ON THE PERIODIC SOLUTIONS OF THE MILCHELSON CONTINUOUS AND DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEM

Applying new results from the averaging theory for discontinuous and continuous differential systems, we study the periodic solutions of two distinct versions of the Michelson differential system: a Michelson continuous piecewise linear differential system, and a Michelson discontinuous piecewise linear differential system.


Introduction and statement of the main result
The Michelson differential system is given bẏ with (x, y, z) ∈ R 3 and the parameter c ≥ 0. The dot denotes derivative with respect to an independent variable t, usually called the time. This system is due to Michelson [13] for studying the traveling solutions of the Kuramoto-Sivashinsky equation. It also arises in the analysis of the unfolding of the nilpotent singularity of codimension three [4,6]. This system has been largely investigated from the dynamical point of view. In the first study of Michelson [13] he proved that if c > 0 is sufficiently large, then system (1) has a unique bounded solution which is a transversal heteroclinic orbit connecting the two finite singularities (− √ 2c, 0, 0) and ( √ 2c, 0, 0). When c decreases there will appear a cocoon bifurcation (see [7,8,13]). A complete description of the phase portrait at infinity of system (1) via the Poincaré compactification was given in [14]. In [12] there is an analytical proof of the existence of a zero-Hopf bifurcation for system (1).
In [3,2] the authors consider a continuous piecewise linear version of Michelson differential system changing the non linear function x 2 in (1) by the linear function |x|. For such system they proved that some dynamical aspects of the Michelson system remains as the existence of a reversible T-point heteroclinic cycle.
Doing the change of variable (x, y, z, c) → (2εX, 2εY, 2εZ, 2εd) with d ≥ 0 and ε > 0 sufficiently small to the differential Michelson system (1), followed by the change of the function X 2 → |X|, we obtain the systemẋ = y, y = z, that we call the Milchelson discontinuous piecewise linear differential system, where we still use x, y, z instead of X, Y, Z.
In this paper first we study analytically the periodic solutions of the Michelson continuous piecewise linear differential system. Thus our first main result is the following.

Moreover this periodic solution is linearly stable.
Theorem 1 is proved in section 3. Its proof uses an extension of the classical averaging theory for smooth differential systems to continuous differential systems given in [10], the first results in this continuous direction appeared in [1].
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 [11] the authors extended the averaging theory to discontinuous differential systems. An improvement of this result for a much bigger class of discontinuous differential systems is given in [9].
Applying these tools we investigate the periodic solutions of the Milchelson discontinuous piecewise linear differential system. In this version of the Michelson system we do the change of variable (x, y, z, c) → (2εX, 2εY, 2εZ, 2εd) with d ≥ 0 and ε > 0 sufficiently small to the differential Michelson system (1), and then the nonlinear function x 2 is changed by the discontinuous piecewise function In short we consider the Milchelson discontinuous piecewise linear differential system given bẏ Again we will work with x, y, z instead of X, Y, Z. We get the following result on the periodic solutions of system (3).
, then it has only the above periodic solution with r − 0 . These periodic solutions are linearly stable.
Theorem 2 is proved in section 4.

Preliminary
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 [10] we get the following result adapted to the next system (4).
Theorem 3. Consider the following system where F i : R × D → R n for i = 0, 1 and R : R × D × (−ε 0 , ε 0 ) → R n and for each t ∈ R the functions F 0 (t, .) ∈ C 1 , F 1 (t, .) ∈ C 0 and D x F 0 is locally Lipschitz in the second variable, 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. Consider ϕ(., z) : [0, t z ] → R n the solution of the unperturbed system such that ϕ(0, z) = z. Denote by M(z) the fundamental solution matrix of the variational equatioṅ and det(D z F (a)) = 0, then for ε > 0 sufficiently small, system (4) has a T -periodic solution x(t, ε) such that x(0, ε) → a as ε → 0. Moreover the linear stability of the periodic solution x(t, ε) is given by the eigenvalues of the matrix D z F (a).
Note that the stability of the periodic solutions of system (4) when it is applied to the Michelson continuous piecewise linear differential system can be obtained from the stability of a differential system associated to it. In fact given the continuous system (1) consider a band of amplitude ε > 0 around the plane x = 0 and a differentiable extension of the continuous system (1) to this band. Studying the limit of this extended differentiable system when ε → 0 we conclude that the linear stability of system (1) is given by the eigenvalues of D z F (a).
2.2. Discontinuous piecewise linear differential systems. Let D ⊂ R n an open subset and h : R × D → R a C 1 function having 0 as regular value. Consider F 1 , F 2 : R × D → R n continuous functions and Σ = h −1 (0). We define the Filippov's system as The manifold Σ is divided in the closure of two disjoint regions, namely Crossing region (Σ c ) and Sliding region (Σ s ), Consider the differential system associated to system (5) where χ + , χ − are the characteristic functions defined as Systems (5) and (6) does not coincides in h(t, x) = 0, but applying the Fillipov's convention for the solutions of systems (5) and (6) (see [5]) passing through a point (t, x) ∈ Σ we have that these solutions do not depend on the value of F (t, x), so the solutions are the same.
Let P be the space formed by the periodic solutions of (6). If dim P = dim D = d then the following result follows directly from Theorem B of [9]. Theorem 4. Consider the differential system with F 1 i ∈ C 1 , for i = 0, 1 and R 1 , R 2 are continuous functions which are Lipschitz in the second variable, and all these functions are Tperiodic functions in the variable t ∈ R.
For z ∈ D and ε > 0 sufficiently small denote by x(., z, ε) Define the averaged function where x(s, z, 0) is a periodic solution of (7) with ε = 0 such that x(0, z, 0) = z, and M(s, z) is the fundamental matrix of the variational systemẏ = D x F 0 (t, x(t, z, 0))y associated to the unperturbed system evaluated on the periodic solution x(s, z, 0) such that M(0, z) = Id. Moreover we assume the following hypotheses.
(H − ) There exists an open bounded subset C ⊂ D such that, for ε sufficiently small, every orbit starting in C reaches the set of discontinuity only at its crossing region.
a such that f (z) = 0, for all z ∈Ū \{a} and det(D z F (a)) = 0. Then for ε > 0 sufficiently small there exists a T -periodic solution x(t, ε) of (7) such that x(0, ε) → a as ε → 0. Moreover the linear stability of the periodic solution x(t, ε) is given by the eigenvalues of the matrix D z F (a).
The same arguments for computing the kind of stability of the obtained periodic solution from the eigenvalues of the Jacobian matrix for the continuous piecewise differential systems, also work for the discontinuous piecewise differential systems.
As G(θ) = 0 if and only if θ = ± arccos x 0 + r 0 r 0 and the function arccos(x) takes real values when x ∈ [−1, 1] we have to consider the following cases.
Proof. In fact, calling A = arccos Replacing the expression of A in (12) we obtain The function f 1 (x 0 , y 0 )| A has two real zeros if −d 4/3 π 2/3 r for all r 0 > 0, then the unique possibility in order that f 2 (r 0 ) to be zero, is when −d 4/3 π 2/3 r 4/3 0 + r 2 0 → 0, i. e. r 0 → d 2 π. Moreover if r 0 → d 2 π then x 0 → −d 2 π, and we have a candidate to zero of the averaged function, which is a zero because F (−dπ 2 , dπ 2 ) = 0 by direct computations. This completes the proof of the claim.
This concludes the proof of Theorem 1.

Proof of Theorem 2
Doing the change to cylindrical coordinates x = x, y = r sin θ and z = r cos θ the Michelson discontinuous piecewise linear differential system becomesẋ = r sin θ, Now taking as new independent variable the angle θ we get the system where the prime denotes the derivative with respect to θ.
Since g(θ) is 2π-periodic and g(θ) = g(−θ) then Analogously to the study of the continuous system, we separate the calculation of the averaged function in four subcases, the same ones as in Theorem 3.