In this study, we introduced the dynamical modeling of the chaos single-screw (ES) extruder and presented the route to chaos in CS verifying this model by the full three-dimensional numerical simulations. A system with infinite barrier zones is described as the integrable system which consists of the seperatrix with two homoclinic orbits and contains tori in the periodic state space. Periodically inserted non-barrier zone is regarded as a perturbation and we constructed an oneparameter solution map. Using this map, we defined the area-preserving Poincare map for the perturbed system. As the perturbation increases, homoclinic tangle leads to the Cantor set near homoclinic fixed point and elliptic rotations change to the resonance bands or KAM tori, depending on whether the corresponding winding number is rational or irrational. The finite element method of the multivariant Q1⁺P0 elements is applied to solve the velocity fields and the 4th order Runge-Kutta method is used for the particle tracing. The resulting Poincare section verifies the proposed dynamical model, showing the resonace band of rotation number x/3. The system with a large perturbation shows the wide stochastic regions where the random particle motions take place.