LIMIT CYCLES IN UNIFORM ISOCHRONOUS CENTERS OF DISCONTINUOUS DIFFERENTIAL SYSTEMS WITH FOUR ZONES

We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems ẋ = −y + x, ẏ = x + xy, and ẋ = −y + xy, ẏ = x + xy, when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively. Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively.


Introduction and Statement of the Main Results
Suppose that q ∈ R 2 is a center of a polynomial differential system in R 2 .Without loss of generality we can assume that q is at the origin of coordinates.Then q is an isochronous center if there exists a neighborhood U q of q such that all periodic orbits in U q have the same period.An isochronous center is uniform if in polar coordinates x = r cos θ, y = r sin θ it can be written as ṙ = G(θ, r), θ = k, k ∈ R \ {0}, for further details see [10].A singular point q is a weak focus if it is a center for the linearized system at q and this singular point is not a center.
Consider the planar polynomial differential system of degree n ẋ = −y + x f (x, y), ẏ = x + y f (x, y), where f (x, y) is a polynomial in x and y of degree n − 1 and f (0, 0) = 0.This differential system has only one singular point at the origin, which is a center for the linear part of the system.Moreover, the solutions of (1) move around the origin with constant angular speed and therefore, the origin is either a uniform isochronous center or a uniform weak focus.
Isochronicity is important in many fields such as physics, chemistry, biology and engineering.Moreover, isochronicity is relevant in stability theory because a region filled of the periodic solutions is Lyapunov stable if and only if the periodic solutions are isochronous, further details on these topics can be found in [7].In the last decades the bifurcation of limit cycles from uniform isochronous centers has attracted attention of several authors, see for instance [1,11,12,16,18].
In this paper we investigate the birth of limit cycles from a uniform isochronous center of discontinuous piecewise quadratic and cubic differential systems with four zones formed when the plane is divided by two perpendicular straight lines.To the best of our knowledge this is the first work that analyzes the bifurcation of limit cycles under these conditions.
Applying the averaging theory of first order we investigate the number of limit cycles which can bifurcate from the periodic orbits of the uniform isochronous center of the following quadratic and cubic differential systems when they are perturbed inside the classes of the following discontinuous quadratic and cubic polynomial differential systems with i = 1, . . ., 4 denoting the quadrant Q i , and for j = 2 the quadratic case, and for j = 3 the cubic case.That is where ε is a real small parameter, and In what follows we state our main results.
Theorem 1.For |ε| = 0 sufficiently small there exist discontinuous piecewise quadratic polynomial differential systems (4) with j = 2 having at least 10 limit cycles bifurcating from the periodic orbits of the uniform isochronous center of system (2), using the averaging theory of first order.
Theorem 2. For |ε| = 0 sufficiently small there exist discontinuous piecewise cubic polynomial differential systems (4) with j = 3 having at least 12 limit cycles bifurcating from the periodic orbits of the uniform isochronous center of system (3), using the averaging theory of first order.
We remark that the lower bounds for the number of limit cycles provided in Theorems 1 and 2 were obtained using the averaging method of first order.These results could possibly be improved using higher orders of the averaging theory for piecewise discontinuous differential systems, for further details see for instance [16].
The limit cycles that bifurcate from the origin of system (1), for some polynomials f (x, y), when it is perturbed inside some classes of continuous polynomial differential systems have been intensively investigated, see [6] and the references therein.Using results from Bautin [2] in [5] it is proved that at most 2 limit cycles bifurcate from the periodic orbits of the uniform isochronous center of the quadratic differential system (2).Applying the averaging theory in [4] it is showed that at least 2 limit cycles bifurcate from the periodic orbits of that center, when it is perturbed inside the class of all polynomial differential systems of degree 2. For the cubic polynomial differential systems (3) in [13] it is proved that the maximum number of limit cycles bifurcating either from the periodic orbits, or from the uniform isochronous center itself is 3 using the averaging theory of order 1 or 6, respectively.
A large number of phenomena can be modeled by discontinuous differential equations, for instance see [3] and the references therein.Hence the study of limit cycles of discontinuous piecewise differential systems has been significantly increasing in recent years.
In [18] it has been proved that at least 5 limit cycles bifurcate from the periodic orbits of the uniform isochronous center (2) when it is perturbed inside the class of all discontinuous quadratic differential systems with the straight line of discontinuity y = 0.In [13] it is showed that using the averaging theory of order 6, the maximum number of limit cycles that can appear in a Hopf bifurcation at uniform isochronous center of a cubic polynomial differential system (1) is 5, and this number can be reached.In the same work the authors proved that for system (3), using the averaging method of first order, the maximum number of limit cycles that can bifurcate from the periodic solutions surrounding the center is 7, and this number can be reached.In both cases studied in [13], the considered discontinuous systems were formed by two cubic polynomial differential systems separated by the straight line y = 0.
In other words in some sense we extend the works presented in the last two paragraphs for continuous and discontinuous quadratic and cubic polynomial differential systems perturbed inside of the class of the discontinuous piecewise differential systems with two pieces to discontinuous piecewise differential systems with four pieces.We recall that the largest number of limit cycles achieved in these previous works were 5 for quadratic differential systems and 7 for cubic differential systems, both in the case of discontinuous polynomial perturbations.Therefore our work provides results which double the number of limit cycles obtained in previous results for quadratic systems and increase in 5 the number of limit cycles achieved in prior results for cubic systems with a uniform isochronous center.
In short, the results on the number of limit cycles that can bifurcate from the periodic orbits of the uniform isochronous center of the quadratic differential system (2) and of the cubic differential system (3) when these systems are perturbed respectively inside the class of all continuous and discontinuous quadratic and cubic polynomial differential systems are summarized in Table 1.
This work is part of a general program for investigating the dynamics of polynomial piecewise discontinuous vector fields in which computer algebra is combined with singularity theory of mappings and numerical techniques, for further information see for instance [21].
We remark that there exist some works that address the problem of estimating the number of limit cycles from the periodic orbits of a uniform isochronous center of planar quartic polynomial differential systems, see for instance [15].

Preliminary results
In this section we present the main results that we shall use to investigate the discontinuous piecewise quadratic and cubic differential systems (4).The next result is well-known.Further details about it can be found in [14].Proposition 3. [14] Suppose that a differential polynomial system of degree n in R 2 has a center that can be placed without loss of generality at the origin of coordinates.Then this center is uniform isochronous if and only if by applying a rescaling of time and a linear change of variables this system can be written as where f (x, y) is a polynomial in x and y of degree n − 1, and f (0, 0) = 0.
We recall that from Proposition 3 and from the fact that it follows that the uniform isochronous center at the origin of coordinates is the only finite singular point of systems ( 2) and (3) (in fact this result is valid for any system (1)) and hence we only need to analyze the bifurcation of limit cycles from the periodic orbits of such center in those differential systems.
A quadratic polynomial differential system with a uniform isochronous center can always be written under the form (2), after a rescaling of time and a linear change of coordinates, see [20].
For the case of cubic differential systems with a uniform isochronous center, there is the following result due to Collins [9].

Theorem 4. [9]
A cubic polynomial differential system in R 2 with a uniform isochronous center that can be placed without loss of generality at the origin can be reduced to one of the two following expressions.
We remark that using these previous results we were able to study the bifurcation of limit cycles from the periodic orbits of any quadratic differential system with a uniform isochronous center, and of every cubic differential systems with a uniform isochronous center which can be reduced to the form (5) after a change of coordinates and a rescaling of time.
The following result is the first-order averaging theory for discontinuous piecewise differential systems developed in [17].
Let D ⊂ R d be an open subset and S 1 = R/T for a period T > 0. Furthermore let (S n ) be a finite sequence of open disjoint subsets of S 1 × D, with n = 1, . . ., M .We assume that the boundaries of each S n are piecewise C k −embedded hypersurfaces, for k ≥ 1.In addition we suppose that all S n together cover the set S 1 × D, and we denote by Σ the union of all boundaries of S n .Finally, consider A ⊂ S 1 × D and let χ A (t, x) be the characteristic function defined as Theorem 5. [17] Consider the following discontinuous piecewise differential system with where We define the averaging function Moreover, assume the following hypotheses.
(HC) There exists C ⊂ D an open bounded subset such that for each z ∈ C the curve {(t, z) : t ∈ S 1 } reaches transversely the set Σ and only at generic points of discontinuity.(Ha1) For j = 1, . . ., M the continuous functions F j 1 and R j are T −periodic with respect to t and locally Lipschitz with respect to x.In addition the boundaries of S j , for j = 1, . . ., M , are piecewise C k −embedded hypersurfaces, k ≥ 1. (Ha2) For a * ∈ C with f 1 (a * ) = 0, there exists a neighborhood U ⊂ C of a * such that f 1 (z) = 0 for all z ∈ Ū \ {a * } and the Brouwer degree of f Then for |ε| = 0 sufficiently small there exists a T −periodic solution x(t, ε) of system (7) such that x(0, ε) → a * as ε → 0.
We note that if the map f 1 of Theorem 5 is C l then it is sufficient to see that the Jacobian of f 1 at a * is not zero for concluding that d B (f 1 , U, 0) = 0, for more details see [19].
We also remark that in this paper the set Σ referred in Theorem 5 is given by the inverse image of zero by the function w(x, y) = xy, i.e., Σ = w −1 (0).
Consider a planar differential system ẋ = P (x, y), ẏ = Q(x, y), with P, Q : R 2 → R continuous functions and suppose that this system has a continuous family of period solutions , where H is a first integral of (9).Now suppose that we perturb (9) as follows ẋ = P (x, y) + εp(x, y), ẏ = Q(x, y) + εq(x, y), (10) with p, q : R 2 → R continuous functions.Then for |ε| = 0 sufficiently small, in order to study the bifurcation of limit cycles in (10) applying the averaging theory, it is necessary to write this system into the normal form (7). The next result provides a method to do that.
[4] Consider the unperturbed system (9) and its first integral H. Assume that for all (x, y) in the period annulus formed by the ovals {Γ h }, we have xQ(x, y)−yP (x, y) = 0.Moreover, for all R ∈ ( Then the differential equation which describes the dependence between the square root of the energy R = √ h and the angle θ for the perturbed system (10) is where x = ρ(R, θ) cos θ, y = ρ(R, θ) sin θ, and µ = µ(x, y) is the integrating factor corresponding to the first integral H of (9).
In order to determine the number of zeros of the averaging function ( 8) we shall apply the following result, for a proof of it see for instance Lemma 4.5 of [8].Proposition 7. Let I be an interval of R and let f 0 , . . ., f n : I → R be analytic functions linearly independent, that is, if k i=0 α i f i (s) = 0 then α 0 = . . .= α k = 0. Assume that one of the functions f i does not change sign in I. Then there exist s 1 , . . ., s n ∈ I and λ 0 , . . ., λ n ∈ R such that for every j ∈ {1, . . ., n} we have f (s j ) .= n i=0 λ i f i (s j ) = 0 and f (s j ) = 0.
In other words, there exist values of λ 0 , . . ., λ n ∈ R and s 1 , . . ., s n ∈ I such that f , which is a linear combination of n + 1 linearly independent functions, has n simple roots.

Proof of Theorem 1
We recall that the period annulus of a center q is formed by the largest set of periodic orbits surrounding q, and having q itself as its inner boundary.
Consider the first integral H = (x 2 + y 2 )/(1 + y 2 ) and its corresponding integrating factor µ = 1/(1 + y) 3 in the period annulus of the uniform isochronous center of the quadratic differential system (2).This system has the invariant straight line y = −1, and therefore the minimal distance between the outer boundary of the period annulus of the center and the center itself is 1.
Calculating these integrals we obtain with ), All the calculations were made using the software Mathematica.We identify the following relations among the coefficients of f .
Taking into account these relations, the function f can be written as. where Moreover applying the trigonometric identity for 0 < R < 1, we identify the following linear combinations The 11 functions G i : (0, 1) → R, i ∈ {1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13} given in ( 14) are linearly independent.Indeed we have the following Taylor expansions in the variable R, around R = 0, for these functions From the above Taylor expansions we construct the 11 × 11 matrix M , which is the coefficient matrix of the variables R i , i = −1, . . ., 10.After that, using the Mathematica command RowReduce we computed the reduced row echelon form of M resulting in the identity square matrix of order 11.Mathematica also provided the rank of M , which is 11 as expected.Since the 11 functions G i , for i ∈ {1, 2, 13} given in ( 14) are linearly independent, by Proposition 7 there exist a linear combination of these functions with at least 10 zeros.In addition, the coefficients of the functions G i are linearly independent, because their Jacobian matrix in the variables a i jk , b i jk , for 1 ≤ j + k ≤ 2 and i = 1, . . ., 4 has maximum rank, which is 11.Thus there exist R l ∈ (0, 1), l = 1, . . ., 10 and coefficients a i jk , b i jk , 1 ≤ j + k ≤ 2, i = 1, . . ., 4 such that f (R l ) = 0 for l = 1, . . ., 10.In summary there exist discontinuous quadratic polynomial differential systems (4) with j = 2 having at least 10 limit cycles which bifurcate from the periodic orbits of the uniform isochronous center of system (2), applying the averaging theory of first order for discontinuous piecewise differential systems.This completes the proof of Theorem 1.
From the Taylor expansions in the variable R, around R = 0, of the functions U i , i = 1, . . ., 13 given in (17) we construct the square coefficient matrix N of the variables R i , i = 1, . . ., 13.The expressions of the Taylor expansions of these functions are very long so we omit them.Since the reduced row echelon form of N is the identity square matrix of order 13 we conclude that the 13 functions U i are linearly independent.We also calculated the rank of N , obtaining 13 as expected.All the calculations were made using Mathematica.By Proposition 7 there exist a linear combination of the functions U i , i = 1, . . ., 13 with at least 12 zeros.Hence there exist R l ∈ (0, 1), l = 1, . . ., 12 and coefficients a i jk , b i jk , 1 ≤ j + k ≤ 3, i = 1, . . ., 4 such that f (R l ) = 0 for l = 1, . . ., 12.
In short, applying the averaging theory of first order for discontinuous piecewise differential systems, there exist discontinuous cubic polynomial differential systems (4) with j = 3 having at least 12 limit cycles which bifurcate from the periodic orbits of the uniform isochronous center of system (3).This completes the proof of Theorem 2.

Conclusion and future works
Applying the averaging theory of first order for discontinuous differential systems we improved previous results about the number of limit cycles that bifurcate from the periodic orbits of the uniform isochronous center of systems ( 2) and (3).
More precisely, we perturbed the differential systems (2) and ( 3) respectively inside all discontinuous quadratic and cubic differential systems with the straight lines of discontinuity x = 0 and y = 0. Comparing our results with previous results for discontinuous quadratic and cubic differential systems with one straight line of discontinuity we obtained in each case 5 more limit cycles surrounding the origin, and comparing with the continuous quadratic and cubic cases we obtained 8 and 9 more limit cycles respectively, see Table 1.
Due to the lack of a clear pattern in the functions obtained in the cases of one and two straight lines of discontinuity in the studied systems it is unclear that it is possible to obtain a general result relating the number of lines of discontinuity and the number of limit cycles for discontinuous polynomial differential systems in the plane, using the averaging theory.
In future works we intend to study discontinuous differential systems with more straight lines of discontinuity.

Table 1 .
Number of limit cycles for continuous and discontinuous quadratic and cubic differential systems