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