have you ever wondered why when we look
have you ever wondered why when we look

around us in the natural world we see
around us in the natural world we see

very few straight lines but the straight
very few straight lines but the straight

line is ubiquitous in the systems we
line is ubiquitous in the systems we

engineer from buildings to circuit
engineer from buildings to circuit

boards it appears to be the default
boards it appears to be the default

position one way to understand this is
position one way to understand this is

that the systems we engineer are based
that the systems we engineer are based

upon a scientific and mathematical
upon a scientific and mathematical

understanding of the world which has
understanding of the world which has

inevitably started by describing the
inevitably started by describing the

simplest and most orderly systems that
simplest and most orderly systems that

is to say those that are composed of
is to say those that are composed of

linear forms and relations from Euclid
linear forms and relations from Euclid

to Newton and on science has been
to Newton and on science has been

focused upon the orderly systems of
focused upon the orderly systems of

perfect squares triangles and linear
perfect squares triangles and linear

relations have caused an effect that can
relations have caused an effect that can

be encoded in beautifully compact
be encoded in beautifully compact

equations
equations

thus we described real world as a kind
thus we described real world as a kind

of approximation to these perfect linear
of approximation to these perfect linear

forms but let's take a look at some of
forms but let's take a look at some of

the basic principles underlining the
the basic principles underlining the

theory of linear systems linear is
theory of linear systems linear is

basically a fancy word for line and a
basically a fancy word for line and a

line is often understood as the shortest
line is often understood as the shortest

or most direct path from one point to
or most direct path from one point to

another to say there is a linear
another to say there is a linear

relationship between two things is to
relationship between two things is to

say that there's a direct relationship
say that there's a direct relationship

of cause and effect between them let's
of cause and effect between them let's

take an example of this say I'm playing
take an example of this say I'm playing

baseball and I hit a ball with a bat if
baseball and I hit a ball with a bat if

we make a model of this system we see
we make a model of this system we see

that their energy inputted by my swing
that their energy inputted by my swing

of the bat will be directly proportional
of the bat will be directly proportional

to the output of the ball's momentum as
to the output of the ball's momentum as

it travels off in the opposite direction
it travels off in the opposite direction

this is a simplified model but it
this is a simplified model but it

illustrates the direct or linear
illustrates the direct or linear

relationship between cause and effect my
relationship between cause and effect my

swinging of the bat and the balls motion
swinging of the bat and the balls motion

in response there are of course many
in response there are of course many

examples of linear systems particularly
examples of linear systems particularly

within physics but we can capture the
within physics but we can capture the

underlining logic of linear systems in
underlining logic of linear systems in

general with a model that has only two
general with a model that has only two

simple rules called the superposition
simple rules called the superposition

principles which states firstly that the
principles which states firstly that the

output to the system will always be
output to the system will always be

directly proportional to the input so if
directly proportional to the input so if

there's a linear relationship between
there's a linear relationship between

the amount of fuel I put in my car and
the amount of fuel I put in my car and

how far it will go well then if I put
how far it will go well then if I put

twice as much fuel in the car it will go
twice as much fuel in the car it will go

twice as far and secondly that if we add
twice as far and secondly that if we add

two or more systems together
two or more systems together

then the output to this combined system
then the output to this combined system

will simply be the sum of the two
will simply be the sum of the two

outputs of the original systems so say
outputs of the original systems so say

we have two tractor factories each
we have two tractor factories each

producing a million tractors a year well
producing a million tractors a year well

if we merge them then we will get a
if we merge them then we will get a

factory that will produce 2 million
factory that will produce 2 million

tractors a year linear systems are
tractors a year linear systems are

deterministic meaning that if we know
deterministic meaning that if we know

their present state we can fully
their present state we can fully

determine their past and future States
determine their past and future States

this can be seen by plotting a linear
this can be seen by plotting a linear

system as a graph where it will always
system as a graph where it will always

be depicted as a straight line
be depicted as a straight line

although linear systems modeling has
although linear systems modeling has

proven highly successful in many areas
proven highly successful in many areas

and is often a very good approximation
and is often a very good approximation

the reality is that we live in a world
the reality is that we live in a world

with eco systems economies societies and
with eco systems economies societies and

physical systems that are not governed
physical systems that are not governed

by the superposition principles and thus
by the superposition principles and thus

are what we call nonlinear an example of
are what we call nonlinear an example of

non-linearity might be listening to two
non-linearity might be listening to two

of your favorite pieces of music at the
of your favorite pieces of music at the

same time because there is a
same time because there is a

relationship of interference between
relationship of interference between

them the results of the experience will
them the results of the experience will

not be a simple equation of adding the
not be a simple equation of adding the

enjoyment from listening to each
enjoyment from listening to each

independently this illustrates how
independently this illustrates how

non-linearity arises whenever there is
non-linearity arises whenever there is

some relationship between elements
some relationship between elements

within the system but can be either
within the system but can be either

synergistic making the output of the
synergistic making the output of the

system greater than some of its parts or
system greater than some of its parts or

one of interference making the output
one of interference making the output

less than the sum of its individual
less than the sum of its individual

components
components

to illustrate this further let's take an
to illustrate this further let's take an

example of four workers producing
example of four workers producing

clothing in isolation
clothing in isolation

each seamstress can sew a given amount
each seamstress can sew a given amount

of clothes within a day now if we put
of clothes within a day now if we put

them together we might get one of three
them together we might get one of three

results firstly they might not interact
results firstly they might not interact

with each other very much meaning the
with each other very much meaning the

whole would simply be the sum of its
whole would simply be the sum of its

individual parts but equally likely they
individual parts but equally likely they

might form some cooperative relationship
might form some cooperative relationship

which lets them each specialize in a
which lets them each specialize in a

particular function making them more
particular function making them more

efficient as a whole and thus the output
efficient as a whole and thus the output

of the system would be greater than the
of the system would be greater than the

sum of its parts
sum of its parts

due to these synergistic relations or
due to these synergistic relations or

inversely they might start all talking
inversely they might start all talking

with each other getting little work done
with each other getting little work done

and thus the total output would be less
and thus the total output would be less

than the sum of the individual outputs
than the sum of the individual outputs

due to these relations of interference
due to these relations of interference

non-linearity can also arise from
non-linearity can also arise from

feedback loops whereby the same process
feedback loops whereby the same process

is iterated with the output fed back as
is iterated with the output fed back as

the input to the next cycle a classical
the input to the next cycle a classical

example of this is compound interest
example of this is compound interest

where at the end of each period the
where at the end of each period the

balance plus interest is fed back into
balance plus interest is fed back into

the formula to compute the next cycle of
the formula to compute the next cycle of

interest accumulation iterative
interest accumulation iterative

functions are an important concept
functions are an important concept

within nonlinear science and have been
within nonlinear science and have been

used to create a whole new type of
used to create a whole new type of

geometry called fractal geometry where
geometry called fractal geometry where

by iterating a simple function generates
by iterating a simple function generates

irregular organic looking patterns that
irregular organic looking patterns that

can model many of the geometric forms we
can model many of the geometric forms we

see in nature from the structure of
see in nature from the structure of

seashells to the rugged formation of
seashells to the rugged formation of

mountains
mountains

in these nonlinear systems superposition
in these nonlinear systems superposition

fails meaning one can operate the system
fails meaning one can operate the system

down into smaller subproblems and then
down into smaller subproblems and then

add their solutions we must consider a
add their solutions we must consider a

nonlinear problem in total it is this
nonlinear problem in total it is this

need to approach nonlinear systems as a
need to approach nonlinear systems as a

whole that is giving rise to new more
whole that is giving rise to new more

holistic approaches to science that are
holistic approaches to science that are

developing under the canopy of the term
developing under the canopy of the term

complex system science non-linearity in
complex system science non-linearity in

all the shapes and forms is at the heart
all the shapes and forms is at the heart

of many of the 21st century challenges
of many of the 21st century challenges

to science as we try to extend the
to science as we try to extend the

scientific framework beyond its
scientific framework beyond its

dependency upon linear systems theory to
dependency upon linear systems theory to

finding new ways to embrace the complex
finding new ways to embrace the complex

world we live in on its own irregular
world we live in on its own irregular

and imperfect terms
and imperfect terms