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 x ⃛ = f ( x , x ̇ , x ̈ ) , where x , x ̇ , x ̈ and x ⃛ represent the position, velocity, acceleration, and jerk, respectively. The jerk differential equation can be written as the jerk differential system x ̇ = y , y ̇ = z , z ̇ = f ( x , y , z ) , in R 3 . In this paper we study the jerk differential system with f ( x , y , z ) = − a x ( 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 .