The Chaos Screw (CS) nonlinear dynamical model is proposed to describe the development of chaos in a single-screw extrusion process and the model is verified by three-dimensional numerical simulations. The only-barrier channel is the unperturbed Hamiltonian system, which consists of two homoclinic orbits and nested elliptic tori of nonlinear oscillation in periodic (extended) state space. A periodically inserted no-barrier zone represents a perturbation. For small perturbations, homoclinic tangle leads to the Canter set near the homoclinic fixed point and elliptic rotations are changed into the resonance bands or KAM tori, depending on the commensurability of frequency ratio of the corresponding orbits. A finite element method of multivariant Q1+P0 elements is applied to solve the velocity fields and a 4th order Runge-Kutta method is used for the particle tracing. The resulting Poincare: section verifies the proposed dynamical model, showing the resonance band corresponding to rotation number 1/3 under small perturbations, As the strength of perturbation increases, the Poincare sections indicate wider stochastic regions in which random particle motions take place.