What is butterfly effect ?

 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 is closely associated with the work of mathematician and meteorologist Edward Lorenz.

What is butterfly effect in simple terms ?

The butterfly effect is an idea that is more commonly used in chaos theory. A small change can make much bigger changes happen; one small incident can have a big impact on the future. The idea started from weather prediction. ... This term can be used in areas other than weather.

What is butterfly effect theory ?

The butterfly effect is the idea that small things can have non-linear impacts on a complex system. The concept is imagined with a butterfly flapping its wings and causing a typhoon. Of course, a single act like the butterfly flapping its wings cannot cause a typhoon.

IS butterfly effect is dangerous?

The butterfly effect is the semi-serious claim that a butterfly flapping its wings can cause a tornado half way around the world. It's a poetic way of saying that some systems show sensitive dependence on initial conditions, that the slightest change now can make an enormous difference later.

Can  butterfly effect cause tsunami ?

Put more simply, this is the so-called butterfly effect: the light fluttering of butterfly wings may cause unpredictable consequences or, more graphically, can lead to large-scale phenomena like tsunamis.

What is the butterfly theory is love ?

Observing the meaning of the butterfly effect, the most common definition of the term I found was, that it is a metaphoric term. However, in theory, it means something small in statures like a bird or a butterfly could flap its wings and make a great event happen in another part of the world.

Mathematical Definition of butterfly effect ?

Recurrence , the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making complex system, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather) since it is impossible to measure the starting atmospheric conditions completely accurately.

A dynamical system displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical.

If M is the state space for the map ${\displaystyle f^{t}}$, then ${\displaystyle f^{t}}$ displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with distance d(. , .) such that ${\displaystyle 0<d(x,y)<\delta }$ and such that

${\displaystyle d(f^{\tau }(x),f^{\tau }(y))>\mathrm {e} ^{a\tau }\,d(x,y)}$

for some positive parameter a. The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive lyapunov exponent 

The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of the logistic map 

${\displaystyle x_{n+1}=4x_{n}(1-x_{n}),\quad 0\leq x_{0}\leq 1,}$

which, unlike most chaotic maps, has a closed form solution 

${\displaystyle x_{n}=\sin ^{2}(2^{n}\theta \pi )}$

where the initial condition parameter ${\displaystyle \theta }$ is given by ${\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})}$. For rational ${\displaystyle \theta }$, after a finite number of iteration   ${\displaystyle x_{n}}$ maps into a periodic sequence . But almost all  are irrational, and, for irrational ${\displaystyle \theta }$, ${\displaystyle x_{n}}$ never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2ⁿ shows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared sine function keeps ${\displaystyle x_{n}}$ folded within the range [0, 1].