• The butterfly effect is the sensitive
dependency on initial conditions in which a
small change at one place in system can result
in large differences in a later state of that
• This was first theorised by
• The theory derives its name from the example
• He showed the details of a hurricane (exact
time of formation, exact path taken) being
influenced by minor perturbations equating to
the flapping of the wings of a distant butterfly
several weeks earlier.
• Thus he proposed that events of large scale
could be largely influenced by insignificant
seeming events of a very small scale.
Talking of geometric limits and dynamic process systems…
• If something has complicated results, it does
not necessarily mean that it had a
complicated input. Chaos may have crept in
(in something as simple as round-off error for
a calculation), producing complicated results.
• Fractal Dimensions are used to measure the
complexity of objects. Like in the geometry of
Its not always annoying to be chaotic…
• It is a mathematical order of the butterfly
• Deals with the behavior of dynamical
systems that are highly sensitive to initial
conditions—a response popularly referred to
as the butterfly effect.
• Small differences in initial conditions (such as
those due to rounding errors in numerical
computation) yield widely diverging outcomes
for such dynamical systems, rendering long-
term prediction impossible in general.
• This happens even though these systems
are deterministic, meaning that their future
behavior is fully determined by their initial
conditions, with no random elements
involved. This behavior is known
as deterministic chaos, or simply chaos.
• Thus its implementation lies
mostly in the interpretation
of philosophy and history.
A double pendulum showing the chaotic
behavior being a product of simple initial
The resulting pattern is an iteration of simple
pattern being looped chaotically over time.
• The theory was summarized by Edward
Lorenz as follows:
“Chaos: When the present determines the
future, but the approximate present does not approximately determine the
HOW BUTTERFLIES MATTER…
Revisting the butterflies…
• Real world application of butterfly effect and
its supplementing theories lies in the study of
highly dynamic systems.
Flight Simulations: Vortex Formations