Non-isothermal melt spinning of materials having a step-like viscosity variation with temperature is studied in this work. A set of nonlinear equations is used to describe the fiber behavior and to obtain the draw ratio, the square of the ratio of the fiber diameter at the entrance to that at the exit of the fiber-spinning device. The fluid-flow equation is based on a slender-jet approximation, and external heating and cooling have been accounted for with a one-dimensional model in order to obtain the fiber temperature and viscosity along the fiber length. The model is similar to that used by Wylie et al. [J. Fluid Mech. 570, 1 (2007)], but accounts for inertia, shear stress at the fiber surface, surface tension, gravity, cooling, and larger heating rates. Steady-state analysis reveals that the draw ratio increases with an increase in the pulling force, passes through a maximum, and then starts increasing again, resulting in three possible pulling forces for the same draw ratio. However, linear stability analysis reveals that depending on the strength of heating and/or cooling, at most two of the steady states are stable. The stability analysis also predicts complicated oscillatory and non-oscillatory dynamical behavior as the pulling force varies. Nonlinear simulations reveal that an unstable system always tends to limit-cycle behavior. Systems predicted stable by the linear stability analysis are also stable for large-amplitude perturbations. External heating is found to dramatically enhance the draw ratio of the melt-spinning process. The addition of a cooling section suppresses the draw ratio, but this can be compensated for with a higher heating strength.