In this subunit I’ll introduce a quantity known as Lyapunov exponent.

This material is perhaps a bit more technical than what’s gone before

and I won’t use this material later in the course

so you can skip this subunit if you want

The idea behind the Lyapunov exponent is as follows

sensitive dependence on initial conditions

as defined so far is a binary distinction

something either has sensitive dependence on initial condition or doesn’t

however it’s reasonable to suspect and indeed it’s the case

that some systems are more sensitive on their initial conditions than others

so we would like a way to describe just how sensitive

a sensitively dependent on initial condition system is

and that’s what the Lyapunov exponent gives us

it’s a number that tells us how sensitive a system is

and so it gives us more information than just a yes or no

it has butterfly effect or it doesn’t

so let’s get start it and look at the Lyapunov exponent

so here is the idea behind the Lyapunov exponent

it’s a way to measure how sensitive

sensitive dependence on initial conditions is

for a particular dynamical system

so let’s say we have two initial conditions

which I call x-naught and y-naught

and these initial conditions initially start off

a certain distance apart I’ll call that D-naught

so this is the initial difference between the two initial conditions

then as I iterate forward using the dynamical system whatever it is

I’ll talk about this as if it’s an iterated function

but this applies to differential equations too

I can ask how does this difference change in time

so what is D of t

and all that would just be alright that’s xt minus yt

so these bars here mean absolute value

so I just want to know how does the difference between the two orbits

absolute value of that difference how does that change in time

does it grow, if it grows then that probably means

they’re sensitive dependence on initial conditions things get pushed apart

if it shrinks that means are probably moving to a stable attractor of some sort

so we might wonder ok what does this function look like

Oh in this was just the blue curve that’s plotted on the online program

that you’ve been using it’s just well it’s the absolute value of the blue curve

but it’s the same idea

so it turns out that form for many systems

with sensitive dependence on initial conditions

or most this will grow exponentially

so let me write a few things to indicate that

So D of t that’s the separation between orbits at time t

and for many systems, many I think really for most systems

this is an exponential function of time

that separation the distance between the two orbits

if it’s chaotic has sensitive dependence will grow exponentially

and if it is not chaotic we’re moving towards a fixed point or cycle stable and attracting

then this will decay exponentially

so D-naught that’s just the initial separation

and how does that grow or shrink

it grows exponentially

it’s conventional although not necessary to use a base 2

here for the exponential

you could certainly use e as well and convert back and forth

but I will use 2 then t is an exponential function of time

and then lambda is the exponential growth or decay rate

so let me, let me show you an example of this for the logistic equation

and then I’ll just say a little bit more about the Lyapunov exponent

so for the logistic equation with r equals 4

we know that has sensitive dependence on initial conditions

that’s a chaotic value the logistic equation is chaotic for this parameter value

It turns out that the Lyapunov exponent lambda is equal to 1 for this system

and that actually can be proven exactly in

it’s also not hard to calculate numerically

So a lambda of 1 means that the distance between the two initial conditions

will grow exponentially as 2 to the t

so this is our starting separation and then it grows 2 to the t

which means by the way that it doubles every time step

so here I just chose two initial conditions .2 and .2001 I think

and the squares here they’re plotting the D of t

this difference between the two orbits

this was the curve that was blue on the program

the dotted line that’s this function

so the point of this is to note that the dotted line

this function is a very good approximation to D of t

that how the difference between these two orbits grows as a function of time

it’s not a perfect fit because this is just an approximate relationship

if we average in a sense overall initial conditions

this would be how on average orbits grow

here is another view of this

now I just went out to t equals 9 instead of t equals 5

again you can see for this one

the black that’s the data from the from the program

that again is growing approximately exponentially

and it’s well approximated by this dashed line

so the main thing is if we have a chaotic system

with sensitive dependence on initial conditions

the nearby for almost all these systems

the difference between those nearby initial conditions grows exponentially

that exponential growth rate is the Lyapunov exponent

and that’s a measure of how chaotic it is

how fast the two orbits are pulling apart

so here’s a Lyapunov exponent to find little bit more carefully

the quantity we’re interested in is D of t

and that’s the absolute value of the difference between two orbits over time

and we then assume that this function will grow exponentially or decay exponentially

so we assume that it has this form D-naught that’s the initial separation

and then it’s 2 to the lambda t

so it’s some exponential function of time it grows as time or shrinks

but for chaos it grows as time gets larger

and the exponential growth rate lambda that’s Lyapunov exponent

now this relationship for functional form really only applies on average

and for small t and small D-naught

so there’s there more formal definitions of this that

that involve some averaging processes

but the main point of this is just to get a sense sort of intuitive

almost physical sense of what the Lyapunov exponent tells you

ok, so lambda that’s Lyapunov exponent

and if lambda is positive then this is an exponential growth

so we have an exponential growth of the difference between the two nearby orbits

and that means we’ll have sensitive dependence on initial conditions

so positive Lyapunov exponent means we have SDIC

and more importantly or equally importantly

a larger lambda means greater sensitivity

the larger lambda is the faster the two nearby orbits are pulled apart

so the more sensitive it is on those initial conditions

on the other hand if lambda is negative then nearby orbits get closer together

they must be getting pulled on average to some attractor

and in that case there’s no sensitive dependence on initial conditions

if two initial conditions start distance apart

that distance actually gets smaller over time not larger

so as I said there is more formal definitions of Lyapunov exponents

and you can extend this to have multiple exponents in higher dimensional systems

but this is the main idea the Lyapunov exponent is the exponential separation rate

for two nearby trajectories of a dynamical system

so these are very commonly calculated quantities

I’m not going to use them in this course I don’t think

but it’s very likely you encounter them if you read other text or primary literature

so I wanted to just give you some idea

of what the Lyapunov exponent means and how to think about it