# Butterfly Models

These models and simulations have been tagged “Butterfly”.

These models and simulations have been tagged “Butterfly”.

Butterfly Effect

Sensitivity To Initial Conditions

*(sensitive dependence on initial conditions)*

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

If*M* is the state space for the map , then displays sensitive dependence to initial conditions if for any x in *M* and any δ > 0, there are y in *M*, with such that

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

Although the butterfly effect may appear to be an esoteric and unlikely behavior, it is exhibited by very simple systems: for example, a ball placed at the crest of a hill may roll into any of several valleys depending on, among other things, slight differences in initial position. Similarly the direction a pencil falls when held on its tip, or an universe during its initial stages.

These attractors apply to social systems and economics showing jumps between potential wells, and showing the strategic scaling behavior of rotating and cyclic systems whether they be social, economic, or complex spin or rotation of planets affecting weather and climate or spin of galaxies or elementary particles, or even a rock on the end of a piece of string.

What Playing with numbers is all about :)

If

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The lost energy associated with delay.

Hysteresis is the dependence of a system not only on its current environment but also on its past environment. This dependence arises because the system can be in more than one internal state. To predict its future development, either its internal state or its history must be known.[1] If a given input alternately increases and decreases, the output tends to form a loop as in the figure. However, loops may also occur because of a dynamic lag between input and output.

Hysteresis is produced by positive feedback to avoid unwanted rapid switching. Hysteresis has been identified in many other fields, including economics and biology.

Economic systems can exhibit hysteresis. For example, export performance is subject to strong hysteresis effects: because of the fixed transportation costs it may take a big push to start a country's exports, but once the transition is made, not much may be required to keep them going.

Hysteresis is used extensively in the area of labor markets. According to theories based on hysteresis, economic downturns (recession) result in an individual becoming unemployed, losing his/her skills (commonly developed 'on the job'), demotivated/disillusioned, and employers may use time spent in unemployment as a screen. In times of an economic upturn or 'boom', the workers affected will not share in the prosperity, remaining long-term unemployed (over 52 weeks). Hysteresis has been put forward[by whom?] as a possible explanation for the poor unemployment performance of many economies in the 1990s. Labor market reform, or strong economic growth, may not therefore aid this pool of long-term unemployed, and thus specific targeted training programs are presented as a possible policy solution.

One type of hysteresis is a simple lag between input and output. A simple example would be a sinusoidal input X(t) and output Y(t)that are separated by a phase lag φ:

Such behavior can occur in linear systems, and a more general form of response is

where χi is the instantaneous response and Φd(t-τ) is the response at time t to an impulse at time τ. In the frequency domain, input and output are related by a complex *generalized susceptibility*.[3]

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

Although the butterfly effect may appear to be an esoteric and unlikely behavior, it is exhibited by very simple systems: for example, a ball placed at the crest of a hill may roll into any of several valleys depending on, among other things, slight differences in initial position. Similarly the direction a pencil falls when held on its tip, or an universe during its initial stages.

These attractors apply to social systems and economics showing jumps between potential wells, and showing the strategic scaling behavior of rotating and cyclic systems whether they be social, economic, or complex spin or rotation of planets affecting weather and climate or spin of galaxies or elementary particles, or even a rock on the end of a piece of string.

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

Although the butterfly effect may appear to be an esoteric and unlikely behavior, it is exhibited by very simple systems: for example, a ball placed at the crest of a hill may roll into any of several valleys depending on, among other things, slight differences in initial position. Similarly the direction a pencil falls when held on its tip, or an universe during its initial stages.

These attractors apply to social systems and economics showing jumps between potential wells, and showing the strategic scaling behavior of rotating and cyclic systems whether they be social, economic, or complex spin or rotation of planets affecting weather and climate or spin of galaxies or elementary particles, or even a rock on the end of a piece of string.

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

\mathrm{e}^{a\tau} \, d(x,y)." src="http://upload.wikimedia.org/math/8/3/5/8355530aeaa6df83fe2a7851508f881a.png" style="border: none; vertical-align: middle;">

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

\mathrm{e}^{a\tau} \, d(x,y)." src="http://upload.wikimedia.org/math/8/3/5/8355530aeaa6df83fe2a7851508f881a.png" style="border: none; vertical-align: middle;">

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

Sensitivity To Initial Conditions

Navier Stokes Equations

Lorenz Attractor

Chaos Theory, Disorder and Entropy

What Playing with numbers is all about :)

If

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