Periodic solutions for a four-dimensional hyperchaotic system

In this paper, we show a zero-Hopf bifurcation in a four-dimensional smooth quadratic autonomous hyperchaotic system. Using averaging theory, we prove the existence of periodic orbits bifurcating from the zero-Hopf equilibrium located at the origin of the hyperchaotic system, and the stability conditions of periodic solutions are given.

Lorenz system, Zarei et al. proposed a new four-dimensional quadratic autonomous hyperchaotic attractor. It can generate double-wing chaotic and hyperchaotic attractors with only one equilibrium point [11]. In 2017, Zhou et al. proposed a four-dimensional smooth quadratic autonomous hyperchaotic system with complex dynamics, and analyzed the stability of the hyperchaotic system, pitchfork bifurcation, Hopf bifurcation and other local dynamics problems by using the central manifold theorem and bifurcation theory [12]. In 2019, Rajagopal et al. proposed an improved hyperchaotic van der Pol-Duffing snap oscillator. Using a Lyapunov exponent, equilibrium point stability analysis and bifurcation diagram, various dynamic properties of the system were studied [13].
Under certain conditions, some complex invariant sets can be separated from the isolated zero-Hopf equilibrium point, so in some cases, the zero-Hopf equilibrium point may mean the generation of local chaos. There have been many studies on zero-Hopf bifurcation of three-dimensional systems. In 2014, by using the averaging theory, Garcia et al. provided an analytic proof of the existence of zero-Hopf bifurcation in systems with two slow speeds and one fast variable, and to describe the stability or instability of periodic orbits in such zero-Hopf bifurcation [14]. In 2017, Ginoux et al. used the second-order averaging theory to prove that there are two types of zero-Hopf bifurcation in the predator-prey Volterra-Gauss system under different parameters. Under the first parameter condition, the system has a periodic orbit, and under the second parameter condition, the system has five periodic orbits [15]. In 2018, Li et al. considered the existence of zero-Hopf bifurcation and periodic solutions for the improved Chua system by applying the averaging theory [16]. In 2018, Salih studied the zero-Hopf bifurcation of the three-dimensional Lotka-Volterra systems [17]. In 2018, Candido et al. studied the zero-Hopf bifurcation of 16 three-dimensional differential systems without equilibrium by using the averaging theory [18]. However, due to the higher dimension and complexity of hyperchaotic systems, few scholars are currently engaged in the analysis of hyperchaotic theory, there is still very little work done on zero-Hopf bifurcation for n-dimensional systems with n ≥ 4. In 2014, Lorena et al. studied the zero-Hopf bifurcation of a class of Lorenz hyperchaotic systems and the generation of periodic solutions with the change of parameters, which was the first work on the zero-Hopf bifurcation problem in four-dimensional systems [7]. In 2015, Maza studied the zero-Hopf bifurcation of hyperchaotic Chen system, and proved that hyperchaotic Chen system has two periodic orbits at the zero-Hopf equilibrium point by using the averaging theory [19]. In 2017, Chen et al. studied zero-Hopf bifurcation of generalized Lorenz-Stenflo hyperchaotic system and obtained two periodic solutions generated from bifurcation points [20].
In order to fully understand the dynamics of a system, it is necessary to study its periodic solutions. In recent years, scholars have studied the periodic solutions of many classical systems. In 2017, Liu et al. studied the existence of periodic solutions for the Newtonian equation of motion with p-Laplacian operator by asymptotic behavior of potential function [21]. In 2018, Wang et al. considered the existence of periodic solutions for a non-autonomous second-order Hamiltonian systems [22]. In 2018, Wang et al. studied the multiplicity of periodic solutions of one kind of planar Hamiltonian systems with a nonlinear term satisfying semi-linear conditions [23]. In 2019, Chiraz proved the existence of periodic solutions for some non-densely non-autonomous delayed partial differential equations [24].
In this paper, from the perspective of local dynamics, a four-dimensional smooth quadratic autonomous hyperchaotic system [12] is studied: where a, b, c, d, j, e are real parameters.
The system (1) is constructed by adding one state variable to the well-known Lorenz system, which has rich and complex dynamic behaviors. With the change of parameters, the system can evolve into periodic, quasi-periodic, chaotic and hyperchaotic states, and attractors in these states are different from ordinary attractors. In this paper, we study the zero-Hopf bifurcation of the system (1) at equilibrium point, and the generation of periodic solutions as parameters change.

Zero-Hopf bifurcation analysis
We can verify that, for any choice of the parameters, E 0 (0, 0, 0, 0) is always an equilibrium point for the hyperchaotic system (1). Moreover, when c = 0, system (1) has a line equilibrium (0, 0, z, 0); when -c(d+j-bd) d-e > 0, system (1) has a pair of symmetrical equilibria: In the next theorem, we will give the zero-Hopf equilibrium point of the system (1).
In the rest of this section, we will study the zero-Hopf bifurcation and periodic solutions of the hyperchaotic system (1) at the equilibrium point E 0 .
is a sufficiently small parameter and a 1 , c 1 , j 1 , d 1 are nonzero real parameters. The following statements hold.

Averaging theory of periodic orbits
The averaging theory is a classical and mature tool for studying the dynamic behavior of nonlinear dynamical systems, especially for the study of periodic solutions. In recent years, it has been improved and applied well. The classical averaging theory [25] is as follows.
Consider differential system: with x ∈ D, where D is an open subset of R n , t ≥ 0. We assume that F(t, x) and G(t, x, ε) are T-periodic in t. We define the averaged function then there exists a T-periodic solution x(t, ε) of system (2) such that x(0, ε) → p as ε → 0. (b) If the eigenvalue of the Jacobian matrix ( ∂f ∂x ) has a negative real part, the periodic solution x(t, ε) is asymptotically stable.

Proofs
In this section we will provide the proofs of Theorem 1 and Theorem 2.
Theorem 1 is proved.
(I) For the first solution s 0 , it has the Jacobian det ∂f ∂x (s 0 ) = 0.
Then, by Theorem 3, we know the periodic solution cannot be determined.
When a 1 = d 1 , det( ∂f ∂S (s 1 )) = 0. Then according to Theorem 3, we see that the system (11) has a periodic solution x 1 (θ , ε) such that x 1 (0, ε) = s 1 + o(ε). Bring the solution back to the system (9), and we have a periodic solution U(t, ε)). Then the system (6) has a periodic solution To determine the stability of the periodic solution εΦ 1 (t, ε), we calculate the eigenvalues of the Jacobian matrix ∂f ∂x (s 1 ). The eigenvalues are given as follows: Now we discuss the stability of the periodic solution when the eigenvalues are real and imaginary, respectively, and obtain the following solutions.
Similarly, we discuss the case where the eigenvalues are real and imaginary, respectively. Then we arrive at the following conclusions.

Conclusion
Four-dimensional hyperchaotic systems have complex dynamic behavior and are widely used. In this paper, we study a four-dimensional smooth quadratic autonomous hyperchaotic system, and we prove that the system has a zero-Hopf bifurcation at the origin of coordinates. The existence of periodic solutions of the system is proved by the classical averaging method, and the stability conditions of periodic solutions are given. In fact, there are many other rich dynamic properties of this hyperchaotic system that are not fully exploited. We hope to have other discoveries about this system in the future work.