Nonlinear dynamic processes are fundamental to the behavior of acoustic musical instruments, as is well explored in the case of sound production. Such processes may have profound and under-explored implications for how musicians interact with instruments, however. Although nonlinear dynamic processes are ubiquitous in acoustic instruments, they are present in digital musical tools only if explicitly implemented. Thus, an important resource with potentially major effects on how musicians interact with acoustic instruments is typically absent in the way musicians interact with digital instruments. Twenty-four interviews with free-improvising musicians were conducted to explore the role that nonlinear dynamics play in the participants' musical practices and to understand how such processes can afford distinctive methods of creative exploration. Thematic analysis of the interview data is used to demonstrate the potential for nonlinear dynamic processes to provide repeatable, learnable, controllable, and explorable interactions, and to establish a vocabulary for exploring nonlinear dynamic interactions. Two related approaches to engaging with nonlinear dynamic behaviors are elaborated: edge-like interaction, which involves the creative use of critical thresholds; and deep exploration, which involves exploring the virtually unlimited subtleties of a small control region. The elaboration of these approaches provides an important bridge that connects the concrete descriptions of interaction in musical practices, on the one hand, to the more-abstract mathematical formulation of nonlinear dynamic systems, on the other.