Properties of an intense nonneutral heavy ion beam propagating through a periodic quadrupole focussing magnetic field are studied analytically and numerically including self‐field effects. For a Kapchinskj‐Vladimirski (K‐V) beam equilibrium with uniform density profile, the two nonlinear coupled envelope equations are solved for the case of amatchedbeam in the smooth‐beam approximation. Space‐charge effects are taken into account self‐consistently. The analytical solutions for the envelope and the (space‐charge‐depressed) phase advance are compared with the (exact) numerical results, and the technique can be applied to a wide range of system parameters and choices of lattice functions &kgr;q(s)=&kgr;q(s+S). Furthermore, a fully nonlinear study of the envelope equations for the case of amismatchedbeam is carried out numerically. In certain parameter regimes, the beam envelope is found to exhibit bifurcation which results in chaotic behavior of the beam envelope when the focussing field strength is increased beyond a particular threshold. The (unstable) chaotic region in the parameter space (&sgr;0, KS/&Vegr;) is determined numerically. Here, &sgr;0is a measure of the focussing strength defined by &sgr;20/S2≡⟨(∫ss0 ds&kgr;q(s))2⟩,Kis the self‐field perveance,sis the periodicity length of the focussing lattice, and &Vegr; is the unnormalized beam emittance. The relationship between chaoticity and well‐known linear instabilities is examined. In addition, for the case of a nonuniform density beam, it is found that the particle trajectory exhibits intrinsic chaoticity. The consequence of chaotic particle orbits is further investigated. © 1995American Institute of Physics