Abstract
This paper is devoted to local bifurcations of three-dimensional (3D) quadratic jerk system. First, we start by analysing the saddle-node bifurcation. Then we introduce the concept of canonical system. Next, we study the transcritial bifurcation of canonical system. Finally we study the zero-Hopf bifurcations of canonical system, which constitutes the core contributions of this paper. By averaging theory of first order, we prove that, at most, one limit cycle bifurcates from the zero-Hopf equilibrium. By averaging theory of second order, third order, and fourth order, we show that, at most, two limit cycles bifurcate from the equilibrium. Overall, this paper can help to increase our understanding of local behaviour in the jerk dynamical system with quadratic non-linearity.
Highlights
Consider the following system of ordinary differential equations dx= f ( x, μ), dt x ∈ Rn, μ ∈ Rs, (1)where f is sufficiently smooth
By perturbing an equilibrium inside the canonical system and using averaging theory up to fourth order, we give a partial answer to the problem
We characterize when the equilibrium localized at the origin of canonical system is a zero-Hopf equilibrium
Summary
In the case of λ3 = 0, small-amplitude limit cycles may be found in some neighborhood of the origin This phenomenon is called the zero-Hopf bifurcation. Some examples of Hopf bifurcation analysis can be found in [3,4]. Some examples of zero-Hopf bifurcation analysis can be found in [5,6,7]. Complex dynamics, such as self-excited and hidden chaotic attractors, can be found in [7,8,9]. In these studies, the qualitative features of equilibria play an important role in determining the complex behaviour of the system.
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