= Butterfly effect
{wiki}
= Sensitive dependence on initial conditions
{synonym}
The butterfly effect is the intuitive idea that an arbitrarily small change in an <initial condition> can eventually change the observed future substantially. In <dynamical systems>, sensitive dependence makes this precise for a <map> $F:X\to X$ on a <metric space> $(X,d)$: there is one separation scale $\delta>0$ such that
$$
\forall x\in X\ \forall\varepsilon>0\ \exists y\in X\ \exists n\geq0:
\qquad d(x,y)<\varepsilon,\qquad d(F^n(x),F^n(y))>\delta.
$$
The same $\delta$ works at every point, whereas $y$ and the separation time $n$ may depend on the point and the requested closeness. For a <flow map>, replace the <iteration of a map> $F^n$ by $\Phi_t$ with some $t\geq0$. The <state space> matters: restricting to a single <fixed point> removes the alternative nearby states needed for sensitivity.
Sensitivity is compatible with <deterministic evolution>. Given an exact <initial condition>, there is still one <orbit>; uncertainty concerns which exact state a measurement represents. It also allows accurate short-term prediction. For a <continuous function> $F$, each fixed finite <iterated function> $F^n$ is continuous, so sufficiently close initial states remain close at any specified finite step. What fails is a single initial accuracy that controls every future step.
The <logistic map> illustrates both sides. At $r=2.8$, nearby <orbits> in the <basin of attraction> of its attracting <fixed point> approach one another. At $r=4$, arbitrarily close states can eventually separate by a macroscopic fraction of $[0,1]$. The definition asserts the existence of separating neighbours, not that every nearby pair must separate: $x$ and $1-x$ have exactly the same next state under the <logistic map> and hence the same subsequent <orbit>. Nor must separation increase at every step.
\Image[/sensitive-dependence.png]
{title=The same deterministic logistic rule can shrink or amplify initial uncertainty}
{description=Each pair starts at $0.2$ and $0.2+10^{-8}$ and uses the same <logistic map> parameter. At $r=2.8$, the <orbits> approach the same attracting <fixed point>. At $r=4$, this pair eventually gives visibly different states. The logarithmic panels expose changes hidden in the upper panels; dashed slopes illustrate the eventual contraction factor $0.8$ and a representative exponential growth factor $2$. They are reference slopes, not exact formulas for these finite separations. Distances fluctuate and remain bounded by one. These computations illustrate the phenomenon; two numerical <orbits> alone do not establish the quantified definition.}
A positive <Lyapunov exponent> describes average exponential growth of an infinitesimal displacement along an <orbit>, when the relevant <limit> exists. It is related to sensitivity but is not a substitute for its uniform definition. For a differentiable <interval map>, the <chain rule> gives
$$
\lambda(x_0)=\lim_{N\to\infty}\frac1N\sum_{j=0}^{N-1}\log|F'(x_j)|.
$$
At $r=4$, the <logistic map> has $\lambda=\log2$ for <Lebesgue measure>-almost every <initial condition>, while exceptional <orbits> can have different exponents. Finite-separation growth is only approximately exponential while the linear approximation remains useful. In a bounded <state space>, a separation cannot grow exponentially forever.
Sensitivity alone is also weaker than <Devaney chaos>, which additionally requires <topological transitivity> and <dense periodic points>. For example, $F(x)=2x$ on the real line has sensitive dependence and a positive <Lyapunov exponent>, but its only <periodic point> is zero, so it is not <Devaney-chaotic>. This distinguishes growth of uncertainty from the richer organization of recurrent <orbits>.
Edward N. Lorenz's "Deterministic Nonperiodic Flow" (1963) studied a <thermal convection> model with three <ordinary differential equations>. Its numerical <orbits> exhibited persistent nonperiodic motion and instability to small changes of the <initial condition>. Lorenz discussed the consequences for very-long-range <weather forecasting>: a deterministic law need not make a detailed distant future reliably recoverable from approximate observations. The historically relevant lesson of the <Lorenz system> is this growth of uncertainty under <deterministic evolution>.https://samizdat.co/works/do-while/lorenz-1963.pdf[Lorenz's 1963 convection and predictability paper]{ref}
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