It's not particularly that there's sensitive dependence on initial conditions: we actually have measurements that should be precise enough to determine what will happen for a long time.
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And as part of his work, Poincaré also saw something else: that at least in particular cases of the three-body problem, there was arbitrarily sensitive dependence on initial conditions—implying that even tiny errors in measurement could be amplified to arbitrarily large changes in predicted behavior (the classic "chaos theory" phenomenon).
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Chaotic dynamics are irregular and bounded and subject to sensitive dependence on initial conditions.
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Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated.
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Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior. Linear vector fields and a few trajectories.
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Note that the estimate is accurate for only a few time steps. This is a general characteristic of chaotic time series. This is a property of the sensitive dependence on initial conditions common to chaotic time series. A small initial error is amplified with time.
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The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in semiclassical and quantum physics including atoms in strong fields and the anisotropic Kepler problem. Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments; however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by Martin Gutzwiller and Delos and co-workers. The random matrix theory and simulations with quantum computers prove that some versions of the butterfly effect in quantum mechanics do not exist. Other authors suggest that the butterfly effect can be observed in quantum systems.
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Many mathematical models of physical systems are deterministic. This is true of most models involving differential equations (notably, those measuring rate of change over time). Mathematical models that are not deterministic because they involve randomness are called stochastic. Because of sensitive dependence on initial conditions, some deterministic models may appear to behave non- deterministically; in such cases, a deterministic interpretation of the model may not be useful due to numerical instability and a finite amount of precision in measurement.
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Devaney is known for formulating a simple and widely used definition of chaotic systems, one that does not need advanced concepts such as measure theory. In his 1989 book An Introduction to Chaotic Dynamical Systems, Devaney defined a system to be chaotic if it has sensitive dependence on initial conditions, it is topologically transitive (for any two open sets, some points from one set will eventually hit the other set), and its periodic orbits form a dense set.. Later, it was observed that this definition is redundant: sensitive dependence on initial conditions follows automatically as a mathematical consequence of the other two properties.. Devaney hairs, a fractal structure in certain Julia sets, are named after Devaney, who was the first to investigate them.. As well as research and teaching in mathematics, Devaney's mathematical activities have included organizing one-day immersion programs in mathematics for thousands of Boston- area high school students, and consulting on the mathematics behind media productions including the 2008 film 21 and the 1993 play Arcadia. He was president of the Mathematical Association of America from 2013 to 2015..
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A plot of Lorenz's strange attractor for values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a dynamical system that, starting from any of various arbitrarily close alternative initial conditions on the attractor, the iterated points will become arbitrarily spread out from each other. Experimental demonstration of the butterfly effect with different recordings of the same double pendulum. In each recording, the pendulum starts with almost the same initial condition.
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In The Vocation of Man (1800), Johann Gottlieb Fichte says "you could not remove a single grain of sand from its place without thereby ... changing something throughout all parts of the immeasurable whole". Chaos theory and the sensitive dependence on initial conditions were described in numerous forms of literature. This is evidenced by the case of the three-body problem by Henri Poincaré in 1890.Some Historical Notes: History of Chaos Theory He later proposed that such phenomena could be common, for example, in meteorology.
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Some nonlinear dynamical systems, such as the Lorenz system, can produce a mathematical phenomenon known as chaos. Chaos, as it applies to complex systems, refers to the sensitive dependence on initial conditions, or "butterfly effect", that a complex system can exhibit. In such a system, small changes to initial conditions can lead to dramatically different outcomes. Chaotic behavior can, therefore, be extremely hard to model numerically, because small rounding errors at an intermediate stage of computation can cause the model to generate completely inaccurate output.
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The Lorenz attractor displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor. Some dynamical systems, like the one-dimensional logistic map defined by x → 4 x (1 – x), are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an attractor, since then a large set of initial conditions leads to orbits that converge to this chaotic region.
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Others exhibit properties familiar from traditional science, such as thermodynamic behavior, continuum behavior, conserved quantities, percolation, sensitive dependence on initial conditions, and others. They have been used as models of traffic, material fracture, crystal growth, biological growth, and various sociological, geological, and ecological phenomena. Another feature of simple programs is that, according to the book, making them more complicated seems to have little effect on their overall complexity. A New Kind of Science argues that this is evidence that simple programs are enough to capture the essence of almost any complex system.
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This illustrates sensitive dependence on initial conditions—the mapping from the truncated initial condition has deviated exponentially from the mapping from the true initial condition. And since our simulation has reached a fixed point, for almost all initial conditions it will not describe the dynamics in the qualitatively correct way as chaotic. Equivalent to the concept of information loss is the concept of information gain. In practice some real-world process may generate a sequence of values {xn} over time, but we may only be able to observe these values in truncated form.
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Mortal Chaos is a series of novels for teenage and young adult readers which was published by Oxford University Press in 2012. Written by Matt Dickinson, the series explores the world of chaos theory and the butterfly effect. Previous published works by Matt Dickinson have been The Death Zone (Random House 1997), High Risk, (Random House 1999) and Black Ice (Random House 2001). Mortal Chaos is inspired by the work of Edward Lorenz and Ray Bradbury's 1952 story "A Sound of Thunder", in which the concept of sensitive dependence on initial conditions is applied to time travel.
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Over time the differences in the dynamics grow from almost unnoticeable to drastic. In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. The term butterfly effect is closely associated with the work of Edward Lorenz. It is derived from the metaphorical example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant butterfly flapping its wings several weeks earlier.
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Moreover, in those nonlinear systems showing chaotic behavior, the evolution of the variables exhibits sensitive dependence on initial conditions: the iterated values of any two very nearby points on the same strange attractor, while each remaining on the attractor, will diverge from each other over time. Thus even on a single attractor the precise values of the initial conditions make a substantial difference for the future positions of the iterates. This feature makes accurate simulation of future values difficult, and impossible over long horizons, because stating the initial conditions with exact precision is seldom possible and because rounding error is inevitable after even only a few iterations from an exact initial condition.
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Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self- organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in China can cause a hurricane in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.
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Stephen Wolfram also notes that the Lorenz equations are highly simplified and do not contain terms that represent viscous effects; he believes that these terms would tend to damp out small perturbations. While the "butterfly effect" is often explained as being synonymous with sensitive dependence on initial conditions of the kind described by Lorenz in his 1963 paper (and previously observed by Poincaré), the butterfly metaphor was originally appliedLorenz: "Predictability", AAAS 139th meeting, 1972 Retrieved May 22, 2015 to work he published in 1969 which took the idea a step further. Lorenz proposed a mathematical model for how tiny motions in the atmosphere scale up to affect larger systems. He found that the systems in that model could only be predicted up to a specific point in the future, and beyond that, reducing the error in the initial conditions would not increase the predictability (as long as the error is not zero).
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Wolfram based his classification of Rule 30 as chaotic based primarily on its visual appearance, and it was later shown to meet more rigorous definitions of chaos proposed by Devaney and Knudson. In particular, according to Devaney's criteria, Rule 30 displays sensitive dependence on initial conditions (two initial configurations that differ only in a small number of cells rapidly diverge), its periodic configurations are dense in the space of all configurations, according to the Cantor topology on the space of configurations (there is a periodic configuration with any finite pattern of cells), and it is mixing (for any two finite patterns of cells, there is a configuration containing one pattern that eventually leads to a configuration containing the other pattern). According to Knudson's criteria, it displays sensitive dependence and there is a dense orbit (an initial configuration that eventually displays any finite pattern of cells). Both of these characterizations of the rule's chaotic behavior follow from a simpler and easy to verify property of Rule 30: it is left permutative, meaning that if two configurations and differ in the state of a single cell at position , then after a single step the new configurations will differ at cell .
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