Zero–Hopf bifurcation in a hyperchaotic Lorenz system

We characterize the zero–Hopf bifurcation at the singular points of a parameter codimension four hyperchaotic Lorenz system. Using averaging theory, we find sufficient conditions so that at the bifurcation points two periodic solutions emerge and describe the stability of these orbits.

(the Lorenz attractor). The origins of this system lies in atmospheric modeling. However, the Lorenz equations also appear in the modeling of lasers; see [11] and for dynamos see [15].
Recently, a so-called hyperchaotic Lorentz system was introduced; see for instance [1,6,10,13,14,25,[27][28][29][30][31][32][33] and the references therein. (MathSciNet presently lists 24 papers on hyperchaotic Lorenz systems.) We remark that not all these hyperchaotic Lorenz systems coincide, as they can vary in one or two terms. A precise definition of a hyperchaotic system comprises (1) an autonomous differential equations system with a phase space of dimension at least four, (2) a dissipative structure, and (3) at least two unstable directions, out of which at least one is due to a nonlinearity. Since it generates multiple positive Lyapunov exponents, the dynamics of hyperchaotic systems is hard to predict and control. Consequently, such systems are of use in secure communication, and thus received a great deal of attention mainly in engineering (circuit and communications systems; see, for instance [24], and references therein).
In this paper, we approach a hyperchaotic system from a dynamical systems point of view. More precisely, we investigate a 4-dimensional zero-Hopf equilibrium (that is, an isolated equilibrium with a double zero eigenvalue and a pair of purely imaginary eigenvalues), and the birth of periodic solutions as parameters vary.
There are several works studying the unfolding of the 3-dimensional zero-Hopf equilibria. Recall that a 3-dimensional zero-Hopf bifurcation is a twoparameter unfolding (or family) of a 3-dimensional autonomous differential system with a zero-Hopf equilibrium. The unfolding has an isolated equilibrium with a zero eigenvalue and a pair of purely imaginary eigenvalues if the two parameters take zero values, and the unfolding has different topological type of dynamics in the small neighborhood of this isolated equilibrium as the two parameters vary in a small neighborhood of the origin. The zero-Hopf bifurcation has been studied by Guckenheimer [8], Guckenheimer, and Holmes [9], Han [12], Kuznetsov [16], Llibre [18], Marsden and Scheurle [23]. It has been shown that, under certain conditions, some complicated invariant sets of the unfolding could be bifurcated from the isolated zero-Hopf equilibrium and hence, in some cases, a zero-Hopf equilibrium could signal a local birth of "chaos" (see [5,23]). Also, recently Li and Wang [17] published a paper on a Hopf bifurcation in a (3-dimensional) Lorentz-type system. Due to the complexity related to the high dimensionality, there is very little work done on the n-dimensional zero-Hopf bifurcation with n > 3.
Here, we study the following hyperchaotic Lorenz system (as given in [6,14]) which displays a zero-Hopf equilibrium for an appropriate choice of the parameters a, b, c, and d. Using the method of averaging and a blow-up of both the variables and the parameters at the zero-Hopf equilibrium, we show that two periodic solutions emerge as parameters vary. Further, we characterize the stability of these periodic solutions.
To our knowledge, zero-Hopf equilibria and bifurcations in systems (which have a central manifold) of dimension n ≥ 4 has not been studied yet. While we focus on the case of a hyperchaotic Lorentz system, we mention that the method used here is amenable to a larger class of nonlinear differential systems in R n . We plan to develop this in future work.

Statements of the main results
One may verify that for any choice of the parameters the origin of coordinates of R 4 is always an equilibrium point for the hyperchaotic Lorenz system (1). Moreover, if In the next proposition, we characterize when the equilibrium point localized at the origin of coordinates of the hyperchaotic Lorenz system (1) is a zero-Hopf equilibrium point.

Proposition 1
There is an one-parameter family of hyperchaotic Lorenz system (1) for which the origin of coordinates is a zero-Hopf equilibrium point. Namely, a = −1, b = d = 0 and c > 1. Moreover, the eigenvalues at the origin for this one-parameter family are 0, 0, and ±i √ c − 1.
We shall study when the hyperchaotic Lorenz system (1) having a zero-Hopf equilibrium point at the origin of coordinates has a zero-Hopf bifurcation producing some periodic orbit.
the hyperchaotic Lorenz system has a zero-Hopf bifurcation at the equilibrium point localized at the origin of coordinates. Two periodic solutions born at this equilibrium and they are stable if (a + 1) > 0, b > 0, and d > 0.

The averaging theory for periodic orbits
The averaging theory is a classical and matured tool for studying the behavior of the dynamics of nonlinear smooth dynamical systems, and in particular of their periodic orbits. The method of averaging has a long history that starts with the classical works of Lagrange and Laplace who provided an intuitive justification of the process. The first formalization of this procedure is due to Fatou [7] in 1928. Important practical and theoretical contributions in this theory were made by Krylov and Bogoliubov [3] in the 1930s and Bogoliubov [2] in 1945. The averaging theory of first order for studying periodic orbits can be found in [26]; see also [9]. It can be summarized as follows.
We consider the problem of the bifurcation of T -periodic solutions from differential systems of the forṁ with ε = 0 to ε = 0 sufficiently small. Here, the , T -periodic in the first variable, and Ω is an open subset of R n . The main assumption is that the unperturbed systeṁ has a submanifold of periodic solutions. A solution of this problem is given using the averaging theory. Let x(t, z, ε) be the solution of system (4) such that x(0, z, ε) = z. We write the linearization of the unperturbed system along a periodic solution x(t, z, 0) aṡ In what follows, we denote by M z (t) some fundamental matrix of the linear differential system (5). We assume that there exists an open set V with Cl(V ) ⊂ Ω such that for each z ∈ Cl(V ), x(t, z, 0) is T -periodic, where x(t, z, 0) denotes the solution of the unperturbed system (4) with x(0, z, 0) = z. The set Cl(V ) is isochronous for the system (3); i.e., it is a set formed only by periodic orbits, all of them having the same period. Then an answer to the bifurcation problem of T -periodic solutions from the periodic solutions x(t, z, 0) contained in Cl(V ) is given in the following result.

Theorem 2 (Perturbations of an isochronous set) We assume that there exists an open and bounded set V with Cl(V ) ⊂ Ω such that for each z ∈ Cl(V ), the solution x(t, z) is T -periodic, then we consider the func-
Then the following statements holds: (a) If there exists a ∈ V with F(a) = 0 and det((∂F /∂z)(a)) = 0, then there exists a T -periodic solution x(t, ε) of system (3) such that type of stability of the periodic solution x(t, ε) is given by the eigenvalues of the Jacobian matrix ((∂F /∂z)(a)).
For an easy proof of Theorem 2(a), see Corollary 1 of [4]. In fact, the result of Theorem 2 is a classical result due to Malkin [20] and Roseau [21].
For additional information on averaging theory, see the book [22].

Proofs
Proof of Proposition 1 The Jacobian matrix evaluated at (x, y, z, w) = (0, 0, 0, 0) is Solving this equation, we find that the eigenvalues of A are λ 1 = −b, λ 2 = d and λ 3,4 = (−(a + 1) ± (a + 1) 2 − 4a(1 − c))/2. To satisfy the conditions for the origin to be a zero-Hopf equilibrium, it must be satisfied that b = d = 0, and for the other two eigenvalues to be purely imaginary we must have a = −1 and 4(c − 1) < 0, this is c > 1. Note that when this conditions are satisfied; the only equilibrium of the system is the origin. Therefore, when a = −1, b = d = 0 and c > 1, we obtain a one parameter family of hyperchaotic Lorenz systems for which the origin is a zero-Hopf equilibrium. Moreover, the eigenvalues for this one parameter family are λ 1 = λ 2 = 0 and λ 3,4 = ±i √ c − 1. (a, b, d) = (−1 + εa 1 , εb 1 , εd 1 ) where ε > 0 is a sufficiently small parameter and a 1 , b 1 , and d 1 are real nonzero numbers. With these substitutions, the Lorenz system becomeṡ

Proof of Theorem 1 Let
Next, we rescale the variables. Let (x, y, z, w) = (εX, εY, εZ, εW ), and by denoting again the variables (X, Y, Z, W ) by (x, y, z, w) the system (7) is written as follows: To describe the behavior of the system (8), we will apply the averaging method described in Theo-rem 2. Using the same notation as in Sect. 3, we have x = (x, y, z, w), and F 2 (t, x, ε) = 0. Let us consider the next initial value problem given by the unperturbed systeṁ The solution to this unperturbed system is Note that all the solutions x(t, z, ε) with z = 0 are periodic, and have the same period T = 2π We know write the linearization of the unperturbed system along a periodic solution x(t, z, 0) asẏ The fundamental matrix M z (t) is given by where r := √ c − 1. Computing the integral established in Theorem 2, where the components of F(z) are given as follows: and solving the nonlinear system given by F(z) = 0 we can conclude that the system has the next three solutions s 0 = (0, 0, 0, 0), The first solution corresponds to the equilibrium point localized at the origin. For the other two solutions, we note that Conditions (2) ensure that and so by Theorem 2(a) there exists two T -periodic solution x 1 (t, ε) and x 2 (t, ε) of system (8) such that x 1 (0, ε) → s 1 and x 2 (0, ε) → s 2 as ε → 0, where its period is T = 2π To determine the type of stability of the two periodic solutions, we look at the eigenvalues of the Jacobian matrices Γ 1 = ((∂F /∂z)(s 1 )) and Γ 2 = ((∂F /∂z)(s 2 )). We have that the eigenvalues are the same for both matrices and given as follows: The stability follows by imposing a negative real part to all eigenvalues and substituting a 1 , b 1 , and d 1 in terms of a, b, and d.