by blue ots an the unstable by re ots. Example 2.2. In this example we consier the case η = 0.3 an η + = 0.5; that is, ρ c = 1 an the pair (η, η + ) R (see Figure 5). For r = 1, we choose ε = 0.5. Accoring to Theorem 1.1, with these values an n = 10, there exists a system (J, J +, Ψ) such that its phase portrait has exactly ten hyperbolic limit cycles. This result is summarize in Figures 6 an Concluing remarks. In this article, we stuy one of the main problems in the qualitative theory of planar ifferential equations: the number an istribution of limit cycles in piecewise linear ifferential systems with two zones in the plane. We give a rigorous proof of the existence of an arbitrary number of limit cycles in piecewise linear ifferential systems with two zones in the plane. See Theorem 1.1. Base on our results we prove the conjecture about the existence of a piecewise linear ifferential system with two zones in the plane with exactly n limit cycles, for any n N. See [3].

12 12 D. D. C. BRAGA, L. F. MELLO EJDE-2015/228 Acknowlegements. The authors are partially supporte by CNPq grant / an by CAPES CSF PVE grant / The secon author is partially supporte by CNPq grant / an by FAPEMIG grant PPM References [1] A. Anronov, A. Vitt, S. Khaikin; Theory of Oscillators, Pergamon Press, Oxfor, [2] D. C. Braga, L. F. Mello; Limit cycles in a family of iscontinuous piecewise linear ifferential systems with two zones in the plane, Nonlinear Dynam., 73 (2013), [3] D. C. Braga, L. F. Mello; More than three limit cycles in iscontinuous piecewise linear ifferential systems with two zones in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), [4] E. Freire, E. Ponce, F. Rorigo, F. Torres; Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), [5] E. Freire, E. Ponce, F. Torres; Canonical iscontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Syst., 11 (2012), [6] J. Llibre, E. Ponce; Piecewise linear feeback systems with arbitrary number of limit cycles, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), [7] J. Llibre, E. Ponce; Three neste limit cycles in iscontinuous piecewise linear ifferential systems with two zones, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), [8] J. Llibre, E. Ponce, X. Zhang; Existence of piecewise linear ifferential systems with exactly n limit cycles for all n N, Nonlinear Anal., 54 (2003), [9] J. Llibre, M. A. Teixeira, J. Torregrosa; Lower bouns for the maximum number of limit cycles of iscontinuous piecewise linear ifferential systems with a straight line of separation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), [10] J. C. Merao, J. Torregrosa; Uniqueness of limit cycles for sewing piecewise linear systems, J. Math. Anal. Appl., 431 (2015), [11] D. D. Novaes, E. Ponce; A simple solution to the Braga Mello conjecture, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25 (2015), Denis e Carvalho Braga Instituto e Matemática e Computação, Universiae Feeral e Itajubá, Avenia BPS 1303, Pinheirinho, CEP , Itajubá, MG, Brazil aress: braga@unifei.eu.br Luis Fernano Mello Instituto e Matemática e Computação, Universiae Feeral e Itajubá, Avenia BPS 1303, Pinheirinho, CEP , Itajubá, MG, Brazil. Tel: , Fax: aress: lfmelo@unifei.eu.br