ZERO-HOPF BIFURCATIONS IN THREE-DIMENSIONAL CHAOTIC SYSTEMS WITH ONE STABLE EQUILIBRIUM

. In (Molaie et al., Int J Bifurcat Chaos 23


Introduction and statement of the main results
The interest in finding and studying chaotic differential systems has aroused considerably since Lorenz reported in 1963 a chaotic attractor in a threedimensional autonomous system [15,20].Recently, chaotic attractors have been categorized in two classes: self-excited and hidden attractors.Whereas a self-excited attractor has a basin of attraction that overlaps with the neighborhood of an unstable equilibrium point, a hidden attractor has a basin of attraction which does not intersect with small neighborhoods of any equilibrium point [11,12,13].
The well-known Lorenz system [15], Rössler system [18] and Chua system [6] are examples of chaotic systems with self-excited attractors.Hidden attractors can be found in chaotic differential systems without equilibria [8,14,22], with only stable equilibrium points [10,17,23,24,25] or with an infinite number of equilibria (curves, surfaces) [7,9] and there is little knowledge about their formation, hence analytical proofs of their existence is yet needed.From the point of view of applications, hidden attractors are potentially important in engineering applications [1,2], chaos-based secure Theorem 1.For a = 0, systems SEa 1 to SEa 23 have one curve of equilibrium points given by a straight line containing the origin.In a neighborhood of the origin, the singularities form two branches of foci with opposite stability connected by a zero-Hopf equilibrium point at the origin.For a > 0 small enough, the SEa 1 -SEa 23 systems have three periodic orbits which bifurcate from the zero-Hopf equilibrium at the origin: one of them is unstable and the other two form a pair of saddle type periodic orbits.
Theorem 1 is proved in Section 3. We recall that an (isolated) equilibrium point of a differential system in R 3 is called a zero-Hopf equilibrium point if the Jacobian matrix of the system at this point has a zero and a pair of purely imaginary eigenvalues.As for a = 0 the origin of the SEa 1 -SEa 23 systems is a non-isolated equilibrium point with eigenvalues 0 and ±i, we call it a non-isolated zero-Hopf equilibrium point.
The existence of periodic orbits plays an important role in the formation of chaotic attractors.Indeed some known routes to chaos are characterized by the existence of this kind of orbits.In [17] the authors stated that the differential systems studied by them (SEa 1 -SE 23 systems in Table 1 of [17]) appear to approach chaotic regime through a succession of perioddoubling bifurcations of periodic orbits and, as an example, they numerically showed this phenomenon for their SE 3 system (which corresponds to our SEa 3 system with a = 1), taking the third equation as ż = −αx − y − 4z + y 2 + xy and varying parameter α from 3.3 to 3.4 (see Fig. 2 of [17]).
Here, by using numerical simulations, we show that a cascade of perioddoubling bifurcations also occur for the classes of differential systems SEa 1 -SEa 23 , as the parameter a is varied near a = 1.We summarize it in the following result.
Conjecture.The hidden chaotic attractors which exist for a = 1 in the SEa 1 -SEa 23 systems displayed in Tables 1, 2 and 3, are formed by the occurrence of period-doubling bifurcations of periodic orbits, as the parameter a is varied.
In order to illustrate the result stated above, in Section 4, we numerically study the formation of hidden chaotic attractors in the SEa 1 and SEa 11 systems.The dynamical behavior is quite similar for the other systems .
Based on the results described above, we conjecture that the hidden chaotic attractors of SEa i systems with a = 1, for i = 1, . . ., 23, are created by a cascade of period doubling bifurcations which start at one of the periodic orbits bifurcated from the zero-Hopf equilibrium point at the origin.Due to the instability of these periodic orbits, the numerical study of their
continuation, from a = 0 to a = 1 is very difficult.However, there is no the occurrence of other Hopf like bifurcations as the parameter a varies between 0 and 1, because systems SEa i have only one equilibrium point whose stability does not change.This fact led us to formulate such a conjecture.
This paper is devoted to prove Theorem 1 and it is organized as follows.In Section 2 we present some basic aspects of the averaging theory of first order, which we shall use to prove Theorem 1 in Section 3. The formation of the hidden chaotic attractors of SEa 1 and SEa 11 systems with a = 1 are numerically studied in Section 4 and these numerical results motivate the conjecture.Finally in Section 5 we give some concluding remarks.

Averaging theory of first order
For the sake of completeness and to fix the notation which will be used ahead, in this section we present the resuelt from averaging theory that we shall use for proving Theorem 1, its proof can be found in [19].
Consider the initial value problems and ẏ = ε g(y), where x = x(t) and y = y(t) are defined in some open subset Ω ⊂ R n for t ∈ [0, ∞), x 0 is a point of Ω and ε ∈ (0, ε 0 ], for some fixed ε 0 > 0 small enough.Suppose that F 1 and F 2 are periodic functions of period T in the variable t, and set Assume that D x g and D xx g are all the first and second derivatives of g respectively.Under these assumptions, we have the following result. Theorem 2. Let F 1 , D x F 1 , D xx F 1 and D x F 2 be continuous and bounded by a constant, which does not depend on ε, in [0, ∞)×Ω×(0, ε 0 ] and suppose that y(t) ∈ Ω for t ∈ [0, 1/ε].Then, the following statements hold.
If p = 0 is an equilibrium point of system (2) such that det[D y g(p)] = 0, then system (1) has a periodic solution φ(t, ε) of period T , which is close to p and such that φ(0, ε) − p = O(ε) as ε → 0. 3. The stability of the periodic solution φ(t, ε) is given by the stability of the equilibrium point p.

Proof of Theorem 1
Consider the SEa 1 system given in Table 1.For a = 0 the x-axis is filled up of equilibrium points.The linear part of SEa 1 system at these equilibrium points are Note that, in a neighborhood of the origin, the x-axis is formed by two branches of foci with opposite stability connected by the origin, which is a non-isolated zero-Hopf equilibrium point with eigenvalues λ 1 = 0 and λ 2,3 = ±i.Now, using Theorem 2, we prove that three periodic orbits bifurcate from the origin for a > 0 small enough.In order to put the SEa 1 system in the appropriate normal form to apply the averaging theorem, first consider the linear change of variables (x, y, z) → (u, v, w), where x = u + w, y = v and z = −u.
In the new variables (u, v, w) the SEa 1 system becomes

ZERO-HOPF BIFURCATIONS IN SYSTEMS WITH ONE STABLE EQUILIBRIUM 7
where and F 2 (u, v, w) = −F 1 (u, v, w).Note that the linear part of system (3) at the origin when a = 0 is in the real Jordan normal form, that is in the form   0 1 0 −1 0 0 0 0 0   .
Now write system (3) in cylindrical coordinates (r, θ, w) doing the change of variables u = r cos θ, v = r sin θ.Then system (3) becomes where Then, taking θ as the new independent variable and doing the Taylor expansion in the variable ε at ε = 0 up to order 2 we get where Using the notation of Theorem 2, consider In this way we have .
Hence g(y) = 0 has the real solutions Then, by Theorem 2, for ε > 0 sufficiently small, the SEa 1 system has the periodic solutions Moreover, the eigenvalues of the matrix [D y g(p 1 )] are 1 and 3/4 and the eigenvalues of the matrices [D y g(p 2 )] and [D y g(p 3 )] are (1 ± √ 7)/2.Thus the SEa 1 system has three periodic orbits bifurcating from the origin, one of them a repeller and a pair of periodic orbits of saddle type.
Going back to the initial variables of the SEa 1 system, we get that, for a > 0 sufficiently small, such system has three periodic orbits of period near to 2π, the first is given by γ 1 (t) = (x 1 (t), y 1 (t), z 1 (t)), where which is a repeller, and the other two are given by γ 2,3 = (x 2,3 (t), y 2,3 (t), z 2,3 (t)), where ), which are a pair of periodic orbits of saddle type.
Following the same steps and doing the suitable changes of variables it can be shown that the same results are valid for the other 22 differential systems of Tables 1, 2 and 3. We give explicit expressions for the change of variables needed to prove Theorem 1 for the SEa 2 -SEa 23 systems in the Appendix, with some pertinent comments.

On the Formation of the Hidden Chaotic Attractors
Periodic orbits in chaotic differential systems are important dynamical elements in the formation of chaotic attractors.This was exhaustively studied in the case of self-exited attractors with the well-known period-doubling route to chaos and it also seems to happen in the case of the hidden attractors of these 23 differential systems here studied.In [3], for example, the authors showed that the hidden chaotic attractors in some NE systems (chaotic systems having no equilibrium points) are formed by a period-doubling of periodic orbits which bifurcate from a zero-Hopf equilibrium point at the origin.
In [17] the authors stated that the differential systems studied by them (SEa 1 -SEa 23 systems in Tables 1, 2 and 3 with a = 1) appear to approach ZERO-HOPF BIFURCATIONS IN SYSTEMS WITH ONE STABLE EQUILIBRIUM 9 chaos through a succession of period-doubling of periodic orbits.Here, by numerical simulations, we observed that this phenomenon also occurs for all the twenty three classes of differential systems given in Tables 1, 2 and 3, as stated in the conjecture, when the parameter a varies.In order to avoid a long list of figures, we illustrate the formation of hidden chaotic attractors in SEa 1 and SEa 11 systems.
In Figures 1 and 2 are shown the formation of the hidden chaotic attractors in SEa 1 and SEa 11 systems as the parameter value a tends to 1.Note that in both cases the formation of the hidden attractor begins with a periodic orbit which doubles period confirming the period-doubling route to chaos.

Concluding remarks
In this paper we provide the expression of twenty three classes of quadratic differential systems in R 3 depending on a real parameter a which encompass the chaotic differential systems having hidden attractors studied in [17].By using the averaging theory of first order presented in Section 2, we proved, in Section 3, that three periodic orbits, one of them being unstable and the other two forming a pair of periodic orbits of saddle type, bifurcate from a non-isolated zero-Hopf equilibrium point located at origin, when the parameter a vary from a = 0 to a > 0 sufficiently small.Doing numerical simulations we observed that the hidden chaotic attractors of all twenty three classes of differential systems given in Tables 1, 2 and 3 are formed by a period-doubling of periodic orbits.In Section 4 we presented the formation of the hidden attractor in the SEa 1 and the SEa 11 systems in order to illustrate this fact.
In this way, the existence of periodic orbits in the SEa 1 -SEa 23 systems play an important role in the occurrence of chaotic behavior evidencing the importance of the zero-Hopf bifurcation stated by Theorem 1.Moreover, the existence of periodic orbits bifurcating from zero-Hopf equilibrium points and period-doubling route to chaos was also observed in [3,4,16] for chaotic differential systems having no equilibrium points (the NE systems).Then, the occurrence of zero-Hopf bifurcation and the formation of hidden chaotic attractors seems to be strictly related.

Figure 1 .
Figure 1.Orbit of the SEa 1 system with initial condition (4a, −2a, 0) for different values of the parameter a.This orbit is in the hidden chaotic attractor when a = 1.

Figure 2 .
Figure 2. Orbit of the SEa 11 system with initial condition (−2a, 0, 0.1a) for different values of the parameter a.This orbit is in the hidden chaotic attractor when a = 1.