Integral characterization for Poincaré half-maps in planar linear systems
IIntegral characterization for Poincar´ehalf-maps in planar linear systems
Victoriano Carmona ∗ Fernando Fern´andez-S´anchez † Dpto. Matem´atica Aplicada II & IMUS.Universidad de Sevilla.
Abstract
The intrinsic nature of a problem usually suggests a first suitablemethod to deal with it. Unfortunately, the apparent ease of applica-tion of these initial approaches may make their possible flaws seemto be inherent to the problem and often no alternative ways to solveit are searched for. For instance, since linear systems of differentialequations are easy to integrate, Poincar´e half-maps for piecewise lin-ear systems are always studied by using the direct integration of thesystem in each zone of linearity. However, this approach is accom-panied by two important defects: due to the different spectra of theinvolved matrices, many cases and strategies must be considered and,since the flight time appears as a new variable, nonlinear complicatedequations arise.This manuscript is devoted to present a novel theory to charac-terize Poincar´e half-maps in planar linear systems that avoids thecomputation of their solutions and the problems it causes. This newperspective rests on the use of line integrals of a specific conservativevector field which is orthogonal to the flow of the linear system. Be-sides the obvious mathematical interest, this approach is attractivebecause it allows to simplify the study of piecewise-linear systems anddeal with open problems in this field.
Keywords:
Piecewise planar linear systems, Poincar´e half-maps, Inverseintegrating factors.
MSC2010 : 34A26, 34A36, 34C05. ∗ Escuela Polit´ecnica Superior. Calle Virgen de ´Africa, 7. 41011 Sevilla. Spain. Email:[email protected] † Escuela T´ecnica Superior de Ingenier´ıa. Camino de los Descubrimientos s/n. 41092Sevilla. Spain. Email: [email protected] a r X i v : . [ m a t h . C A ] O c t ontents Linearization, Lyapunov stability, normal forms and index theory are somefundamental tools for the analysis of dynamical systems. Another one, theconstruction of Poincar´e maps, is specially suitable for the study of existence,uniqueness and stability of periodic orbits, homoclinic connections and het-eroclinic cycles.In the case of piecewise systems, the analysis of orbits that cross theseparation manifolds between different regions leads naturally to the use ofPoincar´e maps that are therefore defined as composition of transition maps,sometimes called the Poincar´e half-maps, between the separation boundaries.Usually, the explicit calculation of these maps is a difficult task because itobviously depends on the equations of the involved systems. However, in thecase of piecewise linear systems, direct integration of the equations may beused in each region to obtain feasible expressions.Unfortunately, this advantage of linear systems, that is, the possibilityof performing direct integration of the equations, has also two importantweaknesses for the construction of Poincar´e half-maps. The first one is thatthe computation of the solutions of the linear systems, together with itssubsequent study, is strongly conditioned by the spectrum of the matrix ofthe system and the final expression of Poincar´e half-maps is written in termsof the eigenvalues. This fact forces the appearance of many different cases tostudy. The second weak point is the inevitable (non-linear) dependence ofthe Poincar´e half-maps on the flight time, namely, the time spent by the orbitbetween two consecutive intersection points with the separation manifolds.2hese two deficiencies are even further important if we take into accountthat almost every works about Poincar´e maps for piecewise linear systems usedirect integration of the systems, what is accompanied by large case-by-casestudies. The valuable works [1, 9, 21, 22, 20], sorted by year of publica-tion, are a few examples of this case-by-case studies from the early years tonowadays. Moreover, each one of these different cases requires individualtechniques. Obviously, this fact hinders and slows the research on dynamicbehavior of piecewise linear systems.The main motivation of this work is how to override the flaws in theanalysis of the transition maps due to performing the integration of planarlinear systems. The obvious procedure is to avoid the computation of theseintegrals. In order to do it, we develop a new technique to characterizePoincar´e half-maps of linear systems in a common, unique and more suitableexpression, without annoying exhaustive divisions into cases and without theunnecessary dependence on the flight time. This new approach is the maingoal of this manuscript.Beyond the importance on its own of the new characterization of Poincar´ehalf-maps, the relevance of this approach is also made evident by the sim-plification of the study of many important issues related to planar piecewiselinear systems; for instance, the analyticity of Poincar´e half-maps at tangencypoints (that is given in this work) or the open problem about providing op-timal upper bounds on the number of limit cycles (see [5, 6]).Loosely speaking, the characterization of the Poincar´e half-map relatedto the Poincar´e section Σ ≡ { x = 0 } for a generic planar linear system inLienard form (cid:26) ˙ x = T x − y, ˙ y = Dx − a where a , T , D are real numbers, says that the following assertions are equiv-alent:a) y is the image of y by means of a Poincar´e half-map,b) y , y ∈ R satisfy y y (cid:54) (cid:90) y y − yDy − aT y + a dy = cT, for three concrete values of constant c ∈ R . Here, PV stands for theCauchy Principal Value defined at (13).Notice that the Cauchy Principal Value is only necessary for the case a = 0,where the integral is improper and divergent due to a singularity at the origin.3or the sake of rigorousness, conditions for the existence of the Poincar´e half-map and the integral must be added to the domains of variables y , y andto the values of the parameters. Moreover, the three values of c dependon the parameters and the relative location of the Poincar´e section and theequilibrium of the system, if exists.The formal statement of the main result needs the definition of concepts,development of ideas and establishment of preliminary results that are se-quentially presented in this manuscript for the sake of better understanding.Its proof is a direct consequence of the reciprocal results Theorem 8 andTheorem 19. Furthermore, in this second theorem an integral expression forthe flight time is also given.Therefore, this work is divided into several parts which are devoted topresent the new concepts and ideas needed to prove the reciprocal resultsTheorem 8 (in Section 3) and Theorem 19 (in Section 5). The first logical stepfor our analysis is to give an accurate definition and a detailed descriptionof Poincar´e half-maps associated to a straight line for planar linear systems.The location of the equilibrium of the linear system, if exist, relative to thestraight line allows to classify all possible Poincar´e half-maps into just threescenarios. This is done in Section 2.Now, the target of Section 3 is the construction of an alternative way towrite the existing relationship between a point y and its image y by thePoincar´e half-map. More specifically, the beginning of this section is a briefsummary of definitions and results about inverse integrating factors, a basictool to obtain the suitable vector field that is integrated on appropriate closedcurves to reach the desired alternative expression. This integral expressionbrings together all possible geometric configurations into a unique commonfunction, in variables y and y , one of whose level curves is the graph ofthe Poincar´e half-map. Moreover, in this section, it is put into evidence anatural relationship between this common function and the index of a closedcurve, what suggests for this function the name of index-like function .Section 4 corresponds to the study of the main properties of the index-like function and its set of level curves. The primary results of this sectionconcern, on the one hand, the analyticity and bijectivity of the implicit func-tions defined by the level curves and, on the other hand, the existence of athird-order differential systems whose orbits are the graphs of these implicitfunctions. All the properties obtained in Section 4 are used in Section 5to prove Theorem 19 and thus to close the characterization of the Poincar´ehalf-maps (and to get the flight time) in terms of the index-like function.Finally, some conclusions and future works are given in Section 6.4 Poincare half-maps for planar linear sys-tems
Let us consider, for x = ( x , x ) T , the autonomous linear system˙ x = M x + b (1)where M = ( m ij ) i,j =1 , is a real matrix and b = ( b , b ) T ∈ R . Let us chosethe Poincar´e section Σ ≡ { x = 0 } .Although the further analysis can be performed directly to system (1),it is a good idea to reduce previously the number of parameters. Note thatif coefficient m vanishes, system (1) is uncoupled in such a way that aPoincar´e half-map on section Σ can not be defined (no return is possible).Therefore, from now on, let us assume that m (cid:54) = 0, what is usually calledthe observability condition [7]. Under this assumption, the linear change ofvariable x = x , y = m x − m x − b , allows to write system (1) into thegeneralized Lienard form, (cid:18) ˙ x ˙ y (cid:19) = (cid:18) T − D (cid:19) (cid:18) xy (cid:19) − (cid:18) a (cid:19) , (2)where a = m b − m b and T and D stand for the trace and the determinantof matrix M respectively. Let us call A the matrix of system (2) and L ( x, y ) = ( T x − y, Dx − a ) (3)the corresponding vector field. In the new coordinates, since x = x , Poincar´esection Σ remains the same.The first equation of system (2) evaluated on section Σ = { x = 0 } isreduced to ˙ x | Σ = − y . Therefore, the flow of the system crosses Σ from half-plane { x > } to { x < } when y >
0, from half-plane { x < } to { x > } when y < A , thereare only three different geometric types of Poincar´e half-maps.Without loss of generality, since system (2) is invariant under the change( x, y, a ) ←→ ( − x, − y, − a ), it is only necessary to define the left Poincar´ehalf-map. That is, let us consider (0 , y ) ∈ Σ with y (cid:62) t ; y ) = (Ψ ( t ; y ) , Ψ ( t ; y )) (4)the orbit of system (2) that satisfies Ψ(0; y ) = (0 , y ). If there exists avalue τ ( y ) > ( τ ( y ); y ) = 0 and Ψ ( t ; y ) < ∈ (0 , τ ( y )), we say that y = Ψ ( τ ( y ); y ) (cid:54) y by theleft Poincar´e half-map, denoted by y = P ( y ), and the value τ ( y ) is thecorresponding left flight time. See Fig. 1(a). Remark 1.
In the case that P (0) can not be defined in this previous waybut for every ε > there exist y ∈ (0 , ε ) and y ∈ ( − ε, such that P ( y ) = y , the left Poincar´e half-map can be extended with P (0) = 0 . This casecorresponds to an equilibrium at the origin or a scenario known as an invisibletangency [18] for half-plane { x < } . See Fig. 1(b).Notice that in this case the left flight time can be also extended to theorigin. For an invisible tangency, since τ ( y ) tends to as y tends to , the left flight time τ (0) should vanish. However, when the origin is anequilibrium, since the existence of the left Poincar´e half-map implies that D − T > (i.e., the equilibrium point is a center or a focus) and it isknown that τ ( y ) = π √ D − T for any y > (see [9]) then the natural choiceis τ (0) = π √ D − T . (a) (b) Figure 1: (a) Schematic drawing of the construction of the left Poincar´e half-map. (b) Left Poincar´e half-map for an invisible tangency or an equilibriumpoint at the origin.From now until the end of this section, we assume that the followinghypothesis holds:(H)
There exist y (cid:62) and τ ( y ) > such that Ψ ( τ ( y ); y ) = 0 and Ψ ( t ; y ) < for every t ∈ (0 , τ ( y )) . y = P ( y ), the segment Γ = { (0 , y ) ∈ R : y ∈ ( y , y ) } , the piece of orbit Γ = { Ψ( t ; y ) : t ∈ [0 , τ ( y )] } and the Jordancurve Γ = Γ ∪ Γ . (5)Note that the set Γ may be the empty set (when y = y = 0). Underassumption (H), linear system (2) has, at most, one equilibrium point. Notethat if the system had infinitely many equilibria (that is, the equality D + a = 0 holds) the straight line y = T x would be foliated by these equilibriumpoints. This fact contradicts the theorem of existence and uniqueness ofsolutions, since the straight line y = T x would intersect the piece of orbit Γ .Depending on the relative position of Γ and the equilibrium point ofsystem (2), if any, the following three mutually exclusive scenarios appear. Definition 2.
Let us name the different scenarios as:(S ) The equilibrium point exists and it belongs to the interior of Jordancurve Γ , Int(Γ) .(S ) The equilibrium point is the origin or, equivalently, the equilibrium ex-ists and it belongs to segment Γ .(S ) All other cases, i.e., the set Γ ∪ Int(Γ) contains no equilibrium pointsof system (2) . The reason of the numbering of the different scenarios comes from thevalues k that will be given in Theorem 8.Let us briefly describe the three scenarios. In scenario (S ), the equilib-rium of system (2) is located at the half-plane { x < } and the only possibleconfigurations for the phase portrait are a center, a stable focus and an un-stable focus (that is, 4 D − T > a < D of the left Poincar´e half-map is the interval [0 , + ∞ ) and itsrange R is the interval ( −∞ , y > P (ˆ y ) = 0, the domain is D = [ˆ y , + ∞ ) and the range is R = ( −∞ , y < P (0) = ˆ y , the domain is D = [0 , + ∞ ) and the range is R = ( −∞ , ˆ y ]. SeeFig. 2.For scenario (S ), the equilibrium of system (2) is located at the originand the only possible configurations for the phase portrait are a center, astable focus and an unstable focus (that is, 4 D − T > a = 0). In allthis cases, the left Poincar´e half-map can be extended to the origin by meansof the definition P (0) = 0 (see Remark 1). Therefore the domain is [0 , + ∞ )and the range is ( −∞ , a) (b) (c) Figure 2: Scenario (S ): (a) center, (b) stable focus, (c) unstable focus. (a) (b) (c) Figure 3: Scenario (S ): (a) center, (b) stable focus, (c) unstable focus.Finally, the scenario (S ) includes many different cases (saddles, nodes,degenerate nodes, foci, centers and degenerated situations without equilib-ria). On the one hand, for all of them there exists an invisible tangency at theorigin and, thus, the left Poincar´e half-map can be extended to P (0) = 0 (seeRemark 1). On the other hand, the domain and range of the left Poincar´ehalf-map for foci and centers are respectively D = [0 , + ∞ ) and R = ( −∞ , ), the existence of invariant straightmanifolds (at most two) restricts these sets. In fact, it is direct to see thatinvariant straight manifolds of system (2) cannot be parallel to section Σand cannot contain the origin. Therefore, the intersections between all theseinvariant manifolds with section Σ divide Σ in at most three open intervalsand one of them contains the origin. Thus, denoting by J this interval,the domain and range of the left Poincar´e half-map for cases with invariant8traight manifolds are respectively D = [0 , + ∞ ) ∩ J and R = ( −∞ , ∩ J .See Fig. 4. (a) (b) (c)(d) (e) (f) Figure 4: Scenario (S ): (a) center, (b) focus, (c) degenerate node, (d) node,(e) saddle, (f) degenerated case without equilibria. The goal of this section is to give an integral expression for the left Poincar´ehalf-map of system (2) corresponding to section Σ = { x = 0 } . In order toachieve this aim, we use line integrals of an specific conservative vector fieldwhich is orthogonal to the flow of the system. The vector field is obtainedin a convenient manner by means of a suitable inverse integrating factor.Since inverse integrating factors are a key tool for the study of classicproblems of planar smooth systems and, as far as we know, they have not9een often used for piecewise systems, it is appropriate to devote a few para-graphs to present, without going into much details, some of the principalfeatures of inverse integrating factors, particularly the basic properties andthose ideas that are going to be applied to transition maps of planar linearsystems. More and deeper information can be found in [12].Let us consider the vector field F ( x, y ) = ( f ( x, y ) , g ( x, y )) and the planarautonomous differential system dxdt = f ( x, y ) ,dydt = g ( x, y ) , (6)where f, g : U −→ R are smooth functions and U is a neighboorhood in R .A smooth function V : U −→ R is an inverse integrating factor of system(6) if its zero set V − ( { } ) = { ( x, y ) ∈ U : V ( x, y ) = 0 } does not contain anynon-empty open set and it satisfies the condition ∇ V ( x, y ) · F ( x, y ) = V ( x, y ) div F ( x, y ) , (7)where ∇ V ( x, y ) = (cid:16) ∂V∂x ( x, y ) , ∂V∂y ( x, y ) (cid:17) is the gradient of V , div F ( x, y ) = ∂f∂x ( x, y ) + ∂g∂y ( x, y ) is the divergence of the vector field F and the dot ( · )stands for the inner product.Note that the reason why a function V that satisfies the condition (7) iscalled an inverse integrating factor of system (6) is that for every ( x, y ) ∈U \ V − ( { } ) the function 1 /V is an integrating factor for the equation ofthe orbits g ( x, y ) dx − f ( x, y ) dy = 0. Equivalently, the system dxds = f ( x, y ) V ( x, y ) ,dyds = g ( x, y ) V ( x, y ) , obtained from system (6) by performing a change of the temporal variablethat satisfies ds = V ( x, y ) dt , is Hamiltonian in every simply connected com-ponent of U \ V − ( { } ).Other important results about inverse integrating factors are related totheir zero sets. Let us denote by Φ( t ; p ), the orbit of system (6) that satisfiesΦ(0; p ) = p . If V is an inverse integrating factor of (6) then it is easy to seethat the relationship V (Φ( t ; p )) = V ( p ) exp (cid:18)(cid:90) t div F (Φ( s ; p )) ds (cid:19) (8)10olds. Thus, if V ( p ) = 0, then V vanishes at the complete orbit and so thezero set of V is composed of trajectories of system (6).In [15], it is proved that all the limit cycles of system (6) included in thedomain of definition of V are contained in the zero set of V . Moreover, undermild conditions, the separatrices of hyperbolic saddle points are also includedin this set [2]. Once more, let us recommend the reading of the survey [12]to deepen in the knowledge of inverse integrating factors.For the analysis in this work, a suitable inverse integrating factor mustbe chosen. In this case, where L is the vector field given by system (3),condition (7) is written as ∇ V · L = T V, (9)since div L = T .It is well-known that homogeneous linear systems have quadratic inverseintegrating factors (see for instance [8]). If the linear system is not homo-geneous but it has an equilibrium, a simple translation converts it into anhomogeneous system so it also has a quadratic inverse integrating factor. Allthe quadratic inverse integrating factors of system (2) are collected in thefollowing proposition. Proposition 3.
The set V of polynomial inverse integrating factors V ( x, y ) of degree less or equal than two for system (2) is a finite dimensional vectorspace whose dimension depends on the parameters a , T and D . Concretely,the following bases B i may be selected: • If a + D (cid:54) = 0 and ◦ T (cid:54) = 0 , then B = { D x − DT xy + Dy + a ( T − D ) x − aT y + a } . ◦ T = 0 , then B = { , Dx + y − ax } . • If a + D = 0 and ◦ T (cid:54) = 0 , then B = { y − T xy, y − T x } . ◦ T = 0 , then B = { , y, y } .Proof. The proof is straightforward by imposing that the generic real quadraticpolynomial in two variables V ( x, y ) = (cid:88) ≤ i + j ≤ α ij x i y j satisfies condition (9). 11xcept in the case a + D (cid:54) = 0, T (cid:54) = 0, the dimension of vector space V is greater than one, so there exist linearly independent inverse integratingfactors. Since the division of two integrating factors is constant on orbits,the division of two linearly independent inverse integrating factors is a firstintegral of the system. In that case, all the orbits of the system can beobtained as the level curves of these quotients.For T = 0, system (2) is reversible (invariant under the change y ↔ − y , t ↔ − t ). It is also hamiltonian. In fact, constant polynomials are integratingfactors (see the constant polynomial 1 in the bases B and B ). From thisand the previous paragraph, any inverse integrating factor is constant alongthe orbits of system (2).In addition to these two comments, let us mention that the linear combi-nation a · D · ( Dx + y − ax ) of elements of the basis B could also beobtained from the unique element of the basis B if T were allowed to vanish.As a conclusion, we choose V ( x, y ) = D x − DT xy + Dy + a ( T − D ) x − aT y + a (10)as the expression of the inverse integrating factor for system (2) under con-dition a + D (cid:54) = 0 . (11) Remark 4.
Trivially, the level curves of the inverse integrating factor V areconics. In particular, when D − T > , they are ellipses whose center isthe equilibrium point of system (2) and the change of variables x = X + aD ,y = αX + βY + aTD (12) for α = T / , β = √ D − T / transforms the inverse integrating factor into (cid:101) V ( X, Y ) = β ( α + β ) ( X + Y ) = D − T D ( X + Y ) . Moreover, in thiscase, when T < the inverse integrating factor V is a Lyapunov function forsystem (2) , see equation (9) . Note that, from now on, the study will be restricted to case (11) since for a + D = 0 Poincar´e half-maps to section Σ = { x = 0 } of system (2) cannotexist. This is an immediate conclusion from the fact that the component y of every solution of system (2) is constant and, therefore, reinjection into Σis not possible. Moreover, as it was said in previous section, for a + D = 0the straight line y = T x is foliated by equilibrium points.12ow that a suitable inverse integrating factor has been chosen, as ithas been said in the introduction, it is important to determine the zero set V − ( { } ). The following proposition describes it. Proposition 5.
Depending on the parameters of system (2) and under con-dition (11) , the zero set V − ( { } ) of function V given in (10) is the emptyset, a single point (the equilibrium point of system (2) ), a single straight line(invariant for system (2) ) or a pair of crossing straight lines (the invariantmanifolds of the equilibrium point of system (2) ). Concretely: • For D = 0 (no equilibrium case) and ◦ T = 0 , then V − ( { } ) = ∅ . ◦ T (cid:54) = 0 , then V − ( { } ) = { ( x, y ) ∈ R : T x − T y + a = 0 } . • For D (cid:54) = 0 (equilibrium at ( x, y ) = ( a/D, aT /D ) ) and ◦ T − D > , then V − ( { } ) = { ( x, y ) ∈ R : 2 D (cid:0) x − aD (cid:1) = (cid:0) T ± √ T − D (cid:1) (cid:0) y − aTD (cid:1) } . ◦ T − D = 0 , then V − ( { } ) = { ( x, y ) ∈ R : 2 D (cid:0) x − aD (cid:1) = T (cid:0) y − aTD (cid:1) } . ◦ T − D < , then V − ( { } ) = { ( a/D, aT /D ) } .Proof. For D = 0, condition (11) implies that a (cid:54) = 0 and the inverse integrat-ing factor is V ( x, y ) = a ( T x − T y + a ). Thus the conclusion is obvious.For D (cid:54) = 0, the inverse integrating factor V ( x, y ) can be written as V ( x, y ) = − D det (cid:32) A (cid:32) x − aD y − aTD (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) x − aD y − aTD (cid:33)(cid:33) . Therefore, V vanishes if, and only if, the vector ( x − aD , y − aTD ) T belongs toa real eigenspace of matrix A . From this, the proof is direct.Since the expression given in this work for the left Poincar´e half-map interms of the inverse integrating factor involves divergent integrals at zero(concretely, when the equilibrium of system (2) is located at the origin), it isnecessary to use the concept of Cauchy principal value that, particularizedto divergences at zero, is defined as follows. Let be h a continuous functionin [ ξ , ξ ] \ { } , where ξ < < ξ . The Cauchy Principal Value (PV) ofintegral (cid:82) ξ ξ h ( ξ ) dξ is the following limit (if it exists):PV (cid:90) ξ ξ h ( ξ ) dξ := lim ε (cid:38) (cid:18)(cid:90) − εξ h ( ξ ) dξ + (cid:90) ξ ε h ( ξ ) dξ (cid:19) . (13)13bviously, if h is also continuous at ξ = 0 then the Cauchy principal valuecoincides with the value of the integral. By convention, it is said thatPV (cid:82) ξ ξ h ( ξ ) dξ = − PV (cid:82) ξ ξ h ( ξ ) dξ . In [13, 14], the Cauchy principal valueis applied to the analysis of monodromy and the study of the centre problemfor some types of planar systems.The first main theorem of the manuscript is based on the integration ofvector field G ( x, y ) = (cid:18) − Dx − aV ( x, y ) , T x − yV ( x, y ) (cid:19) (14)along a suitable Jordan curve for the different scenarios given in Definition2. Notice that vector field G is orthogonal to the flow of system (2) andconservative in every simply connected component of R \ V − ( { } ). Remark 6.
In order to choose an orientation, from now on we denote ( x, y ) ⊥ = ( − y, x ) . Thus, when V does not vanish, it holds that G = L ⊥ V where L is the vector field defined in (3) . Since condition (9) can be equiva-lently written as L ⊥ · ∇ V ⊥ = T V , the equality G · ∇ V ⊥ = T (15) is satisfied. Remark 7.
Let be ∆ any piecewise-smooth planar curve that does not inter-sect the zero set V − ( { } ) (i.e., it does not intersect the invariant manifoldsor the equilibrium of system (2) , if exist).Let us consider the case D − T > and let be (cid:101) ∆ , in coordinates ( X, Y ) ,the image of curve ∆ by the change of variables given in (12) . Let be ( X a , Y a ) and ( X b , Y b ) the first and last points of curve (cid:101) ∆ . Then, it is trivial that (cid:82) ∆ G · d r = − D (cid:90) (cid:101) ∆ X dX + Y dYX + Y + TD √ D − T (cid:18)(cid:90) (cid:101) ∆ X dY − Y dXX + Y (cid:19) = − D log (cid:0) X + Y (cid:1)(cid:12)(cid:12) ( X b ,Y b )( X a ,Y a ) + TD √ D − T (cid:18)(cid:90) (cid:101) ∆ X dY − Y dXX + Y (cid:19) . Notice that when V | ∆ is constant or ∆ is a closed curve then the firstsummand vanishes and (cid:90) ∆ G · d r = TD √ D − T (cid:18)(cid:90) (cid:101) ∆ X dY − Y dXX + Y (cid:19) . Therefore, when ∆ is a closed curve then (cid:73) ∆ G · d r = 2 πTD √ D − T (cid:18) π (cid:73) (cid:101) ∆ X dY − Y dXX + Y (cid:19) , hat is, πT (cid:0) D √ D − T (cid:1) − times the index (or winding number) of curve (cid:101) ∆ around ( X, Y ) = (0 , or, equivalently, times the index of curve ∆ aroundthe equilibrium point.For the case D − T (cid:54) , when ∆ is a closed curve, due to the conser-vativeness of vector field G , it is clear that (cid:73) ∆ G · d r = 0 . This can also be understood as the index of curve ∆ around the equilibrium,if exist, or any other point not surrounded by ∆ . Now, we are in a position to present and prove a theorem that states anew way to write, in terms of an integral expression, the existing relationshipbetween a value and its image by means of the left Poincar´e half-map.
Theorem 8.
Let assume that condition (11) and hypothesis (H) hold. Letbe y = P ( y ) the image of y by the left Poincar´e half-map, V the inverseintegrating factor given in expression (10) , Γ the jordan curve given in Eq. (5) and (S k ), k ∈ { , , } , the corresponding scenario given in Definition 2.Then PV (cid:90) y y − yV (0 , y ) dy = d k , (16) where d k = if k = 0 ,kπTD √ D − T if k = 1 , . Proof.
The proof of this theorem is direct from the computation of the lineintegral of vector field (14) along the jordan curve Γ given in Eq. (5) andpositively oriented. This computation depends on the relative position of Γand the equilibrium of system (2), if any. Therefore, the proof is divided intothree parts.Before that, since vector field G is orthogonal to the flow of system (2)at R \ V − ( { } ) then (cid:90) Γ G · d r = 0 (17)due to Γ is a piece of orbit of the system.Let us begin with the easiest case, that is, scenario (S ) (see Fig. 5(a)).From Proposition 5, it is obvious that V does not vanish in Int(Γ) ∩ Γ. Thus,since vector field G is conservative, the integral (cid:73) Γ G · d r = (cid:90) Γ G · d r + (cid:90) Γ G · d r (cid:73) Γ G · d r = (cid:90) Γ G · d r = (cid:90) y y − yV (0 , y ) dy. The proof for scenario (S ) is finished.Regarding scenario (S ) (see Fig. 5(b)), the unique equilibrium point be-longs to the interior of Jordan curve Γ and it is the only point at which theinverse integrating factor V vanishes. Therefore, by using identity (17) andRemark 7 it is trivial to see that (cid:90) y y − yV (0 , y ) dy = (cid:90) Γ G · d r = (cid:73) Γ G · d r = 2 πTD √ D − T . The last part of the proof, corresponding to scenario (S ), where a = 0and D (cid:54) = 0, is a little bit more complicated. In fact, it is the reason thatmotivates us to use the Cauchy principal value as defined in Eq. (13) becausethe improper integral (cid:90) y y − yV (0 , y ) dy = (cid:90) y y − Dy dy is divergent.Since the unique equilibrium of the system (the origin) is located at thejordan curve Γ, it is not possible to proceed as in previous scenarios. Underthese circumstances, it is usual to choose a Jordan curve (cid:101)
Γ = Γ ∪ Γ ∪ Γ ∪ Γ as shown in Fig. 5(c). Note that Γ intersects the Poincar´e section { x = 0 } atthe symmetric points (0 , − ε ) and (0 , ε ), with ε >
0. Once more, by identity(17) and Remark 7, it follows that (cid:90) − εy − Dy dy + (cid:90) y ε − Dy dy = − (cid:90) Γ G · d r = πTD √ D − T . Now, to finish the proof, it is enough to take limits as ε (cid:38) Remark 9.
Analogously, if y is the image of y by the right Poincar´e half-map, the same expression given in equation (16) is obtained but now d k mustbe changed by − d k . Note that, at this point, the flight time has been removed from the ex-pression for the Poincar´e half-maps and the analysis of the many cases thatappear due to the configuration of the spectra of matrix A have been reducedto the study of the level curves of the single function, F ( y , y ) = PV (cid:90) y y − yV (0 , y ) dy, (18)16 a) (b) (c) Figure 5: Curves used in the proof of Theorem 8. Labels (a), (b), (c) standrespectively for scenarios (S ), (S ) and (S ).where V is the inverse integrating factor given in expression (10).Let be ∆ a piece of an orbit of system (2) that starts at (0 , y ) and endsat (0 , y ), ∆ the segment bounded by y and y in section Σ and ∆ = ∆ ∪ ∆ the corresponding positively oriented closed curve. Reasoning as in the proofof Theorem (8), it is trivial that F ( y , y ) = (cid:73) ∆ G · d r holds. A suitable name for function F comes from Remark 7. Definition 10.
Function F given in equation (18) is called the index-likefunction. Remark 11.
A logical consequence of the direct procedure (that is, not de-pending on the spectra of matrix A ) of construction of an alternative way ofwriting the left Poincar´e half-map is that the index-like function is commonfor all cases and it only depends (through the expression of V (0 , y ) ) on theparameters of the linear system (2) and not on the eigenvalues of matrix A . Since the qualitative information of such a linear system is given by itscharacteristic polynomial P A and the location of the equilibrium, if it exist,it seems to be clear that there must exist a relationship between V (0 , y ) andthe characteristic polynomial. In fact, it is trivial to prove the equality y P A (cid:18) ay (cid:19) = V (0 , y ) for y (cid:54) = 0 . Thus, the index-like function given in Definition 10 (and, conse-quently, the Poincar´e half-maps) can be written in terms of the characteristicpolynomial P A . F , with a focus on the properties than can be extrapolated to Poincar´ehalf-maps. The first important task in the analysis of the index-like function F givenin (18) is to delimit its domain of definition for the purpose of avoidingnontrivial zeros of function V (0 , y ). Although the integral in the definitionof function F could have been extended, via the Cauchy principal value, tobe valid for intervals containing those zeros, in the context of this work isnot necessary to do it because the Poincar´e half-maps of system (2) cannotbe defined in them.In order to delimit the domain of function F , let us consider the openinterval I = (cid:110) y ∈ R : (0 , y ) ∈ int ( C ∪ { (0 , } ) (cid:111) , (19)where int( · ) stands for the topological concept of the interior of a set andthe set C is the unique maximal connected component of R \ V − ( { } )that contains two points (0 , ξ ) and (0 , ξ ) with ξ · ξ <
0. On the onehand, note that for a (cid:54) = 0 interval I could have been equivalently definedas I = { y ∈ R : (0 , y ) ∈ C} because (0 , / ∈ V − ( { } ). On the otherhand, when a = 0, that is, when the origin (0 , ∈ V − ( { } ) or equivalentlythe only equilibrium point is the origin (remember that, from (11), it is a + D (cid:54) = 0), set C and so interval I are C = R \ { (0 , } and I = R whenthe equilibrium is a focus or a center but they are empty when the equilibriumit is a saddle or a node. At this moment, it should have to be obvious thatthe intricate definition of interval I is suitable with the intention of removingthe cases where the existence of a Poincar´e half-map is not allowed.Moreover, it is clear that the inverse integrating factor V is strictly posi-tive on the set C when C (cid:54) = ∅ . Therefore, the inequality V (0 , y ) > y ∈ I when a (cid:54) = 0 and for all y ∈ I \ { } when a = 0.Finally, when I is not empty, it can be written as I = ( µ , µ ) where −∞ (cid:54) µ < < µ (cid:54) + ∞ . Besides that, if −∞ < µ (resp. µ < + ∞ ) then V (0 , µ ) = 0 (resp. V (0 , µ ) = 0).Let us consider now the integrating function from the definition of func-tion F given in (18), h ( y ) = − yV (0 , y ) = − yDy − aT y + a (20)18efined for every y ∈ I when a (cid:54) = 0 and for every y ∈ I \ { } when a = 0.This function h is strictly positive for y < y > i = 1 , µ i of interval I are infinity or satisfy V (0 , µ i ) = 0, any integral of h involving any of these points is an improperintegral and it is simple to see that it is, moreover, divergent. In fact, forevery a (cid:54) = 0 and z ∈ I , the following equalities hold (cid:90) µ z h ( y ) dy = −∞ and (cid:90) zµ h ( y ) dy = + ∞ . (21)Also, for a = 0, it is h ( y ) = − Dy and since D cannot vanish any integralinvolving y = 0 is also a divergent improper integral.In this first step the sign of V has allowed to obtain an open interval I outside of which a Poincar´e half-map could not be defined. Therefore,from now on, the study of function F will be restricted to the open square( y , y ) ∈ I = I × I . The next result describes the region of analiticity offunction F in this square. Lemma 12.
Let us assume that interval I given in (19) is not empty. Func-tion F given in (18) is analytic in:1. the open square I for a (cid:54) = 0 ;2. the open set I \ { ( y , y ) ∈ R : y · y = 0 } for a = 0 .Proof. For a (cid:54) = 0, function h ( y ) given in (20) is analytic in I . Therefore,statement 1 is direct.Trivially, for a = 0, F ( y , y ) = PV (cid:90) y y − Dy dy = 1 D log (cid:12)(cid:12)(cid:12)(cid:12) y y (cid:12)(cid:12)(cid:12)(cid:12) (22)and the proof is finished. Remark 13.
It is obvious that for a = 0 , function F is not defined at set { ( y , y ) ∈ R : y · y = 0 } . Nevertheless, although lim y → y (cid:54) =0 F ( y , y ) = − sign( D ) · ∞ and lim y → y (cid:54) =0 F ( y , y ) = + sign( D ) · ∞ , the directional limit along the straight line y = cy , for c (cid:54) = 0 , is lim y → F ( cy , y ) = log | c | D so, roughly speaking, we could say that the expression F (0 , takes any valueat this point. F . Hence, the next result is devoted to describe the whole setof level curves F ( y , y ) = q ∈ R restricted to I . In fact, we will see thatthese curves can be seen as graphs of real analytic functions defined in I that, moreover, are solutions of the same differential equation. Theorem 14.
Let us consider the index-like function F given in equa-tion (18) and the inverse integrating factor V given in (10) . Let us as-sume that condition (11) holds and that the open interval I = ( µ , µ ) defined in (19) is not empty. For every value q ∈ R there exist two dif-ferent real analytic functions φ q , ϕ q : I −→ I such that the level curve C q = { ( y , y ) ∈ I : F ( y , y ) = q } is the union of the graphs of functions φ q and ϕ q . Moreover, both functions are solutions of the differential equation y V (0 , y ) dy − y V (0 , y ) dy = 0 (23) in I . To be more precise:1. For a = 0 then C q = { ( y , y ) ∈ I : ( y − φ q ( y ))( y − ϕ q ( y )) = 0 } ,where φ q ( z ) = e Dq z and ϕ q ( z ) = − e Dq z for every z ∈ I .2. For a (cid:54) = 0 and q = 0 then C = (cid:8) ( y , y ) ∈ I : ( y − φ ( y ))( y − ϕ ( y )) = 0 (cid:9) where φ = id , ϕ is an involution in I and ϕ (0) = 0 .3. For a (cid:54) = 0 and q (cid:54) = 0 then C q = (cid:8) ( y , y ) ∈ I : ( y − φ q ( y ))( y − ϕ q ( y )) = 0 (cid:9) for q > and C q = (cid:8) ( y , y ) ∈ I : ( y − φ q ( y ))( y − ϕ q ( y )) = 0 (cid:9) for q < , where every function ϕ q is unimodal in I , function sgn( q ) ϕ q has astrictly negative maximum at the origin and the equivalence φ q ≡ ϕ − q holds. Furthermore, the restricted functions ϕ q : [0 , µ ) −→ ( µ , ϕ q (0)] , for q > ,ϕ q : ( µ , −→ [ ϕ q (0) , µ ) , for q < , (24) are bijective. roof. For a = 0 the proof of statement 1 is direct from expression (22)and Remark 13. Moreover, since D (cid:54) = 0, differential equation (23) is just y y ( y dy − y dy ) = 0, which is trivially satisfied by straight lines thatpass through the origin.In the rest of the proof, we assume a (cid:54) = 0 and, as it has been said be-fore, the principal value can be removed from the definition of function F .Therefore, the partial derivatives of function F are ∂ F ∂y ( y , y ) = y V (0 , y ) and ∂ F ∂y ( y , y ) = − y V (0 , y ) (25)and so, for every q ∈ R , any function implicitly defined in I by the equation F ( y , y ) = q , if it exists, satisfies0 = d F = y V (0 , y ) dy − y V (0 , y ) dy , what is equivalent to differential equation (23) since V (0 , z ) > z ∈ I = ( µ , µ ).Let us prove now the existence of functions φ q and ϕ q mentioned in thestatement of the theorem. From the values of the partial derivatives of func-tion F it is clear that for every y ∈ I function F ( · , y ) is strictly decreasingat interval ( µ ,
0) and strictly increasing at interval (0 , µ ). From the equal-ities given in (21) it is clear that for every y ∈ I ,lim y (cid:38) µ F ( y , y ) = lim y (cid:37) µ F ( y , y ) = + ∞ . Moreover, the inequality F (0 , y ) < y (cid:54) = 0 and F (0 ,
0) = 0.Let us consider now q = 0. For every y ∈ I \ { } there exist twounique and different values φ ( y ) , ϕ ( y ) ∈ I such that F ( φ ( y ) , y ) = F ( ϕ ( y ) , y ) = 0. Moreover, the inequality φ ( y ) · ϕ ( y ) < y · φ ( y ) > φ and ϕ could be extended to y = 0, both functions should satisfy φ (0) = ϕ (0) = 0because the only solution to F ( y ,
0) = 0 is y = 0.On the one hand, in view of the trivial equality F ( y , y ) = 0 for every y ∈ I , function φ : y ∈ I −→ φ ( y ) ∈ I must be the identity function and there exist a function F ∗ defined in I such that F ( y , y ) = ( y − y ) F ∗ ( y , y ) and F ∗ ( ϕ ( y ) , y ) = 0 for every y ∈ I \ { } .Function φ is clearly analytic because it is the identity function. From(25), it is immediate that ∂ F /∂y ( ϕ ( y ) , y ) (cid:54) = 0 for every y (cid:54) = 0 and so21unction ϕ is analytic for every y (cid:54) = 0 as a direct consequence of the implicitfunction theorem for real analytic functions (see Lemma 12).Now it will be proved that ϕ can be analytically extended to y =0. From the expressions of the derivatives given in (25), it is clear that ∂ F /∂y (0 ,
0) = ∂ F /∂y (0 ,
0) = 0, all the mixed partial derivatives of F vanish and the equality ∂ n F ∂y n ( y , y ) = − ∂ n F ∂y n ( y , y )is satisfied for every positive integer n . Thus, function F ∗ is analytic in aneighborhood of the origin, F ∗ (0 ,
0) = 0, ∂ F ∗ /∂y (0 ,
0) = ∂ F ∗ /∂y (0 ,
0) = a − (cid:54) = 0. As a direct consequence of the implicit function theorem appliedto F ∗ at the origin, it is obtained that ϕ can be analytically extended to y = 0 and ϕ (0) = 0.On the other hand, taking into account that F ( y , y ) = −F ( y , y ) (26)then function ϕ : y ∈ I −→ ϕ ( y ) ∈ I is an involution.In the proof of statements 1 and 2, variable y has been obtained as afunction of y in the complete interval I for the level curves of function F .Note that, analogously, variable y could have been obtained as a functionof y in the complete interval.The proof of statement 3 (where a · q (cid:54) = 0) is similar to the proof ofstatement 2 (in fact, it is even easier because the level curves do not containthe origin and the singular case does not appear).Let us consider q >
0. For every y ∈ I there exist two unique and dif-ferent values φ q ( y ) , ϕ q ( y ) ∈ I such that F ( φ q ( y ) , y ) = F ( ϕ q ( y ) , y ) = q .Moreover, the inequality φ q ( y ) · ϕ q ( y ) < ϕ q is strictly negative). By means of the im-plicit function theorem for real analytical functions, functions φ q and ϕ q areanalytic in the complete interval I (see Lema 12) and satisfy the inequalities y dφ q dy ( y ) = y V (0 , φ q ( y )) φ q ( y ) V (0 , y ) > , y dϕ q dy ( y ) = y V (0 , ϕ q ( y )) ϕ q ( y ) V (0 , y ) < y ∈ I \ { } . Therefore, both functions are unimodal and their criticalpoint (minimum for φ q and maximum for ϕ q ) are located at y = 0.22or q <
0, an analogous reasoning with a simple interchange of variables y , y and assuming that ϕ q is strictly positive, leads to the correspondingresult. Besides that, since (26) holds then φ q ≡ ϕ − q for every q ∈ R \ { } .Finally, again from the divergence of the integrals shown in (21), it is nowobvious to see that the restricted functions given in (24) are bijective. Remark 15.
In expression (24) , the only shown bijective functions corre-spond to functions whose graphs are located into the fourth quadrant since thisis the natural quadrant for the left Poincar´e half-map. Also the restrictions tothis fourth quadrant of functions ϕ q given in items 1 and 2 of Theorem 14 arebijective. There exist, obviously, bijective restrictions to the other quadrantsfor functions φ q and ϕ q . In Figure 6, several level maps of F ( y , y ) are shown. (a) (b) (c) Figure 6: Level curves of the index-like function F in I for several values q ∈ R when: (a) a = 0; (b) a (cid:54) = 0 and 4 D − T >
0; (c) a (cid:54) = 0 and D < q = 0) and 3 ( q (cid:54) = 0) for values of the parameters such that system (2) hasa focus or a center. Case (c) corresponds to items 2 ( q = 0) and 3 ( q (cid:54) = 0)for values of the parameters such that system (2) has a saddle equilibriumand, therefore, the set I is bounded. Remark 16.
An immediate conclusion of the last Theorem is that every levelcurve given by F ( y , y ) = q , q ∈ R , in I is an orbit of the following thirddegree polynomial planar system of differential equations (cid:26) ˙ x = y V (0 , x ) , ˙ y = x V (0 , y ) . (27)23 eciprocally, for a (cid:54) = 0 every orbit of system (27) in I is a level curve offunction F . For a = 0 , this is also true except for the coordinate axes, whichare foliated by equilibria of system (27) . Furthermore, for a (cid:54) = 0 , the originis the unique equilibrium point in I and it is a saddle point (see Figure 6).A differential equation analogous to (23) appeared in [11] as an equationfor local Poincar´e maps, close to an orbit, between two transversal sectionsto this orbit. As it can be deduced from the rest of this work, equation (23) characterizes the global left (and right) Poincar´e half-map of system (2) asso-ciated to section Σ . Obviously, to obtain the solution corresponding to thesemaps, a suitable initial condition must be added to the differential equationin both cases. Remark 17.
For a (cid:54) = 0 the origin is a saddle point of surface z = F ( y , y ) .From this point of view and roughly speaking, the set of level curves of theindex-like function F can be understood locally as the universal unfolding of y − y (see simple bifurcations at [16]). The analysis of the level curves of the index-like function F led us tothe bijective functions ϕ q restricted to the fourth quadrant that have beenmentioned in Remark 15. For some concrete values of q , these functions (if q (cid:62)
0) or their inverse ones (if q <
0) will give the left Poincar´e half-mapof system (2). The following section is devoted to proof this assertion, whatcloses the integral characterization of Poincar´e half-maps.
This section is devoted to the formulation and proof of Theorem 19, a recip-rocal to Theorem 8. That is, not only a Poincar´e half-map defines an integralrelationship between the initial point and its image (see equation (16)) butthis integral relationship defines, for the correct values of the parameters, thePoincar´e half-map. In order to do so, it is convenient to state and proof thefollowing Lemma.
Lemma 18.
Let us consider the inverse integrating factor V given in (10) and the index-like function F given in (18) . Let us assume that condition (11) holds and that the open interval I defined in (19) is not empty. Let be c ∈ R and y , y ∈ I such that equality F ( y , y ) = cT holds. Then, equality log (cid:18) V (0 , y ) V (0 , y ) (cid:19) = T (cid:18) Dc + (cid:90) y y aV (0 , y ) dy (cid:19) (28) is true. In fact, for D (cid:54) = 0 , both equalities are equivalent. roof. For D = 0, since condition (11) holds, then a (cid:54) = 0 and it is direct tosee that (28) is true for every values y and y (even if F ( y , y ) (cid:54) = cT ).For the generic case D · a (cid:54) = 0, it is trivial that the Cauchy principal valuecan be obviated from the definition of function F . Since (cid:90) y y − yV (0 , y ) dy = − D (cid:90) y y Dy − aTDy − aT y + a dy + 12 D (cid:90) y y − aTDy − aT y + a dy, the proof is direct.For the case a = 0, it is aV (0 ,y ) ≡ D does not vanish. Besides that, from equality (22) it is knownthat 2 DcT = 2 D PV (cid:90) y y − yV (0 , y ) dy = 2 log (cid:12)(cid:12)(cid:12)(cid:12) y y (cid:12)(cid:12)(cid:12)(cid:12) = log (cid:18) V (0 , y ) V (0 , y ) (cid:19) . This concludes the proof.Equation (28) together with equality (8) suggest an expression for the leftflight time. In fact, this expression (shown in (30)) will be a crucial elementfor the proof of the next Theorem, where we take the last step in order todefinitively show the equivalence between Poincar´e half-maps and suitablelevel curves of the index-like function F given in (18). Theorem 19.
Let us consider the inverse integrating factor V given in (10) and the index-like function F given in (18) . Let us assume that condition (11) holds and that the open interval I defined in (19) is not empty. Let be c ∈ R and y , y ∈ I , y (cid:62) , y (cid:54) such that F ( y , y ) = cT. (29) Assume that one of the following conditions holds:(i) c = 0 and a > ,(ii) c = πD √ D − T ∈ R and a = 0 ,(iii) c = 2 πD √ D − T ∈ R and a < .Then y is the image of y by the left Poincar´e half-map of system (2) relatedto the Poincar´e section Σ = { x = 0 } . Moreover, the corresponding left flighttime is τ ( y ) = 2 Dc + (cid:90) y y aV (0 , y ) dy. (30)25 roof. The proof is immediate for the limit cases of the left Poincar´e half-map when y = y = 0 and point ( y , y ) = (0 ,
0) is an invisible tangencyor the equilibrium of system (2) (see Remark 1), that is, when y = y = 0and a (cid:62)
0. From now till the end of the proof, we assume that this situationdoes not occur.Let us suppose that there exist y , y ∈ I , y (cid:62) y (cid:54) τ as the right hand side of the equality given in (30) and let beΨ( t ; y ) = (Ψ ( t ; y ) , Ψ ( t ; y )) the orbit of system (2) defined in (4). Underthese assumptions, the theorem will be proved if the following conditions areverified:(C1) ˜ τ > ( t ; y ) < t ∈ (0 , ˜ τ ).(C3) Ψ(˜ τ ; y ) = (0 , y ).Condition (C1) is trivial for items (i) and (ii). The proof of condition(C1) for item (iii), where a is negative and c = πD √ D − T ∈ R , is a directconclusion from inequalities0 (cid:54) (cid:90) y y − aV (0 , y ) dy (cid:54) (cid:90) + ∞−∞ − aV (0 , y ) dy = 2 π √ D − T = Dc.
For the proof of conditions (C2) and (C3) we will assume that the trace T does not vanish. Notice that if the proof of the treorem is obtained for T (cid:54) = 0, the results are immediately extended to the case T = 0 by using thecontinuity of solutions of differential equations and integrals with respect toparameters. After the proof of this theorem, an alternative proof for the case T = 0 is given in Remark 20, where some interesting observations are alsoadded.The following reasoning proves that Ψ (˜ τ ; y ) (cid:54) ( t ; y ) < t ∈ (0 , ˜ τ ).Let us assume that there exists a value τ ∗ ∈ (0 , ˜ τ ] such that Ψ ( τ ∗ ; y ) =0 and Ψ ( t ; y ) < t ∈ (0 , τ ∗ ). This fact implies that y ∗ :=Ψ ( τ ∗ ; y ) (cid:54)
0. From the definition of the left Poincar´e half-map it is y ∗ = P ( y ) and from Theorem 8 it is clear that PV (cid:82) y y ∗ − yV (0 ,y ) dy = cT for thecorresponding item (i), (ii) or (iii). The bijectivity of functions ϕ q when y (cid:62) y (cid:54) y ∗ = y .Property (8), applied to system (2), implies V (0 , y ∗ ) = exp ( T τ ∗ ) V (0 , y ),because div L = T for vector field L defined in (3). From relationship (28)26t is V (0 , y ) = exp ( T ˜ τ ) V (0 , y ). Since y ∗ = y and T (cid:54) = 0, the equality˜ τ = τ ∗ is true.Therefore, Ψ (˜ τ ; y ) (cid:54) ( t ; y ) < t ∈ (0 , ˜ τ ). Thismeans that condition (C2) holds.Finally, in order to prove condition (C3), we are going to integrate theorthogonal vector field G defined in (14) on a suitable closed curve (cid:101) Γ = (cid:101) Γ ∪ (cid:101) Γ ∪ (cid:101) Γ , constructed by joining a segment (cid:101) Γ contained at the separationline Σ, a piece of an orbit (cid:101) Γ of the system and a piece of a level curve (cid:101) Γ of the inverse integrating factor V (see Figure 7). Concretely, the first twocurves are the segment (cid:101) Γ = { (0 , y ) ∈ R : y ∈ ( y , y ) } and the piece of orbit (cid:101) Γ = { Ψ( t ; y ) : t ∈ [0 , ˜ τ ) } .Figure 7: Schematic drawing of curve (cid:101) Γ = (cid:101) Γ ∪ (cid:101) Γ ∪ (cid:101) Γ defined in the proofof Theorem 19. Curve (cid:101) Γ is drawn as a dotted line because points (˜ x, ˜ y ) and(0 , y ) are proved to coincide.To define the last portion of (cid:101) Γ, let us consider point (˜ x, ˜ y ) = Ψ(˜ τ ; y ),that satisfies the equality V (˜ x, ˜ y ) = exp ( T ˜ τ ) V (0 , y ) = V (0 , y ) as long asproperty (8) and relationship (28) hold. Therefore, since y , y ∈ I , points(˜ x, ˜ y ) and (0 , y ) are connected by a piece of a level curve of the inverseintegrating factor V where, moreover, V is strictly positive. Notice thatthe portion of level curve of function V that connects these points is uniqueexcept for 4 D − T >
0, where the level curves are ellipses (see Remark4), and hence there are two choices of simple curves. Anyway, it is alwaysposible to select this piece of level curve (cid:101) Γ so that the complete closed curve27 Γ encircles no equilibria for case (i) or the equilibrium of system (2) for case(iii). Trivially, in case (ii) the equilibrium is located at segment (cid:101) Γ .By reasoning as in the proof of Theorem 8, the integration of the orthog-onal vector field G along the closed curve (cid:101) Γ together with equality (29) leadsto (cid:90) (cid:101) Γ G · d r = 0 . (31)On the other hand, since V | (cid:101) Γ is constant, the parameterization of (cid:101) Γ ≡ r ( s ) = ( x ( s ) , y ( s )), s ∈ [ s , s ], may be chosen in such a way that its tangentvector is ∇ V ( x ( s ) , y ( s )) ⊥ , where the orthogonal vector is taken as defined inRemark 6. From equalities (31) and (15) it holds0 = (cid:90) (cid:101) Γ G · d r = (cid:90) s s G ( x ( s ) , y ( s )) ·∇ V ( x ( s ) , y ( s )) ⊥ ds = (cid:90) s s T ds = T ( s − s ) . Since T (cid:54) = 0 then s = s . That is, (˜ x, ˜ y ) = (0 , y ) and the proof concludes. Remark 20.
In the proof of Theorem 19, the singular features of case T =0 have forced to analyze it separately. Specifically, since for T = 0 , theinverse integrating factor V is constant along any orbit of system (2) , allthe properties based on the variation of V along the orbits are useless. Forinstance, the expression τ = 1 T log (cid:18) V (0 , y ) V (0 , y ) (cid:19) , derived from equality (8) and used to obtain the left flight time τ between twopoints of the Poincar´e map, can only be used for T (cid:54) = 0 .Besides the presented proof for case T = 0 (based on arguments of con-tinuity with respect of parameters), there are two alternative proofs based onideas we would like to highlight. Notice that the reversibility of system (2) for T = 0 implies that y = − y .The first proof (the classical way for linear systems) requires the integra-tion of system (2) for T = 0 (trivial integration but with several distinguishedcases), the obtention of the left flight time by means of the imposition of theleft Poincar´e half-map conditions and the verification of the equality betweenthe computed left flight time and the one given in (30) .For the second alternative proof for T = 0 (based on the computation ofthe left flight time from the hamiltonian character of the system), we use theconservation of V ( x, y ) = ( Dx − a ) + Dy along the orbits of system (2) ,that is, at any point ( x, y ) of the orbit that passes through point (0 , y ) it olds ˙ y + Dy = ( Dx − a ) + Dy = a + Dy . For case (i) of Theorem 19and the piece of orbit contained in { x < } that connect points (0 , ± y ) , itis true that ˙ y < and so ˙ y = − (cid:112) a + Dy − Dy . Therefore, the left flighttime is τ ( y ) = (cid:90) − y y − dy (cid:112) a + Dy − Dy . (32) It is direct to check that the change of variable w = (cid:115) V (0 , y ) V (0 , y ) y, transforms the integral of expression (30) into (32) .In cases (ii) and (iii), condition T = 0 implies that system (2) correspondsto a linear center and all the orbits are periodic with period π √ D − T . Thus,for case (ii) the proof is immediate (the orbit is a half of a complete periodicorbit) and for case (iii) expression (30) must be understood as the completeperiod minus the right flight time. The immediate and more relevant conclusion of Theorems 8 and 19 is thefollowing Corollary.
Corollary 21.
Let us assume that condition (11) holds and that the openinterval I = ( µ , µ ) defined in (19) is not empty. Let be functions ϕ q , q ∈ R as given in Theorem 14. Assume that c ∈ R satisfies one of conditions (i),(ii), (iii) of Theorem 19.If the trace verifies T (cid:62) then the left Poincar´e half-map of system (2) related to the Poincar´e section Σ is ϕ cT : [0 , µ ) −→ ( µ , ϕ cT (0)] .If the trace verifies T < then the left Poincar´e half-map of system (2) related to the Poincar´e section Σ is ϕ − cT : [ ϕ cT (0) , µ ) −→ ( µ , . Remark 22.
For T = 0 , functions ϕ and ϕ − coincide (see Theorem 14).Therefore, case T = 0 could have been also joined to case T < in thestatement of Corollary 21. Remark 23.
Let us briefly explain the role of the value ϕ cT (0) that acts aslimit point of the range or domain of the left Poincar´e half-map ϕ cT or ϕ − cT respectively. For cT = 0 , it is ϕ (0) = 0 (see Theorem 14) and it obviouslycorresponds to the tangency point for scenarios ( S ), ( S ) and the centercase of ( S ) (see Figures 4, 3 and 2(a)). When cT (cid:54) = 0 and T > , then ϕ cT (0) = ˆ y < (see Theorem 14 and Figure 2(c) corresponding to scenario( S )), in other words, it is the image of y = 0 by means of the left Poincar´ehalf-map. When cT (cid:54) = 0 and T < , then ϕ cT (0) = ˆ y > (see Theorem S )), in other words, it is thepre-image of y = 0 by means of the left Poincar´e half-map. Remark 24.
Theorem 19 and Corollary 21 can be easily extended to theright Poincar´e half-map.
As was said in the introduction, in order to avoid the flaws due to the compu-tation of the solutions of linear systems in the analysis of Poincar´e half-mapsa novel theory has been developed in this manuscript. The key point wasthe introduction and study of the index-like function F given in (18) thatwas obtained from the line integration, on a suitable curve, of an orthogonalvector field written in terms of a good choice of inverse integrating factor.In fact, this index-like function gives a common way to express the Poincar´ehalf-maps.This new approach could be extended to the study of Poincar´e half-mapsfor non-linear planar systems as far as a nice inverse integrating factor maybe found. Another interesting extension of the theory could be an analogousanalysis for higher dimensions, where the important role of inverse integratingfactor should be assumed by inverse Jacobi multipliers (see [3]). We areconvinced that these two ideas will open fruitful lines of study in short ormedium term.However, the true importance of the technique developed in this work iscurrently revealed in its application to the analysis of the dynamical behav-ior of planar piecewise linear systems, in particular, the obtention of optimalupper bounds on the number of limit cycles. On the one hand, for contin-uous planar piecewise linear systems with two zones of linearity it is knownthat this upper bound is one. This result was originally proved in [9] withexhaustive and long case-by-case analysis. By using the index-like function,we have got a direct and short proof (without cases) of this same result (see[5]). On the other hand, with the same technique it is possible to obtain thesame bound for sewing discontinuous planar piecewise linear systems withtwo zones of linearity (see [6]). Previous works (see [10, 22]) give partialresults for these kind of systems by using the case-by-case analysis.It is also interesting to study the optimal upper bound for generic dis-continuous planar piecewise linear systems with two zones of linearity. Thefirst basic open problem is the existence of such a uniform bound for all thesesystems, that is, independent of the value of parameters. Since the use of ournew approach allows to understand this problem as the existence of a uni-form bound for the number of solutions of a common system of polynomial30quations of fixed degrees, the conclusion is obvious (see [6]). Moreover, theoptimal bound, that it is known to be greater or equal than three [17, 19, 4],could be directly stablished (without a case-by-case study) from this systemof polynomial equations. Nowadays, this study is one of our priority lines ofresearch.Regarding other interesting achieved results in this manuscript, we wouldlike to mention the analyticity of the Poincar´e half-map or its inverse functionat the tangency points. Acknowledgements
The authors would like to express their gratitude to professors Douglas D.Novaes and Jos´e A. Rodr´ıguez for valuable and constructive discussions andpriceless encouragement during part of the development and writing of thiswork.This work has been partially supported by the
Ministerio de Econom´ıa yCompetitividad cofinanced with FEDER funds, in the frame of the projectsMTM2014-56272-C2-1-P, MTM2015-65608-P, MTM2017-87915-C2-1-P andPGC2018-096265-B-I00 and by the
Consejer´ıa de Educaci´on y Ciencia de laJunta de Andaluc´ıa (TIC-0130, P12-FQM-1658).
References [1] A. Andronov, A. Vitt, S. Khaikin. Theory of Oscillations (PergamonPress, Oxford), Chapter 8 (1966), 443–582.[2] L.R. Berrone, H. Giacomini.
On the vanishing set of inverse integratingfactors , Qualitative Theory of Dynamical Systems, 1 (2000), 211–230.[3] L.R. Berrone, H. Giacomini.
Inverse Jacobi multipliers , Rendiconti delCircolo Matematico di Palermo Series 2, 52 (2003), 77–130.[4] C. A. Buzzi, C. Pessoa, J. Torregrosa.
Piecewise linear perturbations ofa linear center , Discrete and Continuous Dynamical Systems, 33 (2013),3915–3936.[5] V. Carmona, F. Fern´andez-S´anchez, Douglas D. Novaes.
A new simpleproof for the Lum-Chua’s conjecture , preprint.[6] V. Carmona, F. Fern´andez-S´anchez, Douglas D. Novaes.
Uniqueness andstability of limit cycles in planar piecewise linear differential systems with-out sliding region , preprint. 317] V. Carmona, E. Freire, E. Ponce, F.Torres.
On simplifying and classifyingpiecewise-linear systems , IEEE Transactions on Circuits and Systems IFundamental Theory and Applications, 49 (2002), 609–620.[8] J. Chavarriga, H. Giacomini, J. Gin´e, J. Llibre.
On the integrability oftwo-dimensional flows , Journal of Differential Equations, 157 (1999), 163–182.[9] E. Freire, E. Ponce, F. Rodrigo, F. Torres.
Bifurcation Sets of Contin-uos Piecewise Linear Systems with Two Zones , International Journal ofBifurcation and Chaos, 8,11 (1998), 2073–2097.[10] E. Freire, E. Ponce, and F. Torres.
Planar filippov systems with maximalcrossing set and piecewise linear focus dynamics . In Progress and Chal-lenges in Dynamical Systems, Springer Berlin Heidelberg (2013), 221–232.[11] I.A. Garc´ıa, H. Giacomini, M. Grau.
The inverse integrating factor andthe Poincar´e map , Transactions of the American Mathematical Society,362 (2010), 3591–3612.[12] I.A. Garc´ıa, M. Grau.
A survey on the inverse integrating factor , Qual-itative Theory of Dynamical Systems, 9 (2010), 115–166.[13] A. Gasull, J. Llibre, V. Ma˜nosa, F. Ma˜nosas.
The focus-centre problemfor a type of degenerate system , Nonlinearity, 13 (2000), 699–729.[14] A. Gasull, V. Ma˜nosa, F. Ma˜nosas.
Monodromy and stability of a classof degenerate planar critical points , Journal of Differential Equations, 182(2002), 169–190.[15] H. Giacomini, J. Llibre, M. Viano.
On the nonexistence, existence, anduniqueness of limit cycles , Nonlinearity, 9 (1996), 501–516.[16] M. Golubitsky, D.G. Schaeffer. Singularities and Groups in Bifurca-tion Theory. Vol. 1. Berlin-Heidelberg-New York-Tokyo, Springer-Verlag(1985).[17] S. M. Huan, X. S. Yang.
The number of limit cycles in general planarpiecewise linear systems , Discrete and Continuous Dynamical Systems-A,32 (2012), 2147–2164.[18] Yu. A. Kuznetsov, S. Rinaldi, A. Gragnani.
One-parameter bifurcationsin planar Filippov systems , International Journal of Bifurcation and Chaos,13, 8 (2003), 2157–2188. 3219] J. Llibre, E. Ponce.
Three nested limit cycles in discontinuous piece-wise linear differential systems with two zones , Dynamics of Continuous,Discrete and Impulsive Systems B, 19 (2012), 325–335.[20] J. Llibre, E. Ponce, C. Valls.
Two limit cycles in Linard piecewise lineardifferential systems , Journal of Nonlinear Science, 29 (2019), 1499–1522.[21] J. Llibre, A. E. Teruel. Introduction to the qualitative theory of differen-tial systems: planar, symmetric and continuous piecewise linear systems.Springer-Verlag (2014).[22] J. C. Medrado, J. Torregrosa.