A Succinct Characterization of Period Annuli in Planar Piecewise Linear Differential Systems with a Straight Line of Nonsmoothness

We close the problem of the existence of crossing period annuli in planar piecewise linear differential systems with a straight line of nonsmoothness. In fact, a characterization for the existence of such objects is provided by means of a few basic operations on the parameters.


INTRODUCTION
Determining sufficient and necessary conditions for the existence of a period annulus in planar differential systems is a classical problem in qualitative theory of planar vector fields.For the particular case that the period annulus ends in a monodromic singularity, such a problem is known as Center Problem, which was exhaustively studied for polynomial vector fields (see, for instance, [18]).This problem has also been considered in the context of planar piecewise smooth differential systems (see, for instance, [7,8,12,16,15,17]).
However, due to the complexity imposed by the nonsmoothness, the center problem is not solved even for the simplest family of piecewise smooth differential systems, namely piecewise linear differential systems with two zones separated by the straight line Σ = (x, y) P R 2 Here, x = (x, y) P R 2 , 2 , and the dot denotes the derivative with respect to the independent variable t.The Filippov's convention [9] is assumed for trajectories of (1).
The main goal of this paper is to close the problem of the existence of crossing period annuli for system (1) by providing a characterization for the existence of such objects by means of a few basic operations on the parameters.
Since system (1) is piecewise linear, two obvious conditions implying the existence of period annuli are: (A) T L = 0, D L ą 0, and a L ă 0; or (B) T R = 0, D R ą 0, and a R ą 0, where T L , T R and D L , D R denote, respectively, the traces and determinants of the matrices A L and A R and (2) Indeed, condition (A) implies that system (1) has a linear center (and so a period annulus) in the half-plane t(x, y) P R : x ă 0u, and condition (B) implies that system (1) has a linear center (and so a period annulus) in the half-plane t(x, y) P R : x ą 0u.
Apart the trivial cases above, system (1) admits period annuli whose orbits cross the separation line Σ.Regarding those period annuli we may quote the following papers.In [11], Freire et al. provided sufficient conditions for piecewise linear systems of kind (1) formed by two foci and without sliding set to have a global center around the origin.In [1], Buzzi et al. classified the centers at infinity for piecewise linear perturbations of linear centers.In [14], Medrado & Torregrosa established sufficient conditions in order for a monodromic singularity at the separation line Σ to be a center.Finally, in [10], Freire et al. characterize when systems of kind (1), formed by two foci, have a center at infinity.
In this paper, we present a general and concise characterization of the existence of a crossing period annulus for system (1).This characterization will be given in terms of their parameters and, unlike the mentioned papers, regardless the local nature of each linear system.
Notice that the existence of a crossing periodic orbit for system (1) implies trivially the existence of the Poincaré half-maps associated with Σ.In turn, such maps exist if, and only if, the following set of conditions hold: 12 a R 12 ą 0; a L ď 0 and 4D L ´T2 L ą 0, or a L ą 0; a R ě 0 and 4D R ´T2 R ą 0, or a R ă 0. Indeed, taking into account the direction of the flow on the separation line x = 0, it is straightforward to see that the inequality a L 12 a R 12 ą 0 is necessary for the existence of crossing periodic solutions.The other two conditions will be discussed below (see Propositions 1 and 2).Now, we present the main result of this paper.
Theorem 1.Consider the planar piecewise linear differential system (1).Let T L , T R and D L , D R be, respectively, the traces and determinants of the matrices A L and A R and let a L and a R be the values given in expression (2).Denote Then, the differential system (1) has a crossing period annulus if, and only if, the condition (H) holds, sign(T R ) = ´sign(T L ), and ξ 0 = ξ 8 = β = 0.
At this point, we must clarify the dynamical meanings of the values ξ 0 , ξ 8 , and β and of the relationship sign(T R ) = ´sign(T L ).
First, under the hypothesis a L 12 a R 12 ą 0, system (1) has a sliding region contained in Σ and delimited by the points 0, ´bL 1 /a L 12 and 0, ´bR 1 /a R 12 provided that β does not vanish.Accordingly, the condition β = 0 indicates that system (1) does not have any sliding region.Second, when system (1) does not have a sliding region, as it follows from Proposition 14 of [5], the sign of the value ξ 0 (called by ξ in that work) provides the stability of the origin of system (1) when it is a monodromic singularity.Moreover, from Proposition 15 of [5], under the assumption sign(T R ) = ´sign(T L ) ‰ 0, the sign of the value c 8 = T L ξ 8 determines the stability of the infinity for system (1) when it is monodromic.Finally, since system (1) is linear on each side of the separation straight line Σ, the signs of the traces T L and T R determine the (area) contraction/expansion of the system on each side of Σ and so the condition sign(T R ) = ´sign(T L ) ensures a kind of balance between the contraction of the system in one zone and the expansion of the system in the other zone.
Theorem 1 is proven in Section 3. Its proof is based on a recent integral characterization for Poincaré half-maps for planar linear differential systems introduced in [2] by Carmona & Fernández-Sánchez, which has been successfully used to analyze periodic behavior of piecewise linear systems (see, for instance, [4,5,6]).This characterization as well as some useful properties of the Poincaré half-maps will be introduced in Section 2.

RESULTS
In this section, after introducing a canonical form for system (1) in Subsection 2.1, the definition of the Poincaré half-maps for planar linear differential systems will be presented in Subsection 2.2.Some useful properties of these maps, provided in [3], will be collected in Subsection 2.3.In Subsection 2.4, a displacement function will be given together with some of its main features.

Canonical form.
As it was said in Introduction, the existence of crossing periodic solutions of system (1) implies straightforwardly the first condition of Hypothesis (H), that is, a L 12 a R 12 ą 0.Moreover, under this condition, Freire et al. in [11] stated that the differential system (1) is reduced, by a homeomorphism preserving the separation line Σ = (x, y) P R 2 : x = 0 ( , into the following Liénard canonical form (4) being a L and a R the values given in expression (2), T L , T R and D L , D R , respectively, the traces and determinants of the matrices A L and A R , and b = β/a R 12 , where β is given in expression (3).

Integral characterization of Poincaré half-maps.
The periodic solutions of the piecewise linear differential system (4) are studied via two Poincaré Half-Maps defined on Σ: the Forward Poincaré Half-Map y L : , b].On the one hand, the forward Poincaré half-map takes a point (0, y 0 ), with y 0 ě 0, and maps it to a point (0, y L (y 0 )) by traveling through the flow of (4) in the positive time direction.Clearly, it is determined by the left linear differential system of ( 4) and its formal definition will be given in Proposition 1.
On the other hand, the backward Poincaré half-map takes a point (0, y 0 ), with y 0 ě b, and maps it to a point (0, y b R (y 0 )) by traveling through the flow of (4) in the negative time direction.Clearly, it is determined by the right linear differential system of (4).Notice that the simple translation y Þ Ñ y ´b applied to this right linear system allows us to write The formal definition of the map y R and its domain I R will be given in Proposition 2.
In Propositions 1 and 2, we will need the following concept of Cauchy Principal Value: for y 1 ă 0 ă y 0 and f continuous in [y 1 , y 0 ]zt0u (see, for instance, [13]).Note that if f is also continuous at 0, then the Cauchy principal value coincides with the definite integral.
The forward Poincaré half-map y L refers to the linear system (5) which corresponds with the left linear system of (4).Thus, its definition, its domain I L , and its analyticity are given by Theorem 19, Corollary 21, and Corollary 24 of [2].In the following proposition, we summarize the mentioned results (see [ where (e) The forward Poincaré half-map y L is analytic in Int(I L ).
On the other hand, the backward Poincaré half-map y R refers to the linear system which corresponds with the right linear system of ( 4) for b = 0. Thus, its definition, its domain I R , and its analyticity are obtained from Proposition 1 by means of the change of variables (t, x) Þ Ñ (´t, ´x) and taking (a (e) The backward Poincaré half-map y R is analytic in Int(I R ).
Remark 1.Notice that the integral that appears in (6) (resp.(8)) is divergent for a L = 0 (resp.a R = 0).Nevertheless, in this case, the Cauchy principal value provides In any other case, that is, a L ‰ 0 (resp.a R ‰ 0), the Cauchy principal value can be removed because the integral is a proper integral.

2.3.
Properties of Poincaré half-maps.Some useful properties of the Poincaré halfmaps y L and y R will be collected in the next results.The proofs of these properties for the map y L are given in [3] and they can be extended to y R by means of the change of variables (t, x) Þ Ñ (´t, ´x) and taking (a L , D L , T L ) = (´a R , D R , ´TR ) in system (5).The first one (Proposition 3) provides, as a direct consequence of expressions ( 6) and ( 8), the first derivative of the Poincaré half-maps.The second result (Proposition 4) establishes the relative position between the graph of the Poincaré half-maps and the bisector of the fourth quadrant.The third result (Proposition 5) gives the first coefficients of the Taylor expansions of the Poincaré half-map y R at the origin.The last result (Proposition 6) shows the first coefficient of the Newton-Puiseux series expansion of y L around a point p y 0 ą 0 such that y L (p y 0 ) = 0.
Proposition 3. The first derivatives of the Poincaré half-maps y L and y R are given by y 1 L (y 0 ) = y 0 W L (y L (y 0 )) y L (y 0 )W L (y 0 ) ă 0 for y 0 P int(I L ), where the polynomials W L and W R are given in Propositions 1 and 2, respectively.Proposition 4. The following statements hold.
(a) The forward Poincaré half-map y L satisfies the relationship sign (y 0 + y L (y 0 )) = ´sign(T L ) for y 0 P I L zt0u.
In addition, when 0 P I L and y L (0) ‰ 0 or when T L = 0, then the relationship above also holds for y 0 = 0. (b) The backward Poincaré half-map y R satisfies the relationship sign (y 0 + y R (y 0 )) = sign(T R ) for y 0 P I R zt0u.
In addition, when 0 P I R and y R (0) ‰ 0 or when T R = 0, then the relationship above also holds for y 0 = 0.
For the sake of simplicity, the next result is only given for the map y R , which will be used later on in the proof of Theorem 1.A version for the map y L can be stated in an analogous way.Proposition 5. Assume that 0 P I R and y R (0) = p y 1 ă 0, then the backward Poincaré half-map y R is a real analytic function in I R and its Taylor expansion around the origin writes as Again, for the sake of simplicity, the next result is only provided for the map y L .An analogous result for the map y R can be also stated.Proposition 6. Assume that there exists a value p y 0 ą 0 such that y L (p y 0 ) = 0.Then, a L ă 0, p y 0 = λ L , that is, p y 0 is the left endpoint of the definition interval I l of y L , and the Poincaré half-map y L admits the Newton-Puiseux series expansion around the point p y 0 given by  b (y 0 ) = 0 refers to the lateral derivative.Now, some of the properties of δ b (in particular, relevant expressions for the sign of the derivatives) will be stated in the next proposition.Its proof can be seen in [5].(11) for b = 0. Suppose that y 0 P int(I 0 ) satisfies δ 0 (y 0 ) = 0. Denote y 1 = y R (y 0 ) = y L (y 0 ) ă 0 and define
The set of polynomial functions tW L , W R u, with W L and W R defined in Propositions 1 and 2, is linearly dependent if, and only if, Moreover, the following equalities hold:

CHARACTERIZATION OF CROSSING PERIOD ANNULI
This section is dedicated to the proof of Theorem 1.It starts with a result on partial necessary conditions for the existence of a crossing period annulus.In particular, this result states that if system (1) has a crossing period annulus, then it cannot have a sliding region.This result has already been obtained in [10] by Freire et al. in the case that system (1) is formed by two foci.

Lemma 1.
If the piecewise linear differential system (1) has a crossing period annulus, then the condition (H) holds, the value β defined in (3) vanishes, and sign(T R ) = ´sign(T L ).
Proof.Notice that, if system (1) has a crossing period annulus, then, in particular, a L  12 a R 12 ą 0 and, therefore, system (1) can be transformed into system (4), which will also have a crossing period annulus.Hence, I b = [λ b , µ b ) ‰ H, the Poincaré halfmaps are well defined and so, from Propositions 1 and 2, we have that Hypothesis (H) holds.Now, we show that the existence of a crossing period annulus implies that b = 0 and, consequently, β = 0. Suppose, by reduction to absurdity, that system (4) has a crossing period annulus and b ‰ 0. Let us assume that b ą 0, otherwise, by applying the transformation (t, y) Þ Ñ (´t, ´y), we can change the sign of b.This transformation also changes the signs of T L and T R , but this will not be important in getting a contradiction.In the sequel, our reasoning distinguishes Taking into account that b R I b , the last inequality implies, in fact, that λ L ą b ą 0. Thus, by statement (b) of Proposition 1, we have y L (λ L ) = 0 and, then, by Proposition 6,(17) lim From Remark 2, y L (λ L ) = 0 implies that y R (λ L ´b) = ´b ă 0 which, in turns, from statement (b) of Proposition 2 and taking into account that λ L ą b, implies that λ L ´b P Int(I R ).Hence, Proposition 3 implies that (18) y 1 R (λ L ´b) ă 0. The relationships (17) and ( 18) contradicts the fact that δ 1 b (λ L ) = 0. Therefore, we have shown that the existence of a crossing period annulus implies that b = 0 and, consequently, β = 0.
Finally, b = 0 implies that y L (y 0 ) = y R (y 0 ) for every y 0 P I b and, from Proposition 4, it follows that sign(T R ) = ´sign(T L ) and the proof is finished.
3.1.Proof of Theorem 1.Let us start by assuming that the differential system (1) has a crossing period annulus.From Lemma 1, (H) holds, β = 0, and sign(T L ) = ´sign(T R ).In addition, since T L T R = 0 implies that T L = T R = 0 and, therefore, ξ 0 = ξ 8 = 0, then it only remains to show that ξ 0 = ξ 8 = 0 for the case T L T R ă 0.
Recall that, under the first condition of (H), that is, a L 12 a R 12 ą 0, system (1) can be transformed into system (4), with b = 0, which will also have a crossing period annulus.
From Remark 3, the relationship c 0 = c 1 = c 2 = 0 indicates that the polynomials W L (y) = D L y 2 ´aL T L y + a 2 L and W R (y) = D R y 2 ´aR T R y + a 2 R are linearly dependent.By one hand, if a L = 0, then a R = 0 and so ξ 0 = 0. Furthermore, in this case, from (10), one can see that the existence of a crossing period annulus, that is, the condition y L (y 0 ) = y R (y 0 ) for y 0 ě 0 leads, by a direct computation, to ξ 8 = 0. On the other hand, if a L ‰ 0, then a R ‰ 0. Thus, since c 0 = 0, from (15), one gets ξ 0 = 0 and so any of the relationships in (16) implies ξ 8 = 0, because c 1 = 0.
Note that if T L T R = 0, taking into account that sign(T L ) = ´sign(T R ), we have that T L = T R = 0. Thus, from ( 6) and ( 8), since the integrands are odd functions, it is trivial that y L (y 0 ) = y R (y 0 ) = ´y0 for every y 0 P I 0 .This implies the existence of a crossing period annulus.Thus, for the rest of the proof, we can assume T L T R ă 0.
Since c 0 = c 1 = c 2 = 0, from Remark 3, the polynomials W L (y) = D L y 2 áL T L y + a 2 L and W R (y) = D R y 2 ´aR T R y + a 2 R are linearly dependent, that is, W L = kW R .Moreover, k ą 0. Indeed, if a 2 L + a 2 R ‰ 0, then k ą 0 immediately, otherwise, if a L = a R = 0, from (H), we have D L , D R ą 0 and, again, k ą 0.
Hence, sign(a L ) = ´sign(a R ), because T L T R ă 0. In addition, ξ 8 = 0 implies that D L = (T L /T R ) 2 D R .Thus, k = (T L /T R ) 2 and D L = kD R , T L = ´?k T R , and a L = ´?k a R .y L (y 0 ) ´y W R (y) dy + and the functions q L and q R defined in expressions ( 7) and (9) satisfy q L (a L , T L , D L ) = q L ´?k a R , ´?k T R , kD R = 1 k q R (a R , T R , D R ).
Now, from Propositions 1 and 2, we see that y L and y R have the same integral characterization and, consequently, they coincide, that is, y L (y 0 ) = y R (y 0 ) for y 0 P I L = I R .This implies the existence of a crossing period annulus and the proof is finished.

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(a) The right endpoint µ L of the interval I L is the smallest strictly positive root of the polynomial W L (y) = D L y 2 ´aL T L y + a 2 L , if it exists.Otherwise, µ L = +8.(b) The left endpoint λ L of the interval I L is greater than or equal to zero.If λ L ą 0, then y L (λ L ) = 0, a L ă 0, 4D L ´T2 L ą 0, and T L ă 0.Moreover, if y L (λ L ) ă 0, then λ L = 0 and a L ă 0, 4D L ´T2 L ą 0, and T L ą 0. (c) The polynomial W L verifies W L (y) ą 0 for y P ch(I L Y y L (I L ))zt0u, where ch(¨) denotes the convex hull of a set.(d) The forward Poincaré half-map y L is the unique function y L :

(see [ 5 ,Proposition 2 .
Theorem 2]).The backward Poincaré half-map y R is well defined if, and only if, a R 0 and 4D R ´T2 R ą 0, or a R ă 0. In this case, I R := [λ R , µ R ) ‰ H and the following statements hold: (a) The right endpoint µ R of its definition interval I R is the smallest strictly positive root of the polynomial W R

2. 4 .
Displacement function.Once the Poincaré half-maps have been characterized, a displacement function can be defined for system(4).Suppose thatI b := I L X (I R + b) ‰ H.The displacement function δ b is, then, defined in I b as follows: (11) δ b : I b ÝÑ R y 0 Þ ÝÑ δ b (y 0 ) := y b R (y 0 ) ´yL (y 0 ) = y R (y 0 ´b) + b ´yL (y 0 ).From Propositions 1 and 2, one has I b = [λ b , µ b ), where λ b = maxtλ L , λ R + bu and µ b = mintµ L , µ R + bu.In addition, δ b is continuous on I b and analytic on Int(I b ).

Remark 2 .
Notice that, by the continuity of δ b on I b and the analyticity on Int(I b ), a crossing period annulus exists if, and only if, δ b (y 0 ) = 0 for every y 0 P I b .Obviously, in this case, the ith order derivative satisfies δ (i) b (y 0 ) = 0 for every y 0 P I b and i P N. Of course, when y 0 = λ b , δ (i)

0 y 1 L
whether or not b belongs to the interval I b .On the one hand, let us consider b P I b .Then, 0 P I R and b P I L .If y R (0) = 0, it follows that δ b (b) = y R (b ´b) ´yL (b) + b = ´yL (b) + b ą 0 and this contradicts the fact that δ b (y 0 ) = 0 for every y 0 P I b (see Remark 2).If y R (0) ă 0, then, from Proposition 5, one obtains y 1 R (0) = 0. Thus, if y L (b) ă 0, from Proposition 3, one get y 1 L (b) ă 0; if, on the other hand, y L (b) = 0, then, from Proposition 6 (by taking p y 0 = b), one gets that lim y OEb (y 0 ) = ´8.In both cases, δ 1 b (b) ‰ 0 which contradicts the fact that δ 1 b (y 0 ) = 0 for every y 0 P I b (see Remark 2).On the other hand, consider b R I b .We know that λ b = maxtλ L , λ R + bu.First, let us assume that λ b = λ R + b, which implies that λ L ď λ R + b.Taking into account that b R I b , we have that λ R ą 0. Thus, by statement (b) of Proposition 2, we have y