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In this subunit I’ll introduce a quantity known as Lyapunov exponent.
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This material is perhaps a bit more technical than what’s gone before
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and I won’t use this material later in the course
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so you can skip this subunit if you want
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The idea behind the Lyapunov exponent is as follows
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sensitive dependence on initial conditions
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as defined so far is a binary distinction
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something either has sensitive dependence on initial condition or doesn’t
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however it’s reasonable to suspect and indeed it’s the case
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that some systems are more sensitive on their initial conditions than others
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so we would like a way to describe just how sensitive
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a sensitively dependent on initial condition system is
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and that’s what the Lyapunov exponent gives us
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it’s a number that tells us how sensitive a system is
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and so it gives us more information than just a yes or no
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it has butterfly effect or it doesn’t
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so let’s get start it and look at the Lyapunov exponent
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so here is the idea behind the Lyapunov exponent
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it’s a way to measure how sensitive
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sensitive dependence on initial conditions is
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for a particular dynamical system
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so let’s say we have two initial conditions
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which I call x-naught and y-naught
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and these initial conditions initially start off
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a certain distance apart I’ll call that D-naught
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so this is the initial difference between the two initial conditions
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then as I iterate forward using the dynamical system whatever it is
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I’ll talk about this as if it’s an iterated function
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but this applies to differential equations too
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I can ask how does this difference change in time
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so what is D of t
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and all that would just be alright that’s xt minus yt
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so these bars here mean absolute value
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so I just want to know how does the difference between the two orbits
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absolute value of that difference how does that change in time
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does it grow, if it grows then that probably means
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they’re sensitive dependence on initial conditions things get pushed apart
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if it shrinks that means are probably moving to a stable attractor of some sort
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so we might wonder ok what does this function look like
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Oh in this was just the blue curve that’s plotted on the online program
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that you’ve been using it’s just well it’s the absolute value of the blue curve
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but it’s the same idea
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so it turns out that form for many systems
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with sensitive dependence on initial conditions
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or most this will grow exponentially
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so let me write a few things to indicate that
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So D of t that’s the separation between orbits at time t
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and for many systems, many I think really for most systems
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this is an exponential function of time
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that separation the distance between the two orbits
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if it’s chaotic has sensitive dependence will grow exponentially
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and if it is not chaotic we’re moving towards a fixed point or cycle stable and attracting
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then this will decay exponentially
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so D-naught that’s just the initial separation
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and how does that grow or shrink
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it grows exponentially
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it’s conventional although not necessary to use a base 2
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here for the exponential
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you could certainly use e as well and convert back and forth
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but I will use 2 then t is an exponential function of time
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and then lambda is the exponential growth or decay rate
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so let me, let me show you an example of this for the logistic equation
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and then I’ll just say a little bit more about the Lyapunov exponent
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so for the logistic equation with r equals 4
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we know that has sensitive dependence on initial conditions
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that’s a chaotic value the logistic equation is chaotic for this parameter value
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It turns out that the Lyapunov exponent lambda is equal to 1 for this system
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and that actually can be proven exactly in
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it’s also not hard to calculate numerically
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So a lambda of 1 means that the distance between the two initial conditions
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will grow exponentially as 2 to the t
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so this is our starting separation and then it grows 2 to the t
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which means by the way that it doubles every time step
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so here I just chose two initial conditions .2 and .2001 I think
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and the squares here they’re plotting the D of t
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this difference between the two orbits
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this was the curve that was blue on the program
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the dotted line that’s this function
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so the point of this is to note that the dotted line
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this function is a very good approximation to D of t
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that how the difference between these two orbits grows as a function of time
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it’s not a perfect fit because this is just an approximate relationship
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if we average in a sense overall initial conditions
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this would be how on average orbits grow
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here is another view of this
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now I just went out to t equals 9 instead of t equals 5
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again you can see for this one
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the black that’s the data from the from the program
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that again is growing approximately exponentially
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and it’s well approximated by this dashed line
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so the main thing is if we have a chaotic system
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with sensitive dependence on initial conditions
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the nearby for almost all these systems
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the difference between those nearby initial conditions grows exponentially
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that exponential growth rate is the Lyapunov exponent
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and that’s a measure of how chaotic it is
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how fast the two orbits are pulling apart
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so here’s a Lyapunov exponent to find little bit more carefully
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the quantity we’re interested in is D of t
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and that’s the absolute value of the difference between two orbits over time
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and we then assume that this function will grow exponentially or decay exponentially
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so we assume that it has this form D-naught that’s the initial separation
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and then it’s 2 to the lambda t
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so it’s some exponential function of time it grows as time or shrinks
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but for chaos it grows as time gets larger
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and the exponential growth rate lambda that’s Lyapunov exponent
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now this relationship for functional form really only applies on average
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and for small t and small D-naught
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so there’s there more formal definitions of this that
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that involve some averaging processes
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but the main point of this is just to get a sense sort of intuitive
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almost physical sense of what the Lyapunov exponent tells you
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ok, so lambda that’s Lyapunov exponent
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and if lambda is positive then this is an exponential growth
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so we have an exponential growth of the difference between the two nearby orbits
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and that means we’ll have sensitive dependence on initial conditions
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so positive Lyapunov exponent means we have SDIC
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and more importantly or equally importantly
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a larger lambda means greater sensitivity
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the larger lambda is the faster the two nearby orbits are pulled apart
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so the more sensitive it is on those initial conditions
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on the other hand if lambda is negative then nearby orbits get closer together
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they must be getting pulled on average to some attractor
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and in that case there’s no sensitive dependence on initial conditions
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if two initial conditions start distance apart
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that distance actually gets smaller over time not larger
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so as I said there is more formal definitions of Lyapunov exponents
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and you can extend this to have multiple exponents in higher dimensional systems
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but this is the main idea the Lyapunov exponent is the exponential separation rate
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for two nearby trajectories of a dynamical system
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so these are very commonly calculated quantities
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I’m not going to use them in this course I don’t think
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but it’s very likely you encounter them if you read other text or primary literature
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so I wanted to just give you some idea
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of what the Lyapunov exponent means and how to think about it