New results on averaging theory and applications

The usual averaging theory reduces the computation of some periodic solutions of a system of ordinary differential equations, to find the simple zeros of an associated averaged function. When one of these zeros is not simple, i.e., the Jacobian of the averaged function in it is zero, the classical averaging theory does not provide information about the periodic solution associated to a non-simple zero. Here we provide sufficient conditions in order that the averaging theory can be applied also to non-simple zeros for studying their associated periodic solutions. Additionally, we do two applications of this new result for studying the zero–Hopf bifurcation in the Lorenz system and in the Fitzhugh–Nagumo system.


Introduction and statement of the main result
In this paper we introduce an improvement to the first-order averaging theorem and use it to study the zero-Hopf bifurcations of periodic orbits which take place in the Lorenz and Fitzhugh-Nagumo systems.
The first-order averaging theorem, as presented in [17], can be used to determine periodic orbits coming from the simple zeros of the averaged function, see Theorem 1. Here we use the Lyapunov-Schmidt reduction method (see Lemma 7) in order to make the averaging theorem able to determine periodic solutions coming from degenerated zeros of the averaged function, i.e., zeros for which the Jacobian determinant of the averaged function vanishes, see Theorem 2.
Here a zero-Hopf equilibrium is an equilibrium point of a three-dimensional autonomous differential system, which has a zero and a pair of pure imaginary eigenvalues. In general, the zero-Hopf bifurcation is a two-parameter unfolding of a three-dimensional autonomous differential equation with a zero-Hopf equilibrium. This kind of zero-Hopf bifurcations has been studied by Guckenheimer,Han,Holmes,12], and it was shown that some complicated invariant sets can bifurcate from the isolated zero-Hopf equilibrium doing the unfolding. Due to the lack of a general theory describing all these kinds of bifurcations that the unfolding of a zero-Hopf bifurcation can produce, most of the systems exhibiting this kind of bifurcation must be studied directly.
Using Theorem 2, here obtained, we could detect the bifurcation of a periodic orbit from a zero-Hopf bifurcation in the famous Lorenz system of differential equations. As far as we know this was the first time this periodic solution was detected. We also apply this theorem to detect the bifurcation of new periodic solutions in the Fitzhugh-Nagumo system, improving the results obtained in [4].

Averaging theory
The averaging method is a classical theory for studying nonlinear dynamical systems and their periodic solutions. It was conceived by Lagrange in the eighteenth century, without formal proof, what was only achieved in 1928 by Fatou [5]. After the formalization of the theory important contributions were made by Krylov and Bogoliubov [2] in 1930 and Bogoliubov [1] in 1945. The classical averaging method for computing periodic solutions can be summarized as follows.
We consider the initial value probleṁ with x, y, and For a proof of Theorem 1, see, for instance, Theorem 11.1, 11.5 and 11.6 of [17], where it is stated for ε ∈ [0, ε 0 ) but in fact following the proof the same result works for ε ∈ [−ε 0 , ε 0 ], as it is stated here.
The averaged function (3) is a smooth function. Using the notation of Theorem 2, whenever this function vanishes at Z, we have Df 0 (z) = 0, ∀z ∈ Z. Thus, in order to compute periodic solutions for system (1), we cannot apply Theorem 1 at any singular point z ∈ Z. Our aim is to show that, under the assumptions of Theorem 2, it is possible to use the Lyapunov-Schmidt reduction method in order to still apply first-order averaging for finding periodic orbits to the differential system (1).
Therefore, using the near-identity relation where we do the change of variables x → z and equation (1) becomeṡ where Our main result is: The Jacobian matrix Df 0 (z α ) has in the right up corner the k × (n − k) matrix Γ α , while in the right down corner has the (n − k) × (n − k) matrix Δ α with det(Δ α ) = 0.

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New results on averaging theory and applications Page 3 of 11 106 where πh and π ⊥ h are the decomposition of function h in its first k and last n − k coordinates, respectively. Assume that there exists α * ∈ V satisfying the equation f 1 (α * ) = 0 such that the Jacobian det (Df 1 )(α * ) = 0.
Then for |ε| sufficiently small there exists a T -periodic solution Φ t, ε of the differential system (1), such Since in the assumption of Theorem 2 the averaged function f 0 (z) has a continuum of zeros, the determinant of the Jacobian matrix Df 0 (z) at these zeros is null, so some eigenvalues of this matrix is zero, and the usual study for the asymptotic stability is not easy in this case. In fact, the study of the kind of stability of the periodic solutions given by Theorem 2 will be one of our next works. Now our objective is to apply Theorems 1 and 2 for studying the periodic orbits bifurcating from zero-Hopf equilibrium points of two important differential systems, the Lorenz and the Fitzhugh-Nagumo.

Application to Lorenz system
The Lorenz system of differential equations in R 3 arose from the work of meteorologist/mathematician Lorenz [14], who studied forced dissipative hydro-dynamical systems. It has become one of the most widely studied systems of ODEs because of its wide range of behaviors (see, for instance, [16]). Although the origins of this system lies in atmospheric modeling, the Lorenz equations also appear in other areas as in the modeling of lasers see [6], and dynamos see [15]. The Lorenz equations arė with a, b, c being real coefficients.
The next result characterizes when the origin of coordinates is a zero-Hopf equilibria of system (7).

Proposition 3.
If a = −1, c = 0 and b < 0 then there is a 1-parameter family of the Lorenz differential system for which the origin of coordinates is a zero-Hopf equilibrium point.
Theorem 4. Let a = −1 + a 2 ε 2 and c = c 1 ε. Assume that b > 1, a 2 < 0, c 1 = 0 and ε = 0 sufficiently small. Then the Lorenz differential system (7) has a zero-Hopf bifurcation in the equilibrium point at the origin of coordinates, and the following periodic orbit born at this equilibrium when ε = 0.
Theorem 4 is proved using the averaging result given in the new Theorem 2. Proposition 3 and Theorem 4 are proved in Sect. 3.
As far as we know it does not appear in the literature about the Lorenz system the periodic solution that we have found in Theorem 4.

Application to Fitzhugh-Nagumo system
These equations arise in mathematical biology as a model of the transmission of electrical impulses through a nerve axon. The Fitzhugh-Nagumo equations consist in a simplified version of the Hodgkin-Huxley equations which are described using a nonlinear diffusion equation coupled to an ordinary differential equation − a), 0 < a < 1/2 is a constant, δ > 0 and γ > 0 are parameters. In [11] it was stated that a single nerve impulse appears to tend, as t increases, to a traveling wave, i.e., a bounded solution (u, v)(x, t) = (u, v)(ξ) where ξ = x + ct. Hence, one is led to seek for solutions of (9), not identically zero, of the form (u, v) = (u(ξ), v(ξ)) for some c = 0. Substitution in (9) gives a set of ordinary differential equations which upon introduction of the variables x = u, y = v, and z =u takes the formẋ = z, where the dot denotes the derivative with respect to ξ and (a, b, c, d) ∈ R 4 are parameters. For a detailed study concerning traveling waves in (9) see [7]. Hereafter the differential system (10) will be called Fitzhugh-Nagumo differential system.

Proposition 5.
There are two-parameter families of the Fitzhugh-Nagumo differential systems for which the origin of coordinates is a zero-Hopf equilibrium point, both families are two-parameter. Namely: Proof. A proof for this proposition can be obtained in [4].
(d − 1)a 1 b = 0 and ε = 0 sufficiently small. Then the Fitzhugh-Nagumo differential system (10) has a zero-Hopf bifurcation in the equilibrium point at the origin of coordinates, and the periodic orbit born at this equilibrium when ε = 0. and Then the Fithugh-Nagumo differential system (10) has four periodic solutions emerging from the origin.
Theorem 6 is proved using Theorem 1 and 2. Theorem 6 is proved in Sect. 4. Euzébio et al. studied the zero-Hopf bifurcations of system (10) using the classical averaging theory (see Theorem 11 of [4]). Considering the two-parameter families of zero-Hopf equilibria stated in Proposition 5(i) the authors of [4] find using the first-order averaging method, a periodic solution bifurcating from the origin of the system different from the periodic solution (11), because in Theorem 5 of [4] the order of the periodic solution in the three variables (x, y, z) is O(ε) while in our case is O(ε 2 ) for x and y and O(ε) for z. Moreover, using second-order averaging theory, the authors of [4] in Theorem 6 [4] find of ZAMP New results on averaging theory and applications Page 5 of 11 106 one additional periodic solution bifurcating from the origin, while in our Theorem 6 we find four periodic solutions.

Proof of Theorem 2
In order to prove Theorem 2 we shall use a particular case of Lyapunov-Schmidt reduction method for finite dimensional functions. First we need some definitions and notation. In what follows the functions π : R k × R n−k → R k and π ⊥ : R k × R n−k → R n−k denote the projections onto the first k coordinates and onto the last n − k coordinates, respectively. We have the following lemma: Lemma 7. Assume that k ≤ n are positive integers. Let Ω and V be open subsets of R n and R k , respectively. Let g 1 (z) and β : V → R n−k be C 2 functions and take g : Ω × (−ε, ε) → R n such that and let Z = {z α = (α, β(α))} ⊂ Ω. We denote by Γ α the upper right corner k ×(n−k) matrix of Dg 0 (z α ), and by Δ α , the lower right corner (n − k) × (n − k) matrix of Dg 0 (z α ). Assume that for each z α ∈ Z, det(Δ α ) = 0 and g(z α ) = 0. We consider the function f 1 : V → R k as . If there exists α * ∈ V with f 1 (α * ) = 0 and det(Df 1 (α * )) = 0, then there exists α ε such that g(z αε , ε) = 0 and z αε → z α * when ε → 0.
Proof. A proof for this lemma can be found in [13].
Proof of Theorem 2. First we do the change of variables x → z given in (4) in order to write the differential system (1) into system (5). The solution Φ t, z, ε of equation (5) where Φ t, z, ε is defined in its maximal interval for positive time [0, ω + ) ⊂ [0, ∞). Moreover, the solution Φ t, z, ε is T -periodic, if and only if, Φ 0, z, ε = Φ T, z, ε , which leads to equation Then we have and from (13) we obtain that Since for ε = 0 we have Φ s, z, 0 = z, from (14), (16), and (15) we obtain that respectively. Doing the Taylor expansion of the function H(z, ε) around ε and using the functions in (17) we obtain that Hence, the zeros of (18) correspond to periodic solutions of system (5) and provide, going back through the change of variables (4), periodic solutions of system (1). Notice that function (18) is in the form (12) with Thus, the result follows by Lemma 7. In addition, using the notation of Lemma 7 observe that g 1 (z α ) = h(α), where h(α) is defined in (d) of Theorem 2.
In this section we prove Proposition 3 and Theorem 4.

Application to Lorenz system
Proof of Proposition 3. The Jacobian matrix of system (7) evaluated at the origin is Thus, its characteristic polynomial is Proof of Theorem 4. Assume a = −1 + a 2 ε 2 and c = c 1 ε in system (7). Doing the change of variables (x, y, z) → ε (X, Y, Z) the system becomeṡ