5.2. Characteristics of Chaos 235 After a time t, the two points Mo and No have evolved to Mt and Nt. Sensitive dependence says that the distance d(Mt, Nt) grows exponentially with time for some > (Figure 5.13): d(Mt - Nt) = exit. d(Mo - No) Mt Mo d(0)= do d(t) = delt No Figure 5.13: Exponential divergence over time of nearby trajectories. This is characteristic of chaotic systems. In general, for multivariable systems, in both discrete and continuous time, sensitive dependence means exponential divergence of nearby trajectories: there is a number 1 (greek letter lambda), called the Lyapunov characteristic exponent, such that d(Mt - Nt) = exit. d(Mo - No) Exercise 5.2.3 Derive the expression for exponential divergence in the case of the discrete- time logistic system. (Hint: Here, because the logistic system has a single state variable, the distance between two points X and Y is just the absolute value of their difference, (X - Y|.) But of course, this exponentially fast divergence cannot continue forever, because the whole behavior is contained in a box. Therefore, two nearby trajectories will start by diverging from each other exponentially fast, but then they will ultimately be folded back into the box by the dynamics. This tension between "wanting to diverge" and "staying in the box" creates many of the key properties of chaos. Unpredictability Edward Lorenz, the meteorologist and mathematician who helped discover chaos, gave a talk at the annual meeting of the American Association for the Advancement of Science in 1972, called "Predictability: does the flap of a butterfly's wings in Brazil set off a tornado in Texas?"