Abstract
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.
Highlights
Chaos is a complex dynamic phenomenon in nonlinear dynamical system, which exists widely in nature
Most of the research objects are three-dimensional chaotic systems, the biggest feature is that there is only a positive Lyapunov exponent, which reflects that the trajectory of the nonlinear system only generates instability in a certain direction, and develops exponentially
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]
Summary
Chaos is a complex dynamic phenomenon in nonlinear dynamical system, which exists widely in nature. 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 2018, Candido et al studied the zero-Hopf bifurcation of 16 three-dimensional differential systems without equilibrium by using the averaging theory [18]. 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]. We study the zero-Hopf bifurcation of the system (1) at equilibrium point, and the generation of periodic solutions as parameters change
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