Summary: A general methodology is described for constructing systems that have a slowly converging Lyapunov exponent near zero, based on one-dimensional maps with chaotic attractors. In certain parameter ranges, these relatively simple systems display the properties of intermittent dynamics known as chaotic itinerancy. We show that in addition to the local sensitivity characteristic of chaotic dynamics, these itinerant systems display a global sensitivity, in the sense that fine-scale additive noise may significantly change the natural measure on the large scale.