- What Could Be Worse than the Butterfly Effect?

Our understanding of classical mechanics (CM) has undergone significant growth in the latter half of the twentieth century and in the beginning of the twenty-first. This growth has much to do with the explosion of interest in the study of nonlinear systems in contrast with the focus on linear systems that had colored much work in CM from its inception. For example, although Maxwell and Poincaré arguably were some of the first to think about chaotic behavior, the modern study of chaotic dynamics traces its beginning to the pioneering work of Edward Lorenz (1963). This work has yielded a rich variety of behavior in relatively simple classical models that was previously unsuspected by the vast majority of the physics community (see Hilborn 2001). Chaos is a property of nonlinear systems that is usually characterized by sensitive dependence on initial conditions (SDIC). In CM the behavior of simple physical systems is described using models (such as the harmonic oscillator) that capture the main features of the systems in question (Giere 1988). Given two identical linear models of CM in nearly identical initial states, they will evolve in much the same way over long time scales. Two identical nonlinear models exhibiting SDIC in nearly identical states, however, will evolve in radically different ways in a relatively short time period because of extreme sensitivity to the smallest changes in initial states. **[End Page 519]**

This latter behavior has been popularized in the provocative ‘butterfly effect,’ the idea that the flapping of a butterfly’s wings in Argentina could cause a tornado in Texas three weeks later, say. An even more provocative version of such an effect is the possibility that a quantum fluctuation in Argentina could cause a tornado in Texas five weeks later! Jesse Hobbs (1991) and Stephen Kellert (1993) have argued that quantum events can influence classical systems through SDIC. These sensitive dependence (SD) arguments raise interesting questions about whether CM is indeterministic (Bishop and Kronz 1999) and have tantalizing implications for larger philosophical issues such as free will/determinism debates (e.g., Kane 1996; Bishop 2002a).

This discovery of SDIC in nonlinear models runs counter to the textbook vision of CM, a vision guided by an almost exclusive focus on linear systems. Therefore, it is important to clearly distinguish between linear and nonlinear systems along with establishing some basic terminology (§I). The notions of SDIC and chaos also need clarification, since they play crucial roles in SD arguments. This will require some discussion of Lyapunov exponents as well as the relationship between nonlinear dynamics and chaos (§II and Appendix). For the sake of concreteness, it will also be useful to focus on the Laplacean vision for classical particle mechanics (e.g., Bishop 2002b, 2003 and 2005a), particularly the crucial notion of unique evolution (§III). The SD argument can then be stated in a clear form and its defenses, criticisms and limitations assessed in the more general context of nonlinear dynamics (§IV). Concluding remarks follow (§V).

Before jumping in, there is a fundamental question here about whether a CM model of a pendulum is really just an approximation of a full quantum treatment. This is a very deep issue, but one reason to think that classical and quantum systems should be treated as distinct is that the relationship between quantum mechanics (QM) and CM is much subtler than either an approximation treatment or correspondence principles suggest (Primas 1998; Bishop 2005b). There is no space here to treat this question, but the following analysis will presuppose the distinctness of classical and quantum systems.

## I. Linearity and Nonlinearity

Theoretical models in physics include equations of motion (often referred to as dynamical or evolution equations) describing the change in time of variables taken to adequately describe some physical system of interest. In addition, a complete specification of the initial state referred to as the *initial conditions* (ICs) for the model and/or a characterization of the boundaries for the model domain known as the *boundary* **[End Page 520]** *conditions* (BCs) are also required. As a simple example of a CM model, suppose we wanted to study the firing of a rubber...