PERIODIC ORBITS BIFURCATING FROM A NON–ISOLATED ZERO–HOPF EQUILIBRIUM OF THREE-DIMENSIONAL DIFFERENTIAL SYSTEMS REVISITED

In this paper we study the periodic solutions bifurcating from a non-isolated zero-Hopf equilibrium in a polynomial differential system of degree two in R. More specifically, we use recent results of averaging theory to improve the conditions for the existence of one or two periodic solutions bifurcating from such a zero–Hopf equilibrium. This new result is applied for studying the periodic solutions of differential systems in R having n-scroll chaotic attractors.


Introduction and Statement of the Main Result
In this paper we study the periodic orbits bifurcating from a non-isolated zero-Hopf equilibrium of a three-dimensional autonomous differential system, it means that the differential system has a continuum of equilibria containing a point with a zero eigenvalue and a pair of purely imaginary eigenvalues.The zero-Hopf bifurcation is an interesting subject in differential systems and have been studied by Guckenheimer [1981]; Guckenheimer & Holmes [2013]; Scheurle & Marsden [1984]; Kuznetsov [2013] and many other authors.Usually the zero-Hopf bifurcation is a two-parameter unfolding of a three-dimensional autonomous differential system with an isolated zero-Hopf equilibrium.It is known that some complicated invariant sets can bifurcate from an isolated zero-Hopf equilibrium, for instance zero-Hopf bifurcation can imply local birth of "chaos", see for instance Scheurle & Marsden [1984].
Most papers study isolated zero-Hopf equilibrium.One of the few works about non-isolated zero-Hopf equilibrium was done in 2012 by Llibre & Xiao [2014].The authors studied the periodic orbits bifurcating from the non-isolated zero-Hopf equilibrium located at the origin of the following family of polynomial differential systems of degree two with µ = 1 and where the coefficients functions are of the form When ε = 0, this system has a continuum of equilibria which fill a segment, or a half-straight line.As shown by Llibre & Xiao [2014, Proposition 2.2 ] this continuum of equilibria will have a (non-isolated) zero-Hopf equilibrium at the origin if and only if system (1) satisfies the following hypothesis (H 0 ) : a 0020 = b 0020 = c 0020 = 0.
In a small neighbourhood of the origin the authors reduced system (1) to a 2π-periodic differential system using a kind of cylindrical coordinates and a scaling of the variables.Then second order averaging theory was used for providing explicit conditions for the existence of one or two periodic orbits bifurcating from the non-isolated zero-Hopf equilibrium, see Llibre & Xiao [2014, Theorem 2.4].
We shall use recent results obtained in the averaging theory to weak the hypotheses of Llibre & Xiao [2014, Theorem 2.4] improving those results.Mainly their result was obtained assuming the following hypothesis (H 1 ) : c 2000 = c 1100 = c 0200 = 0.
We extend these results giving sufficient conditions for the existence of ε−families of periodic orbits bifurcating from the origin of system (1) using the weaker hypothesis This will be our main result, see Theorem 1.In section 2 we formulate the averaging theorem (see Theorem 2) used for proving our main result.We use averaging theory in this paper, although the same kind of study could be done by Melnikov method.In particular, the Melnikov functions recently developed by Tian & Han [2017] could also be applied here.In section 3 we shall apply Theorem 1 to study three polynomial differential systems of degree two that cannot be studied using Llibre & Xiao [2014, Theorem 2.4].More precisely these three polynomial differential systems are system (11) proposed by Li [2008], system (12) provided by Pan et al. [2010] and system (10) proposed by Elhadj & Sprott [2013].For certain coefficients values all these systems present n-scroll chaotic attractors.They also have non-isolated zero-Hopf equilibria, and we shall use Theorem 1 to give sufficient conditions for the existence of periodic orbits for these systems.
(i) System (1) has one family of periodic orbits bifurcating from the origin if one of the following conditions holds: (1) has two families of periodic orbits bifurcating from the origin if one of the following condition holds: where

Averaging Theory and Proof of Theorem 1
To find the periodic orbits bifurcating from a non-isolated zero-Hopf equibilbrium of the differential system (1) we use averaging theory.The averaging theory has a long history and for a modern exposition of this topic the reader is addressed to Murdock et al. [2007].
We are interested in the formulation of the averaging theory for systems with non-trivial unperturbed part, i.e., we consider the differential system where There are several recent works developing and improving the averaging theory for this kind of systems, see for instance [Cândido et al., 2016;Coll et al., 2012;Llibre et al., 2014;Giné et al., 2016].For the convenience of the reader we present here the second order averaging theory for system (3).Let Φ(•, z) : [0, t z ] → R n be the solution of the unperturbed system, ẋ(t) = F 0 (t, x), such that Φ(0, z) = z.For i = 1, 2 we define the following averaged functions g i : D → R n of order i as where and M (t, z) is the fundamental matrix of the variational differential equation of the unperturbed system along the solution Φ(t, z), such that M (0, z) is the identity matrix.The next theorem provides the second order averaging theory, for a proof see Llibre et al. [2014].
Theorem 2. Assume the following conditions.

(i)
There exists an open subset W of D such that for any z ∈ W , Φ(t, z) is T -periodic in the variable t.
Then for |ε| = 0 sufficiently small there exists a T -periodic solution x(t, z(ε), ε) Now we shall use Theorem 2 for proving our main result.
Proof.[Proof of Theorem 1] Assuming hypotheses (H 0 ) and (H 1 ), we start by writing system (1) into the normal form for applying the averaging Theorem 2. We use the change of coordinates U = R cos θ, V = R sin θ and W = RZ.Then we scale the system taking R = √ εr and Z = √ εz with ε > 0 a small parameter.These are exactly the same steps used to obtain the equation (2.8) of [Llibre & Xiao, 2014].Taking θ as the new independent variable we obtain the following differential system where ) are 2π−periodic in the variable θ for i = 0, 1, 2, and are given by  Taking (r, z) ∈ R + × R as initial conditions, the unperturbed system has the solution Φ(θ, r, z) = r, rc 1100 sin 2 θ + rc 2000 sin(2θ) + 2ωz 2ω .
Thus the solution φ(θ, r, z) is 2π−periodic for all (r, z) ∈ R + ×R, then system (5) satisfies the hypotheses (i) of Theorem 2. Furthermore, the fundamental matrix associated to the variational equation of the solution φ(θ, r, z) is Thus using (4) the averaged functions of system (5) are g 1 (r, z) = (0, 0) and Now we have to find the simple zeros of the system We are interest only in the solutions (r 0 , z 0 ) ∈ R 2 such that g 2 (r 0 , z 0 ) = 0 and r 0 > 0. We divide the study of these solutions in the following cases: Case 1: a 1010 + b 0110 = 0.If λB 1 < 0 and λ(B 1 + B 4 ) = B 1 µ, then system (7) has the solution Consequently (r 0 , z 0 ) is a simple zero of system (7).
Case 2: a 1010 + b 0110 = 0. Solving the first equation of system (7) with respect to z and r > 0 we have that z = −(B 1 r 2 + 16λω)/(8rω(a 1010 + b 0110 )).Eliminating z in the system (7) we obtain the polynomial, The bi-quadratic polynomial (8) may have one or two real positive roots.Thus we use its discriminant to study such roots and then verify when them provide simple zeros of system (7).The discriminant vanishes if Q 1 λµ = 0 or 3Q 2 2 − 4Q 1 λµ = 0, in this case one can verify the following subcases Subcase 2.1: If λµ = 0 and Q 1 Q 2 < 0, system (7) has the zero .

Applications
The existence of differential systems with only zero, one or fewer than n equilibrium points generating n−scroll chaotic attractors is an important open problem whose solution is not easy.For more information about n − scroll chaotic attractors see Lü & Chen [2006].This type of system has several real word applications, for instance in engineering and secure communication.Elhadj & Sprott [2013] have shown that the simplest family of systems displaying n-scroll chaotic attractors is given by the quadratic polynomial differential system Using Theorem 1 we can find conditions in order that the differential system (10) has two periodic orbits.
Furthermore we also shall use Theorem 1 for proving the existence of ε-families of periodic orbits in the following two differential systems ẋ =a(y − x) + dxz, System (11) was derived from the classical Lorenz system by Li [2008].This system exhibits a threescroll chaotic attractor, with two scrolls symmetric with respect to the z−axis as in the Lorenz attractor, and the third scroll is around the z−axis.System (12) were provided by Pan et al. [2010] with f 0 = f 1 .It was derived from the Chen system and also presents a three-scroll chaotic attractor.The authors show that the parameter m works as a control parameter that can dramatically change the dynamics of the system.Theorem 5 will reveal that the parameter m is also important for the existence of periodic orbits in system (12).
In the next results we used Theorem 1 to provide sufficient conditions for the existence of periodic orbits in systems ( 11) and ( 12) respectively.
Proof.[Proof of Theorem 5] The point p 0 = (0, 0, m 1 /µ) is an equilibrium of system (12).We translate p 0 to the origin doing the following change of variables Where the functions (2) vanishing except for We note that when ε = 0 the origin of system ( 16) is a non-isolated zero-Hopf equilibrium point.Thus we have Consequently system (15) satisfies the conditions (c) of Theorem 1 since we have by hypothesis that This concludes the proof of the theorem.

Appendix
Here we write the coefficients of system (13).