Integrability and periodic orbits of a 3D jerk system with two quadratic nonlinearities

In mechanics jerk is the rate of change of an object’s acceleration over time. Thus a jerk equation is a differential equation of the form $\stackrel{⃛}{x}=f(x,\dot{x},\ddot{x})$, where $x$, $\dot{x}$, $\ddot{x}$ and $\stackrel{⃛}{x}$ represent the position, velocity, acceleration, and jerk, respectively. The jerk differential equation can be written as the jerk differential system $\dot{x}=y,\dot{y}=z,\dot{z}=f(x,y,z),$ in $R^3$. In this paper we study the jerk differential system with $f(x,y,z)=-ax(1-x)-y+b{y}^2$, previously studied by other authors showing that this system can exhibit chaos for some values of its parameters. When the parameters $a=b=0$ the $x$-axis is filled with zero-Hopf equilibria, and all the other orbits are periodic. Here we prove analytically the existence of two families of periodic orbits for sufficiently small values of the parameters $a$ and $b$. One family bifurcates from the non-isolated zero-Hopf equilibrium $(1,0,0)$ of the jerk system with $a=b=0$, while the other family bifurcates from a periodic orbit of the jerk system with $a=b=0$.