This paper examines the onset of chaotic oscillations in the N-layer model of stratified quasigeostrophic flow due to interaction of N vortex sections in different layers, and then considers the possible role of chaotic self-induced motion in the breakup of quasigeostrophic vortices. In order to investigate the nature of the long-time evolution of the vortex using a low-dimensional dynamical system, attention is restricted to vortices with uniform potential vorticity within the core and sufficiently large values of the stratification parameter that the vortex cross section remains nearly circular with time. Computations using contour dynamics in many-layered systems are performed to check that the deformation of the vortex cross section is small. It is shown that for finite-amplitude perturbations of the vortex centerline, the barotropic interaction between sections of the vortex in different layers can cause the vortex to oscillate chaotically. The approach to chaotic motion is initially investigated using a system with only four layers, in which the vortex is approximated by equal-strength segments of either point vortices or circular patches. The onset of chaos is shown to be sensitive to the stratification parameter, the vortex core radius, and the initial configuration. The consequences of chaos on the vortex evolution and breakup are then demonstrated by numerical computations with a large number of layers for vortices with finite-core area.