1.5. Chaos—Or a Butterfly Spoils Laplace’s Dream 25 Exercise 1–20. Prove the above theorem. Hint: If we define u(t) = σp V (t) − σp W (t) + K , then the conclusion may be written as u(t) ≤ K eKt, which follows from Gronwall’s Inequality provided we can prove u(t) ≤ K + K t 0 u(s) ds. To show that, start from u(t) − K = σp V (t) − σp W (t) ≤ t 0 V (σp V (s)) − W (σp W (s)) ds and use V (σp V (s)) − W (σp W (s)) ≤ V (σp V (s)) − V (σp W (s)) + V (σW p (s)) − W (σW p (s)) ≤ (Ku(s) − ) + = Ku(s). 1.5. Chaos—Or a Butterfly Spoils Laplace’s Dream L’´ etat pr´ esent du syst` eme de la Nature est ´ evidemment une suite de ce qu’elle ´ etait au moment pr´ ec´ edent et, si nous concevons une intelligence qui, pour un instant donn´ e, em- brasse tous les rapports des ˆ etres de cet Univers, elle pourra d´ eterminer pour un temps quelconque pris dans le pass´ e ou dans l’avenir la position respective, les motions et g´ en´erale- ment toutes les affections de ces ˆ etres... —Pierre Simon de Laplace, 17731 The so-called “scientific method” is a loosely defined iterative process of experimentation, induction, and deduction with the goal of deriv- ing general “laws” for describing various aspects of reality. Prediction plays a central role in this enterprise. During the period of discovery and research, comparing experiments against predictions helps elim- inate erroneous preliminary versions of a theory and conversely can provide confirming evidence when a theory is correct. And when a theory finally has been validated, its predictive power can lead to valu- able new technologies. In the physical sciences, the laws frequently take the form of differential equations (of just the sort we have been 1 The current state of Nature is evidently a consequence of what it was in the preceding moment, and if we conceive of an intelligence that at a given moment knows the relations of all things of this Universe, it could then tell the positions, motions and effects of all of these entities at any past or future time. . .

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2009 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.