LIMIT CYCLES OF CONTINUOUS AND DISCONTINUOUS PIECEWISE–LINEAR DIFFERENTIAL SYSTEMS IN R3

We study the limit cycles of two families of piecewise–linear differential systems in R3 with two pieces separated by a plane Σ. In one family the differential systems are only continuous on the plane Σ, and in the other family they are only discontinuous on the plane Σ. The usual tool for studying these limit cycles is the Poincaré map, but here we shall use recent results which extend the averaging theory to continuous and discontinuous differential systems. All the computations have been checked with the algebraic manipulator mathematica.


Introduction and statement of the main results
The study of piecewise linear differential systems essentially started with Andronov, Vitt and Khaikin [1] and still continues to receive attention by researchers. The continuous and discontinuous piecewise-linear differential systems plays an important role inside the nonlinear dynamical systems. First they appear in a natural way in nonlinear engineering models, where certain devices are accurately modeled by such differential systems, see for instance the books of di Bernardo, Budd, Champneys and Kowalczyk [3], and Simpson [28], and the survey of Makarenkov and Lamb [26], and the hundreds of references quoted in these last three works. Moreover these kind of differential systems are frequent in applications from electronic engineering and nonlinear control systems, where they cannot be considered as idealized models; they are also used in mathematical biology as well, see for instance [7,29,30,31].
There are many studies of the limit cycles of continuous and discontinuous piecewise-linear differential systems in R 2 with two pieces separated by a straight line, see for instance [2,4,6,9,10,11,12,13,14,15,16,21,22,23,25,27]. But there are few results about the limit cycles of continuous and discontinuous piecewise-linear differential systems in R 3 with two pieces separated by a plane. The objective of this work is to study the limit cycles of some of these last systems.
We consider perturbations of the linear differential system (1)ẋ = −y, y = x, z = ix, with (x, y, z) ∈ R 3 and i a real parameter. The dot denotes derivative with respect to an independent variable t, usually called the time. Straightforward computations show that the solutions of (1) are all periodic with the exception of the z-axis which is filled with equilibria.
In this paper first we study the periodic solution of the following perturbed continuous piecewise linear differential system (2)ẋ = −y + ε(a + bx + cy + d|z|), y = x + ε(e + f x + gy + h|z|), z = ix + ε(j + kx + ly + m|z|), of system (1) with two zones z > 0 and z < 0, where a, b, c, d, e, f, g, h, j, l and m are real parameters and the parameter ε > 0 is sufficiently small. Changing z by −z if necessary, we always can assume that the parameter Our main result on the periodic solutions of the continuous piecewise linear differential system (2) is the following. This result is obtained using the extension of the classical averaging theory for smooth differential systems to continuous differential systems given in [20], see section 2 for more details. Theorem 1. Using the averaging theory of first order for the continuous piecewise linear differential system (2) the following statements hold.
(a) For ε = 0 sufficiently small if i > 0 system (2) has the periodic solution In the proof of this statement we describe how to compute the values of r * and z * in function of the parameters of system (2). (b) If i = 0 system (2) has two periodic solutions Theorem 1 is proved in section 3. Many problems in physics, economics, biology and applied areas are modeled by discontinuous differential systems but there exist only few analytical techniques for studying their periodic solutions. In [19]  the averaging theory to discontinuous differential systems. An improvement of this result for a much bigger class of discontinuous differential systems is given in [18].
Applying these tools we also investigate the periodic solutions of the discontinuous piecewise linear differential system , with two pieces defined by f (z) = z + sign(z) and We get the following result on the periodic solutions of the discontinuous piecewise linear differential system (3).

Theorem 2.
Using the averaging theory of first order for the discontinuous piecewise linear differential system (3), the following statements hold.
hi − m ∈ (4π, γ * ) and h(b + g − hi) < 0 or h(b + g + 3hi) > 0, for ε = 0 sufficiently small system (3) has two periodic solutions given by In the proof of this theorem we describe how to compute the values of r * and γ * in function of the parameters of system (3).
Theorem 2 is proved in section 4.

Basic results on the averaging theory
For proving Theorems 1 and 2 we apply two recent results from the averaging theory, one for the continuous piecewise linear differential systems, and the other for the discontinuous piecewise linear differential systems. In this section we present these results and some remarks necessary for their applications.
2.1. Continuous piecewise linear differential systems. From Theorem B of [20] taking the k which appears in its statement equal to 1 we get the next result. Theorem 3. Consider the following differential system

and R ∈ C 0 and locally Lipschitz in the second variable. Moreover D ⊂ R n is an open subset and ε is a small parameter. Assume that there exists an open and bounded subset of
Denote by M z (t) the fundamental matrix of the variational equation x(t, z))dt and (6) det(D z f (a)) = 0, The next result was obtained in [5], for a definition on the Brouwer degree see [24]. Theorem 4. We consider the following differential system

continuous functions, T -periodic in the first variable, and D is an open subset of
and assume that: Then, for |ε| > 0 sufficiently small, there exists an isolated T -periodic solution x(t, ε) of system (7) such that x(0, ε) → a as ε → 0.
From the proof of Theorem 4 it follows the next remark.
We define the discontinuous diffeential system as h(t, x) < 0}. The manifold Σ is divided in the closure of two disjoint regions, namely Crossing region (Σ c ) and Sliding region (Σ s ) where The differential system (9) can be written as where χ + , χ − are the characteristic functions defined as We apply the Fillipov's convention for the solutions of systems (9) or (10) passing through a point (t, x) ∈ Σ (see [8]). Let P be the space formed by the periodic solutions of (9) or (10). If dim(P ) = dim(D) = d then the following result follows directly from Theorem B of [18]. Theorem 6. Consider the differential system For z ∈ D and ε > 0 sufficiently small denote by x (., z, ε) : [0, t (z,ε) ] → R d the solution of system (11) such that x(0, z, ε) = z. Define the averaged function x(s, z, 0))ds, where x(s, z, 0) is a periodic solution of (11) with ε = 0 such that x(0, z, 0) = Let X, Y : R × D → R n be two continuous vector fields. Assume that the functions h, X and Y are T -periodic in the variable t. Now we define a discontinuous piecewise differential system (12) x The discontinuous differential system (12) can be written using the function sign(u) as (13) x The next result is Theorem A of [19].
LIMIT CYCLES OF PIECEWISE-LINEAR DIFFERENTIAL SYSTEMS IN R 3 7 Theorem 7. We consider the following discontinuous differential system (14) x

continuous functions, T -periodic in the variable t and D
is an open subset of R n . We also suppose that h is a C 1 function having 0 as a regular value.
Define the averaged function f : D → R n as We assume the following conditions.  (14) such that x(0, ε) → a as ε → 0.
If the function f (x) in (15) is C 1 the Remark 5 works for it.
In this case the averaged function (19) is This solution (when it exists) corresponds to the equilibrium x = 0, y = 0, z = ei − j hi − m of the initial system when ε = 0, so the averaging theory in this case does not provide any periodic solution.
Finally for system (2) we get the periodic solution In short we have proved the positive result of Theorem 1, but it remains to show that the averaging theory of first order does not provide more results on the periodic solutions of the continuous piecewise differential system (2).
If b+g = 0 then the Jacobian determinant (6) will be zero, so we can assume b + g = 0. Then we obtain the solution r 0 = 0, which corresponds to an equilibrium point of the unperturbed system (18), therefore the averaging theory does not provide any periodic solution in this subcase. Subcase (ii.4): Suppose now that h = 0 and m − hi = 0. We obtain . The same arguments than in the subcase (ii.2) show that the averaging theory does not provide information about the periodic solutions in this subcase. This completes the study of case (ii).
Proof of statement (b) of Theorem 1. Since i = 0 we have system (17) with i = 0. This system satisfies the assumptions of the averaging Theorem 4, so we must compute the function (23), i.e.
(f 1 (r, z), f 2 (r, z)) = 2π 0 (g 1 (θ, r, z), g 2 (θ, r, z))dθ This function has two simple zeros if and only if (b + g)m = 0 and j/m < 0, namely (r, z) = (0, ±j/m). Going back through the changes of variables as we did in the proof of statement (a) we get the result of statement (b).
We note that the function (f 1 (r, z), f 2 (r, z)) has a unique zero if (b+g)m = 0 and j = 0, but in this case the Brouwer degree is zero, and the averaging Theorem 4 cannot be applied.
If (b + g)m = 0, then it is easy to verify that the averaging Theorem 4 does not provide information about the periodic solutions of system (17). This completes the proof of statement (b) of Theorem 1.
As in the continuous case the unperturbed system is given by r ′ = 0, z ′ = ir cos θ. Now the differential system (22) is in the normal form (11) with F 0 (θ, (r, z)) = (0, ir cos θ), F 1 (θ, (r, z)) = (h 1 (θ, r, z), h 2 (θ, r, z)), satisfying all the assumptions of Theorem 6. So we apply this theorem to system (22) and we must calculate the averaged function f (r 0 , z 0 ) = (f 1 (r 0 , z 0 ), f 2 (r 0 , z 0 )) = Analogously to the study of the continuous system for i > 0, we separate the computation of the averaged function in the same three cases, because in the functions h 1 and h 2 also appears the expression sign(z 0 + ir 0 sin θ).
Clearly if (b + g)m = 0 the averaging Theorem 7 does not provide information about the periodic solutions. This completes the proof of statement (c) of Theorem 2.