"Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?" (Edward N Lorenz, [talk] 1972)
"If a single flap of a butterfly's wing can be instrumental in generating a tornado, so all the previous and subsequent flaps of its wings, as can the flaps of the wings of the millions of other butterflies, not to mention the activities of innumerable more powerful creatures, including our own species." (Edward N Lorenz, [talk] 1972)
"If the flap of a butterfly’s wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado." (Edward N Lorenz, [talk] 1972)
"Mathematicians seem to have no difficulty in creating new concepts faster than the old ones become well understood." (Edward N Lorenz, "A scientist by choice", [speech by acceptance of the Kyoto Prize], 1991)
"The term ‘chaos’ currently has a variety of accepted meanings, but here we shall use it to mean deterministically, or nearly deterministically, governed behavior that nevertheless looks rather random. Upon closer inspection, chaotic behavior will generally appear more systematic, but not so much so that it will repeat itself at regular intervals, as do, for example, the oceanic tides." (Edward N Lorenz, "Chaos, spontaneous climatic variations and detection of the greenhouse effect", 1991)
"Unfortunately, recognizing a system as chaotic will not tell us all that we might like to know. It will not provide us with a means of predicting the future course of the system. It will tell us that there is a limit to how far ahead we can predict, but it may not tell us what this limit is. Perhaps the best advice that chaos ‘theory’ can give us is not to jump at conclusions; unexpected occurrences may constitute perfectly normal behavior." (Edward N Lorenz, "Chaos, spontaneous climatic variations and detection of the greenhouse effect", 1991)
"A state of equilibrium is one that remains unchanged as time advances. An equilibrium is unstable if a state that differs slightly from it, such as one that you might purposely produce by disturbing it a bit, will presently evolve into a vastly different state […]. It is stable if a slight initial disturbance fails to have a large subsequent effect. The concept of equilibrium, stable or unstable, can be extended to include periodic behavior." (Edward N Lorenz, "The Essence of Chaos", 1993)
"According to the narrower definition of randomness, a random sequence of events is one in which anything that can ever happen can happen next. Usually it is also understood that the probability that a given event will happen next is the same as the probability that a like event will happen at any later time. [...] According to the broader definition of randomness, a random sequence is simply one in which any one of several things can happen next, even though not necessarily anything that can ever happen can happen next." (Edward N Lorenz, "The Essence of Chaos", 1993)
"An attractor that consists of an infinite number of curves, surfaces, or higher-dimensional manifolds - generalizations of surfaces to multidimensional space - often occurring in parallel sets, with a gap between any two members of the set, is called a strange attractor." (Edward N Lorenz, "The Essence of Chaos", 1993)
"An example, which, like tossing a coin, is intimately associated with games of chance, is the shuffling of a deck of cards. […] the process is not completely random, if by what happens next we mean the outcome of the next single riffle, since one riffle cannot change any given order of the cards in the deck to any other given order. In particular, a single riffle cannot completely reverse the order of the cards, although a sufficient number of successive riffles, of course, can produce any order." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Attractors are examples of invariant sets - sets that will consist of precisely the same points if each point is replaced by the point to which it is mapped, When there is more than one attractor, each basin of attraction is an invariant set, as is the basin boundary, sometimes called a separatrix. There is still another invariant set, which connects the attractors when there are more than one, and which by analogy ought to be called a 'connectrix', but generally, together with the attractors that it connects, is called the attracting set. Despite its name, the attracting set should not be confused with the set of attractors, which is sometimes only a portion of it."
"Because chaos is deterministic, or nearly so, games of
chance should not be expected to provide us with simple examples, but games
that appear to involve chance ought to be able to take their place. Among the
devices that can produce chaos, the one that is nearest of kin to the coin or
the deck of cards may well be the pinball machine. It should be an
old-fashioned one, with no flippers or flashing lights, and with nothing but
simple pins to disturb the free roll of the ball until it scores or becomes
dead."
"[…] bifurcations - the abrupt changes that can take place in the behavior, and often in the complexity, of a system when the value of a constant is altered slightly." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Clearly, however, a zero probability is not the same thing as an impossibility; […] In systems that are now called chaotic, most initial states are followed by nonperiodic behavior, and only a special few lead to periodicity. […] In limited chaos, encountering nonperiodic behavior is analogous to striking a point on the diagonal of the square; although it is possible, its probability is zero. In full chaos, the probability of encountering periodic behavior is zero." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Dynamical systems that vary continuously, like the pendulum
and the rolling rock, and evidently the pinball machine when a ball’s complete
motion is considered, are technically known as flows. The mathematical tool for
handling a flow is the differential equation. A system of differential
equations amounts to a set of formulas that together express the rates at which
all of the variables are currently changing, in terms of the current values of
the variables."
"Dynamical systems that vary in discrete steps […] are
technically known as mappings. The mathematical tool for handling a mapping is
the difference equation. A system of difference equations amounts to a set of
formulas that together express the values of all of the variables at the next
step in terms of the values at the current step. […] For mappings, the
difference equations directly express future states in terms of present ones,
and obtaining chronological sequences of points poses no problems. For flows,
the differential equations must first be solved. General solutions of equations
whose particular solutions are chaotic cannot ordinarily be found, and
approximations to the latter are usually determined by numerical methods."
"How can deterministic behavior look random? If truly identical states do occur on two or more occasions, it is unlikely that the identical states that will necessarily follow will be perceived as being appreciably different. What can readily happen instead is that almost, but not quite, identical states occurring on two occasions will appear to be just alike, while the states that follow, which need not be even nearly alike, will be observably different. In fact, in some dynamical systems it is normal for two almost identical states to be followed, after a sufficient time lapse, by two states bearing no more resemblance than two states chosen at random from a long sequence. Systems in which this is the case are said to be sensitively dependent on initial conditions. With a few more qualifications, to be considered presently, sensitive dependence can serve as an acceptable definition of chaos [...]" (Edward N Lorenz, "The Essence of Chaos", 1993)
"In practice, it may be impossible to purge a real system of its actual randomness and observe the consequences, but often we can guess what these would be by turning to theory. Most theoretical studies of real phenomena are studies of approximations." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Just as few concrete physical systems are strictly
deterministic in their behavior, so very few are strictly linear. The great
importance of linearity lies in a combination of two circumstances. First, many
tangible phenomena behave approximately linearly over restricted periods of
time or restricted ranges of the variables, so that useful linear mathematical
models can simulate their behavior. A pendulum swinging through a small angle
is a nearly linear system. Second, linear equations can be handled by a wide
variety of techniques that do not work with nonlinear equations."
"Mathematicians have found it advantageous to introduce the concept of a completely random continuous process, but it is hard to picture what such a process in nature might look like." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Mathematicians typically do not feel that they have
completely solved a system of differential equations until they have written down
a general solution - a set of formulas giving the value of each variable at
every time, in terms of the supposedly known values at some initial time."
"Physical complexity and mathematical complexity are not the same thing. It is quite possible to replace one system by another that is physically more complicated but mathematically simpler." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Some fractals come close to qualifying as chaos by being produced
by uncomplicated rules while appearing highly intricate and not just unfamiliar
in structure. There is, however, one very close liaison between fractality and
chaos; strange attractors are fractals."
"Systems that vary deterministically as time progresses, such as mathematical models of the swinging pendulum, the rolling rock, and the breaking wave, and also systems that vary with an inconsequential amount of randomness - possibly a real pendulum, rock, or wave - are technically known as dynamical systems." (Edward N Lorenz, "The Essence of Chaos", 1993)
"The coin is an example of complete randomness. It is the sort of randomness that one commonly has in mind when thinking of random numbers, or deciding to use a random-number generator." (Edward N Lorenz, "The Essence of Chaos", 1993)
"The definition of unstable equilibrium has much in common with that of sensitive dependence - both involve the amplification of initially small differences. The distinction between a system that merely possesses some states of unstable equilibrium and one that is chaotic is that, in a system of the latter type, the future course of every state, regardless of whether it is a state of equilibrium, will differ more and more from the future courses of slightly different states."
"The pinball machine is one of those rare dynamical systems whose chaotic nature we can deduce by pure qualitative reasoning, with fair confidence that we have not wandered astray. Nevertheless, the angles in the paths of the balls that are introduced whenever a ball strikes a pin and rebounds […] render the system some what inconvenient for detailed quantitative study." (Edward N Lorenz, "The Essence of Chaos", 1993)
"Topology deals with those properties of curves, surfaces, and more general aggregates of points that are not changed by continuous stretching, squeezing, or bending. To a topologist, a circle and a square are the same, because either one can easily be bent into the shape of the other. In three dimensions, a circle and a closed curve with an overhand knot in it are topologically different, because no amount of bending, squeezing, or stretching will remove the knot." (Edward N Lorenz, "The Essence of Chaos", 1993)
"When a system has more than one attractor, the points in
phase space that are attracted to a particular attractor form the basin of attraction
for that attractor. Each basin contains its attractor, but consists mostly of
points that represent transient states. Two contiguous basins of attraction
will be separated by a basin boundary."
"When the pinball game is treated as a flow instead of a mapping, and a simple enough system of differential equations is used as a model, it may be possible to solve the equations. A complete solution will contain expressions that give the values of the variables at any given time in terms of the values at any previous time. When the times are those of consecutive strikes on a pin, the expressions will amount to nothing more than a system of difference equations, which in this case will have been derived by solving the differential equations. Thus a mapping will have been derived from a flow." (Edward N Lorenz, "The Essence of Chaos", 1993)
"When a butterfly flutters its wings in one part of the world, it can eventually cause a hurricane in another." (Edward N Lorenz)
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