Limit cycles for two classes of control piecewise linear differential systems

We study the bifurcation of limit cycles from the periodic orbits of $2n$--dimensional linear centers $\dot{x} = A_0 x$ when they are perturbed inside classes of continuous and discontinuous piecewise linear differential systems of control theory of the form $\dot{x} = A_0 x + \varepsilon \big(A x + \phi(x_1) b\big)$, where $\phi$ is a continuous or discontinuous piecewise linear function, $A_0$ is a $2n\times 2n$ matrix with only purely imaginary eigenvalues, $\varepsilon$ is a small parameter, $A$ is an arbitrary $2n\times 2n$ matrix, and $b$ is an arbitrary vector of $\mathbb{R}^n$.


Introduction and statement of the main results
In control theory are relevant the continuous piecewise linear differential systems of the form with A a m × m matrix, x, b ∈ R m , ϕ : R → R is the continuous piecewise linear function (2) ϕ(x 1 ) = where x = (x 1 , . . . , x m ) T , and the dot denotes the derivative with respect to the independent variable t, the time.
Also in control theory are important the discontinuous piecewise linear differential systems of the form (1) where instead of the function ϕ we have the discontinuous piecewise linear function (3) ψ(x 1 ) = For more details on these continuous and discontinuous piecewise linear differential systems see for instance the books [1,3,10,11,16,18].
The analysis of discontinuous piecewise linear differential systems goes back mainly to Andronov and coworkers [2] and nowadays still continues to receive attention by many researchers. In particular, discontinuous piecewise linear differential systems appear in a natural way in control theory and in the study of mechanical systems, electrical circuits, ... see for instance the book [4] and the references quoted there. These systems can present complicated dynamical phenomena such as those exhibited by general nonlinear differential systems.
One of the main ingredients in the qualitative description of the dynamical behavior of a differential system is the number and the distribution of its limit cycles. The goal of this paper is to study analytically the existence of limit cycles for a class of continuous and a class of discontinuous piecewise linear differential of the form (1).
More precisely, first we consider the class of continuous piecewise linear differential systems (4) with |ε| = 0 a sufficiently small real parameter, where A 0 is the 2n × 2n matrix having on its principal diagonal the following 2 × 2 matrices and zeros in the complement, A is an arbitrary 2n × 2n matrix and b ∈ R 2n \ {0}. Note that for ε = 0 system (4) becomes Moreover, the origin of (5) is a global isochronous center in R 2n , i.e. all its orbits different from the origin are periodic with period 2π.
In a similar way we consider the discontinuous piecewise linear differential systems Our main results on the limit cycles of the continuous and discontinuous piecewise linear differential systems (4) are the following ones. Theorem 1. For |ε| > 0 sufficiently small and if the conditions for applying the averaging theory of first order hold, then at most one limit cycle γ ε of the continuous piecewise linear differential system (4) bifurcates from the periodic orbits of system (5), i.e. γ ε tends to a periodic solution of system (5) when ε → 0. Moreover there are systems (4) with |ε| > 0 sufficiently small having a such limit cycle.
Theorem 1 is proved in section 3.

Theorem 2.
For |ε| > 0 sufficiently small and if the conditions for applying the averaging theory of first order hold, then at most one limit cycle γ ε of the discontinuous piecewise linear differential system (6) bifurcates from the periodic orbits of system (5). Moreover there are systems (6) with |ε| > 0 sufficiently small having a such limit cycle.
Theorem 2 is proved in section 4.
If instead of the matrix A 0 we consider the matrix A 1 where A 1 is the 2n × 2n matrix having on its principal diagonal the following 2 × 2 matrices and zeros in the complement, then we have the following results.
Theorem 3. Assume that the conditions for applying the averaging theory of first order hold. Then this theory does not provide any information about the limit cycles of the continuous piecewise linear differential system Theorem 4. Assume that the conditions for applying the averaging theory of first order hold. Then this theory does not provide any information about the limit cycles of the discontinuous piecewise linear differential system Theorems 3 and 4 are proved in section 5.
Note the difference between the matrices A 0 and A 1 , in the matrix A 0 the non-zero entries are only the odd numbers 1, 3, . . . , 2n − 1, while in the matrix A 1 the non-zero entries are the numbers 1, 2, . . . , n. This difference provides that the continuous and discontinuous piecewise linear differential systems (4) and (6) can have limit cycles detected by the averaging theory, while for the continuous and discontinuous piecewise linear differential systems (7) and (8) the averaging theory cannot detect limit cycles.
According to the results of the averaging theory used it follows that for the control differential systems here studied, the limit cycles that we obtain bifurcate from some periodic orbit of the 2n-dimensional linear differential center (5). This technique of finding limit cycles bifurcating from centers has been intensively studied in dimension 2, see for instance the book of Christopher and Li [9] and the hundreds of references quoted therein.
Other results different to the ones presented here, but which also study the limit cycles of control systems of the form (1) using averaging theory, can be found in [6,7,8,12,14].
The main tools for proving the previous theorems are the extensions of the classical averaging theory for computing periodic solutions of C 2 differential systems to continuous and discontinuous differential systems. In section 2 we summarize the extensions of the averaging theory that we shall use here for proving our results.

First order averaging theory
For the classical averaging theory for finding periodic orbits of differential systems of class C 2 see for instance the chapter 11 of the book of Verhulst [17].
In this section we present first the result on the continuous averaging theory that we will use for proving our Theorems 1 and 3. This theory uses the Brouwer degree of a continuous function and its proof can be found in [5].
Theorem 5. We consider the following differential system and assume that Then, for |ε| = 0 sufficiently small, there exists an isolated T -periodic solution x(t, ε) of system (9)such that x(0, ε) → p as ε → 0.

Remark 6.
Let h : D → R n be a C 1 function with h(p) = 0, where D is an open bounded subset of R n and p ∈ D. If the Jacobian of h at p is not zero, then there exists a neighborhood V of p such that h(z) = 0 for all z ∈V \ {p}, and the Brouwer degree For a proof of Remark 6 see for instance [15].
For proving Theorems 2 and 4 we shall need the following extension of the averaging theory for computing periodic solutions to discontinuous differential systems done in [13].
Theorem 7. We consider the following discontinuous differential system (11) x with is an open subset of R n . We also suppose that g is a C 1 function having 0 as a regular value.
Define the average function h : D → R n as We assume the following conditions.
there exists an open bounded subset C ⊂ D such that, for |ε| > 0 sufficiently small, every orbit starting in C reaches the set of discontinuity only at its crossing regions.

Proof of Theorem 1
The main tool for proving Theorem 1 is the averaging theory of first order for continuous differential systems presented in Theorem 5. In order to use this theorem we need to write the differential system (4) in the normal form (9), and for obtaining this we need to some changes of variables.
With the results presented previously we are able to prove Theorem 1. Since we can choose m sufficiently large to find the zeroes of the average function h in D m it is sufficient to look for them in (0, ∞) × [S 1 × (0, ∞)] n−1 . To calculate the expression of the average function we consider again two cases.
Case 1: 0 < r ≤ 1. In this case the system whose zeros can provide limit cycles of system (4) is (14) h 1 = (a 11 + a 22 + b 1 )πr, h 2 = (a 33 + a 44 )πr 1 , Note that the variables θ 1 , θ 2 , . . . , θ n−1 does not appear explicitly into system (14). Hence, if this system has zeros, it has a continuum of zeros. Therefore the assumption (ii) of the averaging theory, presented in Theorem 5, is not satisfied and this theorem does not provide any information about the limit cycles of system (13).
Case 2: r > 1. Now the system whose zeros can provide limit cycles of system (13) is (15) h 1 = (a 11 + a 22 )πr + b 1 K(r), h 2 = (a 33 + a 44 )πr 1 + (b 3 cos θ 1 + b 4 sin θ 2 )L 2 (r), For each j ∈ {2, 3, . . . , n} we will study the zeros of the system which implies that K is a strictly decreasing function. Moreover lim r→∞ K (r) = 0 what means that K (r) has a horizontal asymptote given by the axis r and then K (r) ≥ 0. Suppose that there exists an r 0 ∈ (1, ∞) such that K (r 0 ) = 0. Then for all r > r 0 we have K (r) < K (r 0 ) = 0, contradiction. Therefore it follows that K (r) = 0 for all r ∈ (1, ∞) and the Inverse Function Theorem guarantees that K is a local diffeomorphism and since that K is a injective function we obtain the global diffeomorphism, ending the proof of this claim.
First we note that in order that the equation h 1 = 0 has solutions with r > 1 it is necessary that b 1 (a 11 + a 22 ) < 0. Moreover K (r) < 0 implies that the graph of K is convex. In the plane of the graph of K(r) the graph of (a 11 + a 22 )πr is a straight line passing through the origin and then both graphs can intersect at most in two points. But if some straight line intercept the graph of K(r) in two points then it cannot pass through the origin, as we can see in Figure 1. Then the equation h 1 = 0 has at most one solution if r > 1, and since that K(r) is a diffeomorphism we can choose the coefficients a 11 , a 22 and b 1 so that this solution exists. We denote this solution by r 0 and we substitute it into the equations h 2(j−1) = 0 and h 2j−1 = 0. Defining Using the two first equations we obtain Substituting these two expressions in the third equation we get Therefore at most there is one solution r j−1 > 0, which provide a unique u j and v j . Since we fixed an arbitrarily j to solve this system, the same reasoning can be applied to each pair of equations h 2(j−1) = 0 and h 2j−1 = 0, concluding that system (15) has at most one solution. Moreover taking conveniently the parameters of the initial system (4) this solution exists and its Jacobian is not zero. Hence at most one limit cycle can bifurcate from the periodic orbits of the center of system (5) when we perturbe it as in system (4), and there are systems for which a such limit cycles exist. This completes the proof of Theorem 1.

Proof of Theorem 2
This section is devoted to prove Theorem 2. According to Theorem 7 the same kind of arguments used for proving Theorem 1 can be applied to the discontinuous system (6), obtaining that the average function h = (h 1 , h 2 , . . . , h 2(n−1) , h 2n−1 ) : D m → R n−1 defined in (12) is (18) for j = 2, 3, . . . , n, where if j is even, if j is odd.
In fact if we define Note that I j is a constant real number different from zero, and hence h 1 is a straight line, and then system (18) has at most one positive zero. Moreover if we choose conveniently the coefficients b 1 a 11 and a 22 we can find a simple positive zero of system (18). This completes the proof of Theorem 2.