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