In the investigation, we explore different nonliner behaviors occur in sigma delta modulators (SDMs). The necessary and sufficient conditions for the occurrence of fixed point or limit cycle of bandpass SDMs are derived. Then the periodicity of the output sequences, the stability of these elliptical trajectories and the admissible set of periodic output sequences of bandpass SDMs are discussed. The phenomenon that the ellipses or elliptic fractal regions are the global attractors of the second order marginally stable bandpass SDM with sum of the numerator and denominator polynomials of the loop filter equal to one has been analytically explained and proved. The conditions for the occurrence of the trapezoids in bandpass SDMs are also given. Fractal patterns may also be exhibited in the phase plane when the system matrices of bandpass SDMs are strictly stable. This occurs when the sets of initial conditions corresponding to convergent or limit cycle behavior do not cover the whole phase plane. When the leading coefficient of the numerator of the loop filter of bandpass SDMs is positive, the limit cycle (if it exists) will become unstable, and the global attractor region does not exist. Hence, the trajectory will diverge for some initial conditions. The growth is linear and the spectrum of the output sequence has a pole at the natural frequency of the loop filter. This result can be used to distinguish the spectra of elliptic fractal patterns confiend in the trapezoids or irregular chaotic patterns from that of the divergent patterns. Finally, the global stability criterion of high order lowpass SDMs is investigated. The global stability of SDMs does not only depend on the existence of an invariant set or whether the range of the system mapping form a partition on the invariant set or not, but it also depends on the size of the hyperplane formed from the projection of the invariant set onto its first co-ordinate (based on the direct form realization), and the sign of the product of the leading filter coefficients in the numerator and that in the denominator as well.