Around 1960, mathematician Edward Lorenz found unexpected behavior in apparently simple equations representing atmospheric air flows. Whenever he reran his model with the same inputs, different outputs resulted—although the model lacked any random elements. Lorenz realized that tiny rounding errors in his analog computer mushroomed over time, leading to erratic results. His findings marked a seminal moment in the development of chaos theory, which, despite its name, has little to do with randomness.

To understand how unpredictability can arise from deterministic equations, which do not involve chance outcomes, consider the non-chaotic system of two poppy seeds placed in a round bowl. As the seeds roll to the bowl's center, a position known as a point attractor, the distance between the seeds shrinks. If, instead, the bowl is flipped over, two seeds placed on top will roll away from each other. Such a system, while still not technically chaotic, enlarges initial differences in position.

Chaotic systems, such as a machine mixing bread dough, are characterized by both attraction and repulsion. As the dough is stretched, folded, and pressed back together, any poppy seeds sprinkled in are intermixed seemingly at random. But this randomness is illusory. In fact, the poppy seeds are captured by “strange attractors,” staggeringly complex pathways whose tangles appear accidental but are in fact determined by the system's fundamental equations.

During the dough-kneading process, two poppy seeds positioned next to each other eventually go their separate ways. Any early divergence or measurement error is repeatedly amplified by the mixing until the position of any seed becomes effectively unpredictable. It is this “sensitive dependence on initial conditions” and not true randomness that generates unpredictability in chaotic systems, of which one example may be the Earth's weather. According to the popular interpretation of the “Butterfly Effect,” a butterfly flapping its wings causes hurricanes. A better understanding is that the butterfly causes uncertainty about the precise state of the air. This microscopic uncertainty grows until it encompasses even hurricanes. Few meteorologists believe that we will ever be able to predict rain or shine for a particular day years in the future.

The main purpose of this passage is to

(A) explore a common misconception about a complex physical system
(B) trace the historical development of a scientific theory
(C) distinguish a mathematical pattern from its opposite
(D) describe the spread of a technical model from one field of study to others
(E) contrast possible causes of weather phenomena



In the example discussed in the passage, what is true about poppy seeds in bread dough, once the dough has been thoroughly mixed?

(A) They have been individually stretched and folded over, like miniature versions of the entire dough.
(B) They are scattered in random clumps throughout the dough.
(C) They are accidentally caught in tangled objects called strange attractors.
(D) They are bound to regularly dispersed patterns of point attractors.
(E) They are in positions dictated by the underlying equations that govern the mixing process.



According to the passage, the rounding errors in Lorenz's model

(A) indicated that the model was programmed in a fundamentally faulty way
(B) were deliberately included to represent tiny fluctuations in atmospheric air currents
(C) were imperceptibly small at first, but tended to grow
(D) were at least partially expected, given the complexity of the actual atmosphere
(E) shrank to insignificant levels during each trial of the model



The passage mentions each of the following as an example or potential example of a chaotic or non-chaotic system EXCEPT

(A) a dough-mixing machine
(B) atmospheric weather patterns
(C) poppy seeds placed on top of an upside-down bowl
(D) poppy seeds placed in a right-side-up bowl
(E) fluctuating butterfly flight patterns



It can be inferred from the passage that which of the following pairs of items would most likely follow typical pathways within a chaotic system?

(A) Two particles ejected in random directions from the same decaying atomic nucleus.
(B) Two stickers affixed to a balloon that expands and contracts over and over again.
(C) Two avalanches sliding down opposite sides of the same mountain.
(D) Two baseballs placed into a device designed to mix paint.
(E) Two coins flipped into a large bowl.



The author implies which of the following about weather systems?

Indicate \(all\) that apply.


A. They illustrate the same fundamental phenomenon as Lorenz's rounding errors.
B. Experts agree unanimously that weather will never be predictable years in advance.
C. They are governed mostly by seemingly trivial events, such as the flapping of a butterfly's wings.



Select the sentence in the second or third paragraph that illustrates why “chaos theory” might be called a misnomer.