A note on linear differential equations with periodic coefficients.

A note on linear differential equations with periodic coefficients. Maite Grau (1) and Daniel Peralta-Salas (2) (1) Departament de Matemàtica. Universitat de Lleida. Avda. Jaume II, 69. 251 Lleida, Spain. E mail: mtgrau@matematica.udl.es (2) Departamento de Matemáticas. Universidad Carlos III. 28911 Leganés (Madrid), Spain. E mail: dperalta@math.uc3m.es Abstract We consider a linear homogeneous differential equation of the form ẋ = A(t)x where A(t) is a square matrix of C 1, real and T -periodic functions, with T >. We give several criteria on the matrix A(t) to prove asymptotic stability of the trivial solution to equation ẋ = A(t)x. These criteria allow us to show that any finite configuration of cycles in R n can be realized as hyperbolic limit cycles of a polynomial vector field. 2 AMS Subject Classification: 34A3, 34D8, 37D5, 34C7. Key words and phrases: linear differential equations, characteristic multipliers, hyperbolicity, limit cycles. 1 Introduction In this note we are concerned with stability properties of the trivial solution to linear differential equations with periodic coefficients ẋ = A(t) x, (1) where x R n and A(t) is an n n square matrix of C 1 real functions, which is T -periodic in t, with T >. As usual the dot means derivation with respect to the real independent variable t. Let us briefly recall the definition of Lyapunov stability of the trivial solution of (1) as given in [1]. We denote by x the modulus or distance of the vector x to the origin. The solution x = is called Lyapunov stable if for any ε >, there exists a δ = δ(ε) > such that if x < δ, the solution x(t) of (1) with The first author is partially supported by a DGICYT grant number MTM25-698- C2-2. The second author acknowledges financial support from the Spanish MEC through the Juan de la Cierva program. 1

initial condition x() = x satisfies x(t) < ε for any t. The solution x = is called asymptotically stable if there exists a b > such that x < b implies x(t) as t +. The solution x = is unstable if it is not stable. The stability of the trivial solution to (1) can be studied by the modulus of the eigenvalues associated to its monodromy matrix. Let Φ(t) be the fundamental matrix solution of (1) such that Φ() is the identity matrix. Usually, Φ(t) is called the matriciant of (1). The monodromy matrix of (1) is given by Φ(T ). As described in [1], if all the eigenvalues of Φ(T ) have modulus lower or equal to 1, then the trivial solution of (1) is Lyapunov stable and if all the eigenvalues of Φ(T ) have modulus strictly lower than 1, then the trivial solution of (1) is asymptotically stable. If all the eigenvalues of Φ(T ) have modulus different from 1, then the trivial solution is structurally stable and it is said to be hyperbolic. We recall that a real square matrix A is negative (positive) definite if x T A x < (> ) for any nonzero vector x, where the superscript T denotes transposition. We observe that if A is negative definite, then the real part of the eigenvalues of A is strictly negative. The reciprocal is not true: there are matrices with the real part of all their eigenvalues negative but which do not satisfy x T A x < for any x, for instance, the matrix A = ( 1 4 2 ), has its both eigenvalues 1 and 2 real and negative, but the quadratic form (x 1, x 2 ) A (x 1, x 2 ) T = x 2 1 +4x 1 x 2 2x 2 2, is not negative for all nonzero (x 1, x 2 ) (e.g. take the vector (1, 1)). Of course a symmetric matrix whose eigenvalues are negative satisfies to be negative definite. The stability properties of the matrix A(t) and the monodromy matrix Φ(T ) are not generally related. In fact there are several examples which show that the sign of the real part of the eigenvalues of the matrix A(t) does not determine the stability of the solution x = of (1), see e.g. [4, 5, 6]. Therefore the only general procedure to study the stability of x = is solving Eq. (1), a task which is generally extremely difficult. This fact suggests the interest of providing sufficient conditions on the matrix A(t) ensuring that the trivial solution of Eq. (1) is hyperbolic and asymptotically stable. The idea of this work is to obtain some of these sufficient conditions over the matrix A(t). Apart from some theorems that we will review in sections 2 and 3 the literature on this problem is rather scarce. Furthermore no results in this direction appear in the classical textbooks on dynamical systems, as Chicone [4], Hale [1], Lefschetz [11] or Perko [14]. The goal of this paper is to fill this gap and to bring the reader s attention to this important problem. Equations of type (1) are very important in applications because they appear when studying the stability of limit cycles of vector fields in R n+1. Indeed let X be a C 1 vector field with a periodic orbit γ. Let us endow a neighborhood of γ with a local coordinates system defined by: s S 1 is a coordinate on the cycle and x R n are coordinates on a section Σ orthogonal to γ, in particular γ = {x = }. The normal variational equation expressed in these coordinates 2

has the form (1) and it is well known [1] that the stability of the trivial solution of Eq. (1) determines the stability and hyperbolicity of the limit cycle γ of X. Combining Theorem 2 with some ideas introduced in Reference [13] and a Theorem obtained in Reference [8] we can prove the following result, which solves the open problem stated in [13]: Can any finite configuration of cycles be realized as hyperbolic limit cycles of a polynomial vector field? Theorem 1. Any finite configuration of (smooth) cycles in R n can be realized (up to global diffeomorphism) as hyperbolic and asymptotically stable limit cycles of a polynomial vector field. This paper is organized as follows. Section 2 contains some preliminary material on the problem that we study. Several results giving sufficient conditions over the matrix A(t) in order to ensure the hyperbolicity and asymptotic stability of the corresponding Eq. (1) are proved in Section 3. Finally, in Section 4 we prove Theorem 1. 2 Preliminary results The following theorem appears in some books related to control theory and Liapunov stability and is contained in Persidskiĭ and Malkin s theorems, see [12]. Theorem 2. [12] satisfying: Let Q(t) and P (t) be two square, of class C 1, real matrices Q = P + P A + A T P, (2) and such that Q(t) Q, Q being a negative definite constant matrix, and P 1 P (t) P 2, P 1, P 2 being two positive definite constant matrices, for any t R. Then the trivial solution to Eq. (1) is hyperbolic and asymptotically stable. The proof of Theorem 2 goes through considering the function G = x T P (t)x which gives a strict Liapunov function for Eq. (1). That is, we have that Ġ = x T Q(t)x < for all nonzero x. The existence of a strict Liapunov function gives the hyperbolicity and asymptotic stability of Eq. (1), because all its nonzero solutions x(t) need to satisfy that x(t ) < x(). The assumptions on P (t) and Q(t) are necessary in order to apply Liapunov s stability theorem for non-autonomous systems, see e.g. [11]. These assumptions are automatically satisfied if P (t) and Q(t) are T -periodic matrices, positive definite and negative definite respectively for any t [, T ). In the particular case in which the matrix A is constant, Liapunov Theorem ensures that if the eigenvalues of A have negative real part then given any matrix Q constant and negative definite, there exists a matrix P positive definite such that Q = P A + A T P, see for instance [9, 11]. Moreover, this matrix P can be 3

computed through: P = e sat Q e sa ds. The following theorem is stated and proved in paper [7] and is a generalization of Liapunov theorem to matrices A(t) with periodic entries. Theorem 3. [7] Suppose A(t) is a continuous and T -periodic matrix. 1. If all the eigenvalues of the monodromy matrix of Eq. (1) lie in the unit disk { λ < 1} then, for every continuous matrix Q(t) on [, T ], there is a unique solution P (t) to the boundary value problem: P + P A + A P = Q, P () = P (T ). < t < T Moreover, if Q(t) = Q (t) is negative definite, then P (t) = P (t) is positive definite, for t [, T ]. 2. Assume Q(t) is a continuous and negative definite matrix on [, T ] and the solution P (t) to (3) is such that P (t) = P (t) and P () is positive definite, then all the eigenvalues of the monodromy matrix for Eq. (1) lie in the unit disk { λ < 1}. This theorem is stated for complex matrices and A is the conjugate transpose of A. Since we are only considering real matrices, we have that A = A T. We remark that in the proof of Theorem 3 the matrix P (t) is constructed in terms of the fundamental matrix Φ(t) solution to Eq. (1). It establishes the equivalence between asymptotic stability of the trivial solution to Eq. (1) and the existence of matrices Q(t) (negative definite), P (t) (positive definite) satisfying Eq. (3). However this theorem does not give a way to construct these matrices unless Eq. (1) is solved. Note that the second part of Theorem 3 complements Theorem 2, although the matrices Q(t) and P (t) do not need to be symmetric and periodic to apply Theorem 2. In light of these theorems an effective technique to study the stability of the trivial solution to Eq. (1) is to prove the existence of such matrices P (t) and Q(t). In the next section we will exploit this idea to obtain several stability criteria. 3 Criteria for hyperbolicity and asymptotic stability of Eq. (1) This section contains several criteria to ensure the existence of the matrices Q(t) and P (t) described in Theorem 2 and, thus, we can deduce that all the eigenvalues of the monodromy matrix associated to Eq. (1) have modulus strictly lower than 1. (3) 4

A simple observation, which appears in Hahn s stability book [9], although it is not well known, is the following Proposition. Proposition 1. If A(t) is negative definite for any t [, T ) then x = is hyperbolic and asymptotically stable. Proof. Just note that P (t) = 1 and Q(t) = A(t) + A(t) T verify the assumptions in Theorem 2. Most of the known sufficient conditions on the matrix A(t) to ensure the asymptotic stability of the trivial solution to (1) assume a decomposition of the matrix A(t) in the form A(t) = A +A 1 (t), where A is a constant matrix whose eigenvalues have all negative real part and A 1 (t) is generally a small matrix satisfying certain conditions, see [1, 3, 7, 11, 12]. Our criteria do not involve such a decomposition and concern the relation between A(t) and the solutions of (1). Proposition 2. If there exists a real and constant matrix Q, symmetric and negative definite such that Ȧ T Q + QA = and the eigenvalues of A(t) have all negative real part for t [, T ), then the solution to (2) is: P (t) = e sa(t)t Qe sa(t) ds, which is symmetric and positive definite. Accordingly x =, solution to (1), is hyperbolic and asymptotically stable. Proof. Let us define the function φ(s; t) = e sa(t)t Qe sa(t). We note that φ(; t) = Q and, since the eigenvalues of A(t) have all negative real part for t [, T ), lim s φ(s; t) =. Therefore, P (t) is well-defined for all t (just apply Liapunov theorem to each value of t). Let us consider P + P A + A T P. Note that, under the hypotheses: P (t) = e sa(t)t s[ȧ(t)t Q + QȦ(t)]esA(t) ds =. On the other hand, we have that s φ(s; t)e sa(t)t [A(t) T Q+QA(t)]e sa(t). Therefore, P A + A T P = = e sa(t)t [QA(t) + A(t) T Q]e sa(t) ds s φ(s; t) ds = φ(; t) lim φ(s; t) = Q. s We have that the matrix P satisfies Eq. (2). The fact that it is symmetric is proved by: P (t) T = (e sa(t)t Qe sa(t)) T ds = e sa(t)t Q T e sa(t) ds P (t), 5

because Q T = Q. In order to show that P (t) is positive definite, we consider x and the product x T P x = (esa(t) x) T ( Q)(e sa(t) x)ds. We take y(s; t) = e sa(t) x and we have x T P x = y(s; t) T ( Q)y(s; t)ds. Since the matrix Q is negative definite, we have that y(s; t) T ( Q)y(s; t) > for all s and t. When integrating, we get that x T P x = y(s; t) T ( Q)y(s; t)ds >. So, P (t) is positive definite. Proposition 3. Assume that x T A(t)x ɛ (t) + ɛ 1 (t)x 2 and x T C(t)x ɛ (t) + ɛ 1 (t)x 2 for x small enough, where C(t) := Ȧ(t) + A2 (t), ɛ (t), ɛ (t) and ɛ 1 (t) < ɛ 1 (t)/(2k) with K := δ + max t [,T ) ɛ 1 (t), for any t [, T ) and some δ >, then x = is hyperbolic and asymptotically stable as solution of Eq. (1). Proof. Let P (t) = k 1 I + k 2 A(t) with k 1 R + and k 2 R. We have that G = x t P (t)x = k 1 x 2 + k 2 x t A(t)x (k 1 k 2 ɛ 1 (t))x 2 k 2 ɛ (t) (k 1 k 2 ɛ 1 (t))x 2. If we take k 1 = k 2 K, we have that G >. Hence, G is a Lyapunov function (depending on t). We study now the matrix Q(t) = P (t)+p (t)a(t)+a T (t)p (t). Since x T A T Ax = (Ax) T Ax we deduce that x T ( A + A T A + A 2 )x x T (Ȧ + A 2 )x ɛ (t) + ɛ 1 (t)x 2. We get that: x T Q(t)x 2k 1 x T A(t)x + k 2 x T C(t)x (2k 1 ɛ (t) k 2 ɛ (t)) + (2k 1 ɛ 1 (t) k 2 ɛ 1 (t))x 2 (2k 1 ɛ 1 (t) k 2 ɛ 1 (t))x 2. Taking into account the definition of k 1 and the hypotheses, we conclude that x T Q(t)x k 2 (2Kɛ 1 (t) ɛ 1 (t)) <. Applying Theorem 2 the result follows. Corollary 1. If A is negative semidefinite, i.e. x t Ax, and Ȧ + A2 is positive definite, then x = is hyperbolic and asymptotically stable as solution to Eq. (1). Corollary 2. If the minimum of ɛ 1 (t) in t [, T ) is positive, then for ɛ 1 (t) small enough (in t [, T )), x = is hyperbolic and asymptotically stable as solution to Eq. (1). We note that ɛ 1 (t) small enough in Corollary 2 means that A(t) is not negative definite but it is nearly so. The following examples illustrate some of the criteria previously obtained and show that they are easily verifiable in particular cases. 6

Example. We consider the periodic matrix cos A(t) = 4 t sin 2 t 2 cos 4 t + 2 sin 2 t 1/2 2 cos 4 t + 2 sin 2 t 1/2 2 It is symmetric and its eigenvalues are both real and negative for any value of t, hence it is negative definite. Proposition 1 implies that {x = } is a hyperbolic and asymptotically stable equilibrium of Eq. (1). However, obtaining this result solving Eq. (1) cannot be done by any elementary method. A similar discussion applies to the periodic matrix A(t) = 1 cos 2 t 1+cos 4 t 2+3 cos 2 t cos 4 t sin t/2 1+cos 4 t e 2+3 cos 2 t cos 4 t 1+cos 4 t e sin t/2 3e sin t Example. Let us consider any T -periodic function a(t) of class C 1, and let σ, σ 1, σ 2 be three real numbers such that σ <, σ 1 <, σ 2 < and σ 1 σ 2 > 1. We define the matrices ( ) ( ) σ a(t) σ A(t) = 2 a(t) σ1 1, Q =, σ 1 a(t) σ + a(t) 1 σ 2 and we note that A(t) has eigenvalues σ ± a(t) 1 σ 1 σ 2, which have negative real part, and that Q is symmetric and negative definite. Moreover, A(t) satisfies that Ȧ T Q + QȦ =. Therefore, we deduce by Proposition 2, that the trivial solution to the system ẋ = A(t)x is hyperbolic and asymptotically stable. We note that this result cannot be obtained directly solving ẋ = A(t)x due to the generality in the form of the matrix A(t). A particular case of Eq. (1) which usually appears in applications is the second order differential equation ẍ + p(t)ẋ + q(t)x =, (4) where p(t) and q(t) are C 1 and T -periodic functions. Let us finish this section establishing a criterion for asymptotic stability of second order differential equations. An example is also provided to show that the assumptions of the criterion can be easily verified. Proposition 4. Let p(t) and q(t) be C 1 and T -periodic functions which verify 1. T p(t)dt > 2. T r(t)dt 3. T r(t) dt 4 T where r(t) q(t) 1 2ṗ(t) 1 4 p(t)2 is non-identically zero. Then the solution x = to Eq. (4) is hyperbolic and asymptotically stable... 7

Proof. First let us simplify Eq. (4) with the following change of variable x(t) = y(t)e 1 2 This transforms the equation into the form t p(t)dt. ÿ + r(t)y =, (5) where r(t) is defined in the statement of the proposition. Under hypotheses 2 and 3 Bellman proves ([1], pages 123-125) that all solutions y(t) to Eq. (5) are bounded when t ±. Since p(t) satisfies hypothesis 1 we get that e 1 t 2 p(t)dt when t +. Combining both results we conclude that any solution x(t) to Eq. (4) verifies x(t) as t + and hence all the eigenvalues of the monodromy matrix are 1. Hyperbolicity follows because otherwise there would be some solutions for which x(t) is constant for any t R. Example. In order to illustrate Proposition 4, let us consider Eq. (4) with p(t) = cos 2 (t) and q(t) = q R. The first hypothesis of Proposition 4 is fulfilled since π p(t) dt π/2. The value of the corresponding function r(t) gives that π r(t)dt = π(q 3/32). Therefore, the second hypothesis is satisfied when q 3/32. Moreover, an easy numerical analysis shows that if q 343/75, the third hypothesis is also satisfied. Hence, we deduce that if 3/32 q 343/75, the trivial solution is hyperbolic and asymptotically stable, as a consequence of Proposition 4. 4 Proof of Theorem 1 Let C be any finite configuration of smooth cycles in R n+1, possibly linked and knotted when n = 2. Following [13] we apply a modification of Nash-Tognoli s theorem [2] to show that there is a global diffeomorphism H : R n+1 R n+1 such that H(C) is a non-singular algebraic set, so it is defined by f 1 (x 1,..., x n+1 ) =,. f n (x 1,..., x n+1 ) =, where f 1,..., f n are polynomials satisfying that rank(df 1,..., df n ) H(C) = n. (6) Define the following polynomial vector field in R n+1 : X = [ (df 1... df n )] i F, 8

where is the Hodge star operator and i denotes the index raising operator. The function F is defined as F = 1 2 (f 2 1 +... + f 2 n). This vector field is a slight modification of the one used in [13]. Furthermore, following the procedure of [13], it is not difficult to prove that the cycles in H(C) are the only periodic orbits, in fact asymptotically stable limit cycles, of X. Let us now show that these limit cycles are hyperbolic and hence structurally stable under small perturbations of X. First compute the following derivative with respect to the variable t: f i = X f i = n ( f i f j )f j, j=1 i = 1,..., n. This equation can be written in matrix notation as where and Df(x)X(x) = k(x)f(x), f(x) = f 1 (x).. f n (x), ( f 1 ) 2 f 1 f n k(x) =.., f 1 f n ( f n ) 2 which is called the cofactor matrix. Lemma 1. The matrix k(x) evaluated at any limit cycle γ H(C) is negative definite. Proof. Condition (6) implies that the functions A k := det( f i f j ) H(C), 1 i, j k have ( 1) k sign for k = 1,..., n. Since k(x) is symmetric it follows that the matrix k(γ(t)) is negative definite for any limit cycle γ H(C). In [8] it was proved that the stability properties of the trivial solution to ẋ = k(γ(t))x, (7) determine the stability properties of the limit cycle γ of X. Applying Lemma 1 and Theorem 2, with P = I/2 and Q(t) = k(γ(t)), we get that the trivial solution to (7) is hyperbolic, thus concluding that the limit cycles of X are hyperbolic. 9

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