Around 1960, mathematician Edward Lorenz found unexpected behavior in

Question

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.    1 
 1. The main purpose of this passage is to

(A) explore a common misconception about a complexphysical 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

2  
 2. 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.

3  
 3. 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

4  
 4. 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

5  
 5. 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

6  
 6. 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.

7

“staggeringly complex pathways whose tangles appear accidental but are in fact determined by the system’s fundamental equations.”   
 7. Select the sentence in the second or third paragraph that illustrates why “chaos theory” might be called a misnomer.

Sajjad1994 · Answer

OE

1. The main purpose of this passage is to

The first paragraph introduces chaos theory by describing a historical moment in its development. The Point comes at the end of the first paragraph:  “chaos theory ... has little to do with randomness.”  The next three paragraphs focus on further explaining this mystery with analogies involving poppy seeds and bread dough to illustrate the explanations. Finally, as a minor addendum, the last paragraph mentions how this understanding of chaos theory might be applied to the weather, as a possible specific case of a chaotic system.

Taking all of these roles together, the main purpose of the passage is to introduce chaos theory and explain how chaotic systems seem to be random (a common misconception) but actually are governed by very complex equations. explore a common misconception about a complex physical system trace the historical development of a scientific theory distinguish a mathematical pattern from its opposite describe the spread of a technical model from one field of study to others contrast possible causes of weather phenomena

A) CORRECT.   The complicated aspects are the characteristic features of chaotic systems, such as  “sensitive dependence on initial conditions and staggeringly complex pathways. ” The Point of the passage is to explain such features.

B)  The first paragraph, as an introduction, describes a particular milestone in the historical development of chaos theory However, the passage does not go on to describe other developments of this theory over time.

C)  Perhaps the behavior of chaotic systems could arguably be described as a  “mathematical pattern.”  However, the passage does not discuss any category of systems that is categorized clearly as the opposite of chaotic systems. Certain non-chaotic systems are described in the second paragraph, but it is not clear whether these systems would be the opposite of chaotic systems, or whether random systems would be the opposite.

D)  If chaos theory is the technical model mentioned in the answer choice, the passage never describes how that model spreads from one field of study to any other.

E)  In the fourth paragraph, the  “Butterfly Effect”  is mentioned as a popular explanation for at least some hurricanes. However, no other causes of weather phenomena are ever discussed.

Answer: A

Sajjad1994 · Answer

OE

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

The  third paragraph  describes what happens to these poppy seeds: they  “are intermixed seemingly at random.”  But the positions of the seeds are not random, as the next sentences emphasize. Rather, the seeds “are captured by  “strange attractors,”  staggeringly complex pathways whose tangles ... are in fact totally determined by the system’s fundamental equations.” Thus, the positions of the seeds are themselves  “determined by the system’s fundamental equations.”

A)    The passage mentions nothing about any stretching or folding of the poppy seeds themselves; rather, the dough is stretched and folded.

B)    The poppy seeds are scattered throughout the dough, but not in random clumps.

C)    The poppy seeds are caught in strange attractors, but there is nothing accidental about their capture. Moreover, the strange attractors described in the passage are not physical objects but rather mathematical pathways.

D)    Point attractors are not mentioned in relation to the doughmixing process. Also, the poppy seeds, which have been “intermixed seemingly at random,” are not placed at regular intervals.

E)   CORRECT.    The poppy seeds may seem to be scattered at random, but they follow the pathways of the strange attractors. These pathways, and thus the seeds’ positions, have been  “determined by the system’s fundamental equations.”   
 
  Answer: E

Sajjad1994 · Answer

OE

3. According to the passage, the rounding errors in Lorenz’s model

Use the key words  “rounding errors”  and  “Lorenz’s model”  to find the relevant text. The reference to Lorenz leads to the first paragraph:  “Lorenz realized that tiny rounding errors in his analog computer mushroomed over time, leading to erratic results.”  In other words, the rounding errors started out small but became larger.

A)  Although these rounding errors are in fact errors, nothing in the passage indicates or implies that the model overall was built incorrectly.

B)  The errors were not deliberately included in the model. The passage’s first sentence states that Lorenz found  “unexpected behavior”  in his model. It may be argued that the role of these errors is similar to the role of  “tiny fluctuations in atmospheric air currents” —that is, they both introduce uncertainty that grows over time. However, this answer choice claims incorrectly that the errors were inserted on purpose. 
  
  C) CORRECT.    This answer choice corresponds very closely to the statement in the passage. Some synonyms have been used, but the meaning is the same:  “were imperceptibly small at first”  substitutes for  “tiny,”  and  “tended to grow”  substitutes for  “mushroomed over time.”

D)    The passage indicates that the behavior of the model was unexpected. Nothing in the passage indicates that Lorenz expected the errors at all.

E)  The errors did not shrink, but rather  “mushroomed over time.”   
 
  Answer: C

Sajjad1994 · Answer

OE

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

The passage mentions several examples of systems, both chaotic and non-chaotic, to illustrate the special characteristics of chaos. This question is an exercise in finding the references to the four wrong answers quickly.

A)  A dough-mixing machine is first mentioned at the beginning of the third paragraph as an example of chaos in action:  “Chaotic systems, such as a machine mixing bread dough ...”

B)    Atmospheric weather patterns as a system to be studied are mentioned in both the first and the last paragraphs. In the last paragraph, the passage states that the Earth’s weather may be an example of a chaotic system.

C)    Poppy seeds placed on an upside-down bowl are described in the second paragraph as an example of a non-chaotic system that creates divergence.

D)    Poppy seeds placed in a bowl that is right-side-up are described in the second paragraph as an example of a non-chaotic system that creates convergence.

E) CORRECT.    Butterfly flight patterns are not discussed as examples of systems themselves. According to the last paragraph, the “Butterfly Effect” is caused by the flapping of a single butterfly’s wings, and this effect can potentially affect atmospheric systems.  
 
  Answer: E

Sajjad1994 · Answer

OE

5. 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?

Stripped down to its essence, the question asks you to infer which of the five choices describes a system that is the most chaotic, according to the characteristics of chaos outlined in the passage. The most important proof sentence is at the beginning of the third paragraph:  “Chaotic systems, such as a machine mixing bread dough, are characterized by both attraction and repulsion.”  Thus, you should look for the system that is the most analogous to the dough-mixing machine. Moreover, the system should contain both attractive and repulsive elements: in other words, the two items embedded within the system should sometimes come near each other and then separate again.

At the beginning of the fourth paragraph, there is a  “red herring”  sentence:  “During the dough-kneading process, two poppy seeds positioned next to each other eventually go their separate ways.”  This sentence could lead you to think that the defining characteristic of chaotic systems is simply that two embedded items move away from each other. The question is asked in such a way as to focus your attention on the two items, so that you might then use this proof sentence alone and choose an incorrect answer.

A)  The two particles ejected from a nucleus do diverge, but they do not approach each other again. Moreover, there is no implication of any activity analogous to mixing bread dough.

B)  The stickers on the balloon separate and come together repeatedly. This behavior meets the criterion of  “both attraction     and repulsion.”  However, there is no mixing, and as a result, the   system cannot be said to be analogous to a machine mixing   dough.

C)  As in answer choice  (A) , the two items in question (avalanches) separate but never draw near each other again. Likewise, there is no mixing in the system.

D) CORRECT.    Two baseballs placed into a device designed to mix paint is analogous to two poppy seeds placed in bread dough being mixed by a machine: parts of the system are separated, intermingled, and brought back together again in regular, though complex, ways, as determined by the laws of physics. The pathways of the two baseballs will diverge and converge repeatedly, as in any other chaotic system.

E)  The two coins flipped into a bowl are closely analogous to the example in the second paragraph of the passage of two poppy seeds placed in a bowl and allowed to fall; this system is presented as non-chaotic.

Answer: D

Sajjad1994 · Answer

OE

6. The author implies which of the following about weather systems?

This is a Select-One-or-More question of a very specific variety. It is mostly relevant to the last paragraph of the passage, so you should make sure you have a solid understanding of the few sentences devoted to weather systems. 
  
  A. CORRECT.    Lorenz’s rounding errors are actually found in the   first paragraph, where you read that  “Lorenz realized that tiny     rounding errors ... mushroomed over time.”  Similarly, in the final   paragraph, you read:  “this microscopic uncertainty grows until it     encompasses even hurricanes.”  These are both examples of   chaotic systems.

B. The last sentence of the passage says:  “few metereologists     believe that we will ever be able to predict rain or shine for a     particular day years in the future.”  The sentence does not indicate   that meteorologists are unanimous; in fact,  “few meteorologists”    indicates that at least one actually believes that such predictions   might be able to be made in the future.

C.  While you are told that the wings of a butterly can affect   weather systems, you are never told that this is the most   important contributing factor. Likely, major climatic events are   more important than seemingly trivial events, such as a butterfly   taking flight.  
 
  Answer: A

Sajjad1994 · Answer

OE

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

Misnomer means that something has been given an incorrect or misleading name. You learned in the first paragraph that chaos theory, despite its name,  “has little to do with randomness.”  So you want to find a sentence in the second or third paragraph that illustrates this point.
 
 The final sentence of the third paragraph uses poppy seeds to show that even the bread-mixing machine, which appears to be mixing things at random (in a “chaotic” manner), is actually moving the seeds through “staggeringly complex pathways whose tangles appear accidental but are in fact determined by the system’s fundamental equations.” In other words, there’s nothing chaotic at all, only a very complex organization.   This is a perfect example of why “chaos theory” is a kind of misnomer.

Note that the second-to-last sentence of the third paragraph,  “But this randomness is illusory,”  does indicate that chaos theory might be called a misnomer. This sentence, though, does not illustrate why this is so.

Around 1960, mathematician Edward Lorenz found unexpected behavior in

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