Persistence of periodic solutions for higher order perturbed differential systems via Lyapunov–Schmidt reduction

In this work we first provide sufficient conditions to assure the persistence of some zeros of functions having the form g(z,ε)=g0(z)+∑i=1kεigi(z)+O(εk+1), for |ε|≠0 sufficiently small. Here gi:D→Rn, for i=0,1,…,k, are smooth functions being D⊂Rn an open bounded set. Then we use this result to compute the bifurcation functions which allow us to study the periodic solutions of the following T-periodic smooth differential system x′=F0(t,x)+∑i=1kεiFi(t,x)+O(εk+1),(t,z)∈S1×D. It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions Z, dim⁡(Z)⩽n. We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5.

for |ε| = 0 sufficiently small. Here g i : D → R n , for i = 0, 1, . . . , k, are smooth functions being D ⊂ R n an open bounded set. Then we use this result to compute the bifurcation functions which allow us to study the periodic solutions of the following T-periodic smooth differential system It is assumed that the unperturbed differential system has a sub-manifold of periodic solutions Z, dim(Z) n. We also study the case when the bifurcation functions have a continuum of zeros. Finally we provide the explicit expressions of the bifurcation functions up to order 5.
Keywords: Lyapunov-Schimidt reduction, bifurcation theory, periodic solution, limit cycle, nonlinear differential system Mathematics Subject Classification numbers: 34C29, 34C25, 37G15 (Some figures may appear in colour only in the online journal)

Introduction
This work contains two main results. The first one (see theorem A) provides sufficient conditions to assure the persistence of some zeros of smooth functions g : R n × R → R n having the form The second one (see theorem B) provides sufficient conditions to assure the existence of periodic solutions of the following differential system Here S 1 = R/T , for some T > 0, and the assumption t ∈ S 1 means that the system is T-periodic in the variable t. As usual δ 1 (ε) = O (δ 2 (ε)) means that there exists a constant c 0 > 0, which does not depends on ε, such that |δ 1 (ε)| c 0 |δ 2 (ε)| for ε sufficiently small (see [16]). It is assumed that either g(z, 0) vanishes in a submanifold of Z ⊂ D, or that the unperturbed differential system x = F 0 (t, x) has a submanifold Z ⊂ D of T-periodic solutions. In both cases dim(Z) n. The second problem can be often reduced to the first problem, standing as its main motivation.
Regarding the first problem, assume that for some z * ∈ Z, g(z * , 0) = 0. We shall study the persistence of this zero for the function (1), g(x, ε), assuming that |ε| = 0 is sufficiently small. By persistence we mean the existence of continuous branches χ(ε) of simple zeros of g(x, ε) (that is g(χ(ε), ε) = 0) such that χ(0) = z * . It is well known that if the n × n matrix ∂ x g(z * , 0) (the Jacobian matrix of the function g with respect to the variable x evaluated at x = z * ) is nonsingular then, as a direct consequence of the implicit function theorem, there exists a unique smooth branch χ(ε) of zeros of g(x, ε) such that χ(0) = x * . However if the matrix ∂ x g(x * , 0) is singular (has non trivial kernel) we have to use the Lyapunov-Schmidt reduction method to find branches of zeros of g (see, for instance, [8]). Here we generalize some results from [4,5,14], providing a collection of functions f i , i = 1, . . . , k, each one called bifurcation function of order i, which control the persistence of zeros contained in Z.
The second problem goes back to the works of Malkin [17] and Roseau [18], whose have studied the persistence of periodic solutions for the differential system (2) with k = 1. Let x(t, z, ε) denote its solution such that x(0, z, ε) = z. In order to find initial conditions z ∈ D such that the solution x (t, z, ε) is T-periodic we may consider the function g(z, ε) = z − x(T, z, ε), and then try to use the results previously obtained from the first problem. Indeed, if Z ⊂ D is a submanifold of T-periodic solutions of the unperturbed system x = F 0 (t, z) then g(z, 0) vanishes on Z. When dim(Z) = n this problem is studied at an arbitrary order of ε, see [9,11], even for nonsmooth systems. When dim(Z) < n, this approach has already been used in [4], up to order 1, and in [5,6], up to order 2. In [14] this approach was used up to order 3 relaxing some hypotheses assumed in those previous 3 works. In [10], assuming the same hypotheses of [4][5][6], the authors studied this problem at an arbitrary order of ε. Here, following the ideas from [11,14], we improve the results of [10] relaxing some hypotheses and developing the method in a more general way.
In summary, in this paper we use the Lyapunov-Schmidt reduction method for studying the zeros of functions like (1) when the implicit function theorem cannot be directly applied. Another useful tool that we shall use to deal with this problem is the Browder degree theory (see the appendix B), which will allow us to provide estimates for these zeros. Then we apply these previous results for studying the periodic solutions of differential systems like (2) through their bifurcation functions, provided by the higher order averaging theory. This paper is organized as follows. In section 2 we state our main results: theorem A, in section 2.1, dealing with bifurcation of simple zeros of the equation g(z, ε) = 0; and theorem B, in section 2.2, dealing with bifurcation of limit cycles of the differential equation x = F(t, z, ε). In sections 3 and 4 we prove theorems A and B, respectively. In section 5, as an application of theorem B, we study the birth of limit cycles in a 3D polynomial system. Finally, in section 6, we study the case when the averaged functions have a continuum of zeros. In this last situation we also provide some results about the stability of the limit cycles.

Statements of the main results
Before we state our main results we need some preliminary concepts and definitions. Given p, q and L positive integers, γ j = (γ j1 , . . . , γ jp ) ∈ R p for j = 1, . . . , L and z ∈ R p . Let G : R p → R q be a sufficiently smooth function, then the Lth Frechet derivative of G at z is denoted by ∂ L G(z), a symmetric L-multilinear map, which applied to a 'product' of L p-dimensional vectors denoted as · L j=1 γ j ∈ R pL gives The above expression is indeed the Gâteaux derivative We take ∂ 0 as the identity operator.

The Lyapunov-Schmidt reduction method
We consider the function As usual Cl(V) denotes the closure of the set V.
As the main hypothesis we assume that (H a ) the function g 0 vanishes on the m-dimensional submanifold Z of D.
Using the Lyapunov-Schmidt reduction method we shall develop the bifurcation functions of order i, for i = 1, 2, . . . , k, which control, for |ε| = 0 small enough, the existence of branches of zeros z(ε) of (3) bifurcating from Z, that is from z(0) ∈ Z. With this purpose we introduce some notation. The functions π : R m × R n−m → R m and π ⊥ : R m × R n−m → R n−m denote the projections onto the first m coordinates and onto the last n − m coordinates, respectively. For a point z ∈ D we also consider z = (a, b) ∈ R m × R n−m .
For i = 1, 2, . . . , k, we define the bifurcation functions f i : Cl(V) → R m of order i as where γ i : V → R n−m , for i = 1, 2, . . . , k, are defined recurrently as Here S l is the set of all l-tuples of non-negative integers (c 1 , c 2 , · · · , c l ) satisfying c 1 + 2c 2 + · · · + lc l = l, L = c 1 + c 2 + · · · + c l , and S i is the set of all (i − 1)-tuples of nonnegative integers satisfying . We clarify that S 0 = S 0 = ∅, and when c j = 0, for some j, then the term γ j does not appear in the 'product' · l j=1 γ j (α) cj .
Recently in [15] the Bell polynomials were used to provide an alternative formula for recurrences of kind (5) and (7). This new formula can make easier the computational implementation of the bifurcation functions (6).
The next theorem is the first main result of this paper. For sake of simplicity, we take f 0 = 0. Theorem A. Let ∆ α denote the lower right corner (n − m) × (n − m) matrix of the Jacobian matrix D g 0 (z α ). In additional to hypothesis (H a ) we assume that satisfying F k (a ε , ε) = 0; (iv) there exist a constant P 0 > 0 and a positive integer l (k + r + 1)/2 such that Then, for |ε| = 0 sufficiently small, there exists z(ε) such that g(z(ε), Theorem A is proved in section 3.
In the next corollary we present a classical result in the literature, which is a direct consequence of theorem A. Corollary 1. In addiction to hypothesis (H a ), assume that f 1 = f 2 = · · · = f k−1 = 0, that is r = k, and that for each α ∈ Cl(V), det(∆ α ) = 0. If there exists α * ∈ V such that f k (α * ) = 0 and det (Df k (α * )) = 0, then there exists a branch of zeros z(ε) with g(z(ε), ε) = 0 and Corollary 1 is proved in section 3.

Continuation of periodic solutions
As an application of theorem A we study higher order bifurcation of periodic solutions of the following T-periodic C k+1 , k 1 , differential system Here . . } and the prime denotes derivative with respect to time t. Now the manifold Z, defined in (4), is seen as a set of initial conditions of the unperturbed system In fact we shall assume that all solutions of the unperturbed system starting at points of Z are T-periodic, recall that the dimension of Z is m n. Formally, let x(·, z, 0) : [0, t z ) → R n denote the solution of (9) such that x(0, z, 0) = z, we assume that (H b ) Z ⊂ D and for each α ∈ Cl(V) the solution x(t, z α , 0) of (9) is T-periodic.
As usual x(·, z, ε) : [0, t (z,ε) ) → R n denotes a solution of system (8) such that x(0, z, ε) = z. Moreover, let Y(t, z) be a fundamental matrix solution of the linear differential system For sake of simplicity when z = z α ∈ Z we denote Y α (t) = Y(t, z α ). Given fundamental matrix solution Y(t, z), the averaged functions of order i, g i : Cl(V) → R n , i = 1, 2, . . . , k, of system (8) is defined as where Using now the functions g i as stated in (11) we define the functions f i , F k , and γ i given by (5)-(7), respectively. Recently in [15] the Bell polynomials were used to provide an alternative formula for the recurrence (12). This new formula can also make easier the computational implementation of the bifurcation functions (11).
The next theorem is the second main result of this paper. Again, for sake of simplicity, we take f 0 = 0.
In the next corollary we present a classical result in the literature, which is a direct consequence of theorem B.

Corollary 2. In addiction to hypothesis (H b ) we assume that
Corollary 2 is proved in section 4. An application of theorem B is performed in section 5. It is worth to emphasize that theorem B is still true when m = n. In fact, assuming that V is an open subset of R n then Z = Cl(V) ⊂ D and the projections π and π ⊥ become the identity and the null operator respectively. Moreover, in this case the bifurcation functions f i : V → R n , for i = 1, 2, . . . , k, are the averaged functions f i (α) = g i (α) defined in (11). Thus we have the following corollary, which recover the main result from [11].
Corollary 3. Consider m = n, z α = α ∈ Z and the hypothesis (H b ). Thus the result of theorem B holds without any assumption about ∆ α .

Proof of theorem A and corollary 1
A useful tool to study the zeros of a function is the Browder degree (see the appendix B for some of their properties).
is the Jacobian determinant of g at z. This assures that the set Z g is formed by a finite number of isolated points. Then the Brouwer degree of g at 0 is As one of the main properties of the Brouwer degree we have that: The next result is a key lemma for proving theorem A.

Lemma 4. Let V be an open bounded subset of R m . Consider the continuous functions
The above lemma provides a stratagem to track zeros of the perturbed function f (x, ε) using a shrinking neighborhood around the zeros of g(x, ε) that preserves its Brouwer degree. The way it works can be blurry at first, so to make it clear we present the following example: (13)). Further- Therefore from the previous lemma we know that d B ( f (·, ε), V ε , 0) = 1. From the above property of the Brouwer degree we conclude that there exists α ε ∈ V ε such that f (α ε , ε) = 0. Now we recall Faà di Bruno's Formula (see [12]) about the l th derivative of a composite function.

Faà di Bruno's Formula
If u and v are functions with a sufficient number of derivatives, then The remainder of this section consists in the proof of theorem A, which is split in several claims, and the proof corollary 1 Proof of theorem A. We consider g = (πg, π ⊥ g), Thus applying the implicit function theorem it follows that there exists an open neighborhood From here, this proof will be split in several claims.
Firstly, it is easy to check that (∂β/∂ε)(α, 0) = γ 1 (α). Now, for some fixed i ∈ {1, 2, . . . , k}, we assume by induction hypothesis that ( In what follows we prove the claim for s = i. Consider Expanding each function ε → π ⊥ g i α, β(α, ε) in Taylor series we obtain Applying Faà di Bruno's formula we obtain Substituting (15) in (14) we get Since the previous equation is equal to zero for |ε| sufficiently small, the coefficients of each power of ε vanish. Then for 0 i k and (α, This equation can be rewritten as Finally, using the induction hypothesis, equation (16) becomes This concludes the proof of claim 1.
So computing its ith-derivative, 0 i k, in the variable ε, we get Taking ε = 0 and l = i − j we obtain Finally using Faà di Brunno's formula and claim 1 we have This concludes the proof of claim 2.
Using claim 2 the function δ(α, ε) can be expanded in power series of ε as and, from hypothesis (ii), we have From the continuity of the functions δ and G k and from the compactness of the set Cl(V) × [−ε 0 , ε 0 ] we know that R(ε 0 ) < ∞ and R(0) = 0. In order to study the zeros of δ(α, ε) we use lemma 4, for proving the following claim.
Applying lemma 4 for g = δ , as defined in (17) This concludes the proof of theorem A. □ Proof of corollary 1. The basic idea of the proof is to show that F k (α) satisfies all the hypotheses of theorem A. From hypotheses, Hence the proof follows directly from theorem A. □

Proof of theorem B and corollary 2
The next result is needed in the proof of theorem B.

Lemma 5 (Fundamental lemma)
. Let x(t, z, ε) be the solution of the T-periodic C k+1 differential system (8) such that x(0, z, ε) = z. Then the equality Proof. The solution x(t, z, ε) can be written as , and Moreover the result about differentiable dependence on parameters implies that ε → x(t, z, ε) is a C k+1 map. So, for i = 0, 1, . . . , k − 1, we compute the Taylor expansion of F i (t, x(t, z, ε)) around ε = 0 as Using Faà di Bruno's formula to compute the l-derivatives of F i (t, x(t, z, ε)) in the variable ε we get where

The integral equation (28) is equivalent to the Cauchy problem
which has a unique solution given by we obtain This concludes the proof of the lemma □ Proof of theorem B. Let x(·, z, ε) : [0, t (z,ε) ) → R n denote the solution of system (8) such that x(0, z, ε) = z. By theorem 8.3 of [1] there exists a neighborhood U of z and ε 1 sufficiently small such that t (z,ε) > T for all (z, ε) ∈ U × (−ε 1 , ε 1 ). Let h(z, ε) : U × (−ε 1 , ε 1 ) → R n be the displacement function defined as Clearly x(·, z, ε), for some (z, ε) ∈ U × (−ε 1 , ε 1 ), is a T-periodic solution of system (8) if and only if h(z, ε) = 0. Studying the zeros of (29) is equivalent to study the zeros of From lemma 5 we have for all (t, z) ∈ S 1 × D, where y i is defined in (12). Hence substituting (31) into (30) it follows that and, for i = 1, 2, . . . , k, the function g i is defined in (11). From hypothesis (H b ) we know that g 0 (z) vanishes on the manifold Z, therefore hypothesis (H a ) holds for the function (32). Moreover which from hypothesis has its lower right corner (n − m) × (n − m) matrix as being a nonsingular matrix ∆ α . Hence the result follows directly by applying theorem A. □

Birth of a limit cycle in a 3D polynomial system
Consider the following 3D autonomous polynomial differential systeṁ In the next proposition, as an application of theorem B, we provide sufficient conditions for the existence of an isolated periodic solution for the differential system (33).

Proposition 6.
For |ε| > 0 sufficiently small system (33) has an isolated periodic solution We emphasize that the expression (34) is not saying that the period of the solution ϕ(t, ε) is 2π. That is because we cannot assure the period of the order ε functions.

Averaged functions with a continuum of zeros
One of the main difficulties in applying the averaging method for finding periodic solutions is to compute the zeros of the averaged function associated to the differential system. In this section we are going to show how theorems A and B can be combined in order to deal with this problem. To be precise, consider the T-periodic differential system x = F(t, x, ε) as defined in (8), with F 0 = 0. Note that Y(t, z) = Id for every t ∈ S 1 and z ∈ D.
As shown in the proof of theorem B, x(t, z, ε) is a T-periodic solution of (8) if and only if z is a zero of the displacement function h, defined in (29). In this case h(z, ε) = g(z, ε), which reads where the averaged functions g i (z), for i = 1, 2, . . . , k, are defined in (11). In order to apply theorem B we first compute as defined in (6), and then we try to find a ε ∈ V such that F k (a ε , ε) = 0. After that, if all the hypotheses of theorem B are fulfilled we obtain, from its proof, the existence of a branch of zeros z(α) of the displacement function (38). This task can be very complicate because there is no general method to find a ε . Although if there exist r ∈ {1, . . . , k}, an open subset V ⊂ D, and a smooth function β : Cl( V) → D such that g 1 = . . . = g r−1 = 0, g r = 0, and g r α, β α = 0 for all α ⊂ V then theorem A may be used to reduce the dimension of system (39), helping then to find the solution a ε . This strategy is a general method which generalizes the results obtained in [7]. This procedure will be illustrated in the next section.

Maxwell-Bloch system
In nonlinear optics, the Maxwell-Bloch equations are used to describe laser systems. For instance, in [2], these equations were obtained by coupling the Maxwell equations with the Bloch equation (a linear Schrödinger like equation which describes the evolution of atoms resonantly coupled to the laser field). Recently in [13], it was identified weak foci and centers in the Maxwell-Bloch system, which can be written aṡ For c = 0 the differential system (40) has a singular line {(u, v, w)|u = 0, v = 0}; for c = 0 and ac(δ − ab) 0 the differential system (40) has one equilibrium p 0 = (0, 0, δ); and for c = 0 and ac(δ − ab) > 0 the differential system (40) has three equilibria p ± = ±u * , ±v * , w * and p 0 where Using the above strategy we shall prove the following result: and ε a small parameter. Then for |ε| = 0 sufficiently small the Maxwell-Bloch differential system (40) has an isolated periodic , and We emphasize again that the expression (41) does not imply that the period of the solution ϕ(t, ε) is 2π. That is because we cannot assure the period of the order ε 2 functions.

Stability
We have seen that the averaged functions (45) up to order 2 were sufficient for detecting the existence of a periodic solution of the differential system (40). Now we show that the higher order averaged functions may play an important role for studying the stability of the periodic solution ϕ(t, ε) provided by theorem B. Clearly the stability of the periodic solution ϕ(t, ε) can be derived from the eigenvalues of the Jacobian matrix of the displacement function D z h(z(ε), ε) evaluated at z(ε) = ϕ(0, ε). From equation (46) we can write z(ε) = z 0 + O(ε 2 ). Moreover, since in this case Y(t, z) = Id So a first approximation of the eigenvalues λ ± of the Jacobian matrix D z h(z(ε), ε) is given by Clearly the stability of the periodic solution ϕ(t, ε) cannot be completely described by these expressions. Now we show how the higher order bifurcation functions and averaging functions can be used to better analyses the stability of the periodic solution.
We recall that, after some changes of coordinates, the differential system (40) can be transformed into the standard form (43). Expanding it in power series of ε up to order 3, the differential system (43) becomes where F 1 and F 2 are given in (44) and From (12) and (5) we compute the third averaged function and the second bifurcation function, respectively, as As shown in the previous section a ε = α 0 is a simple root of the function f 1 (α). Using the implicit function theorem we find a branch of zeros of the equa- Note that a ε satisfies the hypotheses (iii) and (iv) of theorem A for r = 1, l = 1 and k = 2.

Appendix B. Basic results on the Brouwer degree
In this appendix, following Browder's paper [3], we present the existence and uniqueness result from the degree theory in finite dimensional spaces. Moreover the degree function d( f , V, y 0 ) is uniquely determined by the three above conditions.