Periodic orbits of continuous and discontinuous piecewise linear differential systems via first integrals

Our main concern is to discuss the existence of periodic orbits for certain piecewise linear differential systems in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^ n$$\end{document}, where the discontinuity set is a hypersurface Σ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma .$$\end{document} Using the fact that every linear differential system is always completely integrable, we illustrate for two piecewise linear differential systems, one in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^3$$\end{document} and the other in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^4$$\end{document}, how to study analytically their periodic solutions.

1 Introduction and statement of the main result 1

.1 The goal
Integrability theory for smooth systems has classically been well developed, mainly establishing conditions for the existence of minimal sets and persistence results. However, as far as the authors know, there are few techniques or tools within the nonsmooth differential systems Theory to study differential equations that are piecewise integrable B Marco Antonio Teixeira teixeira@ime.unicamp.br Jaume Llibre jllibre@mat.uab.cat 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain and so new techniques in this field are notoriously welcome. In the present article we intend to open a window in this direction.
The nonsmooth piecewise linear differential systems appear in a natural way in problems of mechanics, electrical circuits, control theory, ... see for more details and additional references the classical book of Andronov et al. [2], or the more recent books of di Bernardo et al. [9] and of Simpson [19]. In these recent years these kind of differential systems have been studied intensively because they are easier to analyze than the nonlinear smooth systems and can exhibit almost all the complicate dynamics of the nonlinear smooth ones. Our goal will be to study the periodic solutions of these kind of nonsmooth differential systems.
On the other hand, this paper is part of a general program involving the study of continuous and discontinuous piecewise differential systems in R n of the forṁ where x = (x 1 , . . . , x n ), and f, g : R n → R n are smooth functions. Note that we have two different differential systems, one in the half-space x 1 > 0 and the other in x 1 < 0. Eventually on the hyperplane = {x 1 = 0} the piecewise differential system can be continuous or discontinuous.
Our main concern is to discuss the existence of periodic orbits for such piecewise differential systems assuming that both differential systems, the one in x 1 > 0 and the other in x 1 < 0, have first integrals.

Historical facts and motivations
Control theory is a natural source of mathematical models of these systems (see, for instance, [1,3,7,8]). It is worth mentioning that Anosov [4,5] studied a class of relay systems in R n of the formẋ = A x + sign(x 1 ) k where A is an n × n real matrix, x = (x 1 , . . . , x n ) ∈ R n , and k = (k 1 , . . . , k n ) is a constant vector in R n . Jacquemard and Teixeira [14] analyzed the simplest model of a relay system ....
The focus was to detect the existence and robustness of a oneparameter families of periodic orbits through reversible polynomial perturbations.
It is worthwhile to cite Ekeland [10] and Klok [15], where the main problem in the classical calculus of variations was carried out to study discontinuous Hamiltonian vector fields.

Integrability for smooth systems
Consider C k differential systems of the forṁ where k ∈ N ∪ {∞, ω}, the dot denotes derivative with respect to the independent variable t, is an open subset of R n , and f (x) = ( f 1 (x), . . . , f n (x)) is a C k function defined in . As usual N denotes the set of positive integers, and C ∞ and C ω denote the sets of infinitely derivable functions and analytic functions, respectively. A first integral of system (1) is a continuous function H (x) defined in a full Lebesgue measure subset 1 of , which is not locally constant on any positive Lebesgue measure subset of 1 ; moreover H (x) is constant along each orbit of system (1) in The easiest class of completely integrable systems are the linear differential systems in R n , i.e. the differential systems of the forṁ where x, b ∈ R n and A is an n × n real matrix. Clearly the domain of definition of a linear differential system is the whole R n . In fact, every linear differential system is completely integrable with Darboux first integrals. Here a first integral is Darboux if it is a real function which can be written in the form where f i (x) for i = 1, . . . , k, g(x) and h(x) are complex polynomials, and the λ i for i = 1, . . . , k are complex numbers. For more details on Darboux first integrals see, for instance, [11,18].

First integrals for nonsmooth systems
We investigate in this paper global stability problems of vector fields defined in R n expressed as where F 1 and F 2 are smooth vector fields in R n . If we have a discontinuous differential system (2) in R n separated by the hyperplane x 1 = 0, given by the vector field X = F 1 + F 2 in x 1 > 0, and the vector field Y = F 1 − F 2 in x 1 < 0. Assume that H 1 and H 2 are first integrals of X and Y respectively. Then it is easy to check that is a first integral of the discontinuous piecewise differential system (2).
These piecewise smooth first integral H can be very useful in detecting periodic orbits. The main goal of this paper, is to present a discussion of this situation by means of two apparent simple nonsmooth differential systems.
It is worth to point out that, as long as the authors know, there is not available any refined technique to deal with integrability of nonsmooth vector fields (mainly when it is defined on non-compact manifolds).

Setting the problem
To study analytically the periodic solutions of a differential system is a very difficult task, usually impossible of doing. Our goal here is to show that the periodic orbits of piecewise differential systems, continuous or discontinuous, which are completely integrable in each piece, "sometimes" can be studied analytically using the first integrals of these systems. This "sometimes" means that for such differential systems in theory always can be studied their periodic orbits using the first integrals, but in the practice the computations that must be done using the first integrals may be difficult and cannot allow to do this study.
More precisely, in this work we study analytically the periodic orbits of the following two discontinuous piecewise linear differential systems: andẋ = y, using their first integrals. As usually the sign function is defined as follows So x = 0 is the plane or hyperplane of discontinuity of the piecewise linear differential systems (3) or (4), respectively. Hence, both discontinuous piecewise linear differential systems are formed by two linear differential systems, one defined in the half-space x ≥ 0 and the other in the half-space x ≤ 0.
We point out that equations expressed as are commonly found in many applications such as Control Theory and Engineering (see [6,9]). The discontinuous piecewise linear differential system (4) is one of the basic models of a semi-linear vector field, for more details on these vector fields see the book [20].
If the discontinuous piecewise linear differential systems (3) or (4) have a periodic solution which cross twice the plane or hyperplane of discontinuity, then in its neighborhood it is defined a Poincaré return map. Then the usual way of studying the periodic solutions of these differential systems is looking for the fixed points of such Poincaré return map. Usually this is done splitting the Poincaré return map as the composition of two trnasition maps, one from x = 0 to itself defined only in the half-space x ≥ 0, and the other from x = 0 to itself defined only in the half-space x ≤ 0. For studying each one of these two transition maps we must compute the solutions of the linear differential systems in x ≥ 0 and in x ≤ 0, with initial conditions at a point p in x = 0, and compute the times that these two solutions need for reaching again the plane or hyperplane x = 0 by first time, for one solution in forward time and for the other in backward time. Once these twice are known, we must impose that the two solutions starting at the point p of x = 0, one in forward time and the other in backward time, reach after these twice the same point of the plane x = 0, and in this way we get a closed orbit, i.e. a periodic orbit. The problem for applying this algorithm usually comes from the fact that we cannot compute explicitly the required times. For illustrating the described algorithm see for instance [17].
There are other ways for computing the periodic solutions which cross twice the plane or hyperplane of discontinuity. Thus the recent developments of the averaging theory for discontinuous differential systems allow to study these periodic solutions, but again for applying this theory we need to know some explicit values of the independent variable of the system, which sometimes are not possible to compute. See for more details [16].
The goal of this work is to study the periodic solutions which cross twice the plane or hyperplane of discontinuity of the discontinuous piecewise linear differential systems (3) and (4) using the first integrals of their linear differential systems. As we said before this way of computing the mentioned periodic solutions also have frequently problems of computation. In any case we shall see that sometimes it works very well, as we shall show in the study of the periodic solutions of the two discontinuous piecewise linear differential systems (3) and (4). We remark that the study of the existence or non-existence of the periodic solutions of system (4) for the other two methods here mentioned (Poincaré return map or averaging theory) is extremely difficult. More precisely, for computing the periodic orbits using the Poincaré return map we need to compute the time that a solution contained in the half-space x ≥ 0 (respectively x ≤ 0) and starting at a point in x = 0 needs for reaching again x = 0, and in general such time cannot be computed analytically. On the other hand, for using the averaging theory for computing periodic orbits it is necessary to compute analytically some integrals, which not always is possible. We have chosen discontinuous piecewise linear differential systems for illustrating how to use the first integrals for computing periodic solutions, but the same technique can be used for continuous piecewise linear differential systems.

Statement of main results
Our main results are the following two theorems.

Theorem 1 The following statements hold.
(a) Let γ be a periodic orbit of the discontinuous piecewise linear differential system there exists a periodic orbit γ of the discontinuous piecewise linear differential system (3) which intersects the discontinuous plane x = 0 in these two points.
Theorem 1 is proved in Sect. 3. The proof of Theorem 1 only uses the first integrals of this differential system.

Corollary 2
The set of periodic orbits of the discontinuous piecewise linear differential system (3) described in Theorem 1 has a boundary surface filled with homoclinic orbits to the equilibrium points (−y, 0, 0) for all y ∈ R with y > 0. The homoclinic orbit to the equilibrium point (−y, 0, 0) with y > 0 intersect the discontinuous plane x = 0 first at the point (0, y, y), and after at the point (0, −y, y).
Corollary 2 is proved at the end of Sect. 3.
Theorem 3 is proved in Sect. 4. The proof of Theorem 3 uses the first integrals of the system and the fact that we know explicitly in function of the time the solutions of a linear differential system.
Remark One of the characteristic properties of smooth (symmetric) integrable systems is that closed orbits of such systems typically appear in k-parameter families, in contrast to periodic orbits of general systems which are typically limit cycles, i.e. they are isolated. Following the results of this work we are tempted to say that the same occurs in the nonsmooth universe. So we are in position to enumerate some questions, such as (i) How do branches of periodic orbits originate or terminate? (ii) Can one prove the existence of one branch of periodic orbits bifurcating from another such branch? and (iii) How is the bifurcation diagram of this process with respect to the parameters?

Proof of Theorem 1
The discontinuous piecewise linear differential system (3) in R 3 is formed by the following two linear differential systemṡ in the half-space x > 0, andẋ in the half-space x < 0. Note that both systems have the points of the x-axis contained in the half-spaces where they are defined as equilibrium points.
Since both differential systems are linear we always can compute two independent first integrals for each system. In fact, linear differential systems always are completely integrable. The integrability of such systems do not depend on the fact that they have either attracting or repelling singular points, or not, see for instance [11]. Thus two independent first integrals for system (5) are So the orbits of system (5) are contained in the sets for all (h 1 , h 2 ) ∈ R 2 when these sets are non-empty.
In a similar way two independent first integrals for system (6) are So the orbits of system (6) are contained in the sets for all ( f 1 , f 2 ) ∈ R 2 when these sets are non-empty. We note that the set γ h 1 h 2 , when it is non-empty and h 2 > 0, is formed by the piece of the connected curve obtained from the intersection of the plane H 1 = h 1 with the cylinder H 2 = h 2 and with the half-space x > 0. So under these assumptions the set γ h 1 h 2 is a connected arc contained in x > 0 which does not contain equilibria, so it is an orbit of system (5). If γ h 1 h 2 is non-empty and h 2 = 0, then γ h 1 h 2 is an equilibrium point. In short, always that the set γ h 1 h 2 is non-empty it is formed by a unique orbit of system (5).
It is easy to check that the set γ f 1 f 2 is always non-empty. If f 2 = 0, it is formed by one or two arcs of the curve obtained from the intersection of the plane F 1 = f 1 with the hyperboloid cylinder F 2 = f 2 contained in the half-space x < 0. Moreover these one or two arcs do not contain equilibria (because the equilibria need that f 2 = 0), so γ f 1 f 2 is formed by one or two orbits of system (6). If f 2 = 0, then γ f 1 f 2 is the intersection of the plane F 1 = f 1 with the two planes F 2 = 0 and with the half-space x < 0. The intersection {F 1 = f 1 } ∩ {F 2 = 0} is formed by two straight lines intersecting at the equilibrium point ( f 1 , 0, 0). The intersection of these two straight lines with the half-space x < 0, can contain either 5 orbits (one of them is the equilibrium point  ( f 1 , 0, 0)), or 2 orbits.
We want to study when an orbit of γ h 1 h 2 and an orbit of γ f 1 f 2 can connect forming a periodic orbit of the discontinuous piecewise linear differential system (3), i.e. when the two orbits reach the plane x = 0 in the same two points. In such a case they form a periodic solution because from system (3) these two points are crossing points. More precisely, let X + (resp. X − ) be the vector field associated to the linear differential system (5) (resp. (6)) in the half-space x ≥ 0 (resp. x ≤ 0). Let p be a point of the discontinuity plane x = 0, when the segment connecting the endpoints of the vectors X + ( p) and X − ( p) does not intersect the plane x = 0, then p is a crossing point. For more details on crossing points see [12,13].
From the previous study done on the orbits of systems (5) and (6) it follows that an orbit of γ h 1 h 2 and an orbit of γ f 1 f 2 can connect forming a periodic orbit only if h 2 > 0 and f 2 = 0.
We take an arbitrary point of the plane of discontinuity, for instance the point (0, y 0 , z 0 ). We evaluate the four first integrals H 1 , H 2 , F 1 and F 2 in this point and we get the four values h 1 = z 0 , h 2 = y 2 0 + z 2 0 , f 1 = −z 0 and f 2 = y 2 0 − z 2 0 , respectively. Now we study how many points of the orbit {H 1 = h 1 } ∩ {H 2 = h 2 } ∩ {x ≥ 0} are in the plane x = 0, solving the system we get two points (0, ±y 0 , z 0 ). We also analyze how many points of the orbits again we get the two points (0, ±y 0 , z 0 ). If these two points belong to the same orbit of the set , then we have a periodic orbit of the discontinuous piecewise linear differential system (3).
If we parameterize the orbit Note that this orbit is symmetric with respect to the y-axis, it is contained in x ≥ 0 and its endpoints are the two points (0, ±y 0 , z 0 ) on the plane x = 0. Again we parameterize the curve {F 1 = f 1 } ∩ {F 2 = f 2 } in the half-space x ≤ 0 using the variable x. This curve is formed by • the two orbits x, x 2 + y 2 0 , x : x ≤ 0 and x, − x 2 + y 2 0 , x : x ≤ 0 if z 0 = 0, each orbit has only one endpoint in the plane x = 0; x ≤ 0} if either z 0 < 0 or z 0 > 0 and y 2 0 − z 2 0 > 0, each orbit has only one endpoint in the plane x = 0; • the two orbits and if z 0 > 0 and y 2 0 − z 2 0 < 0, the first orbit has its two endpoints at the points (0, ±y 0 , z 0 ) of the plane x = 0, and the second orbit has its endpoints at infinity; We note that y 2 0 − z 2 0 = 0 otherwise f 2 = 0. In short, if z 0 > 0 and y 2 0 − z 2 0 < 0 then the orbit (7) of the linear differential system (5) together with the orbit (8) of the linear differential system (6) form a periodic orbit of the discontinuous piecewise linear differential system (3), and this periodic orbit intersects the plane of discontinuity x = 0 at the two points (0, ±y 0 , z 0 ). This completes the proof of Theorem 1.

Proof of Corollary 2
It is immediate to check that the x-axis is filled of equilibrium points of the differential system (3). Connecting the previous three pieces of orbit we obtain the homoclinic orbit starting at the equilibrium (−y, 0, 0) with y > 0, passing through the points (0, y, y) and (0, −y, y), and ending again in the equilibrium (−y, 0, 0). This completes the proof of the corollary.

Proof of Theorem 3
The discontinuous piecewise linear differential system (4) in R 4 is formed by the following two linear differential systemṡ in the half-space x > 0, andẋ in the half-space x < 0. Note that both systems have no equilibrium points.
Since both differential systems are linear we always can compute three independent first integrals for each system, see for instance [11]. Thus three independent first integrals for system (9) are So the orbits of system (5) are contained in the sets for all (h 1 , h 2 , h 3 ) ∈ R 3 when these sets are non-empty.
In a similar way three independent first integrals for system (10) are So the orbits of system (6) are contained in the sets for all ( f 1 , f 2 , f 3 ) ∈ R 3 when these sets are non-empty. Again we are looking for the periodic orbits of the discontinuous piecewise linear differential system (4) which have two points on the hyperplane of discontinuity x = 0. Let (0, y 0 , z 0 , u 0 ) one of these two points. We evaluate the six first integrals H 1 , H 2 , H 3 F 1 , F 2 and F 3 in this point and we get the six values h 1 = −8y 0 u 0 + 4z 0 u 2 0 − u 4 0 , h 2 = 3y 0 −3z 0 u 0 +u 3 0 , h 3 = 2z 0 −u 2 0 , f 1 = 8y 0 u 0 +4z 0 u 2 0 +u 4 0 , f 2 = 3y 0 +3z 0 u 0 +u 3 0 and f 3 = 2z 0 +u 2 0 , respectively. Now we study how many points of the set γ h 1 h 2 h 3 ∩γ f 1 f 2 , f 3 are in the hyperplane x = 0. Of course, if the system has periodic orbits having two points in the hyperplane x = 0, there must exist sets γ h 1 h 2 h 3 ∩ γ f 1 f 2 , f 3 having at least two points in the hyperplane x = 0. We claim that all the sets γ h 1 h 2 h 3 ∩ γ f 1 f 2 , f 3 have exactly two points in the hyperplane x = 0 if u 0 = 0, the points (0, y 0 , z 0 , u 0 ) and (0, −y 0 , z 0 , −u 0 ); if u 0 = 0 the sets γ h 1 h 2 h 3 ∩ γ f 1 f 2 , f 3 have a unique point in the hyperplane x = 0, the point (0, y 0 , z 0 , 0). So when u 0 = 0 the discontinuous piecewise linear differential system (4) has no periodic solutions through the point (0, y 0 , z 0 , 0). Now we shall prove the claim. The points of the set γ h 1 h 2 h 3 ∩ γ f 1 f 2 , f 3 which are on the hyperplane x = 0 are the solutions of the system H k = h k , F k = f k , for k = 1, 2, 3, restricted to x = 0, i.e. we must solve the system Adding the third and the sixth equations we get that z = z 0 . Then from the third we get u = ±u 0 . Now assume u = u 0 , then from the second it follows that y = y 0 .
In short, we have proved that if the discontinuous piecewise linear differential system (4) has some periodic orbit intersecting the hyperplane x = 0 in two points, these two points must be of the form (0, y 0 , z 0 , u 0 ) and (0, −y 0 , z 0 , −u 0 ) with u 0 = 0.
Recall that u 0 cannot be zero. Then the solutions of systems (12) and (13) with u 0 = 0 are y 0 = −u 3 0 /3 and z 0 = 0. Summarizing if a periodic solution of the discontinuous piecewise linear differential system (4) exists having two points in the hyperplane x = 0, these two points must be of the form Therefore it follows that x + (t) > 0 in the interval of time with endpoints 0 and t + , and that x − (t) < 0 in the interval of time with endpoints 0 and t − . This implies that the unique periodic solutions of the discontinuous piecewise linear differential system (4) having two points in the hyperplane of discontinuity x = 0 are the ones passing through the two points (0, −u 3 0 /3, 0, u 0 ) and (0, u 3 0 /3, 0, −u 0 ) for all u 0 = 0. This completes the proof of Theorem 3.