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so in this video we want to talk about drawing a phase portrait so I am here I'm in my mouth 195 MATLAB folder and I'm going to create a new script this script is going to mimic sample code provided by syama now he provides this in Python and we're going to adapt this to MATLAB so on the first line I have put the title of my script and so I'm just going to copy that and I'm going to save remember always save your m-files in your math 195 mountain folder and make sure that is your current working directory okay second line I'll leave blank and then I'm going to include some comments so this script makes a phase portrait for X sub N equals 0.5 times X sub n minus 1 plus y sub n minus 1 and y sub N equals negative 0.5 times X sub n minus 1 plus y sub n minus 1 so you might want to take a moment to think about this this is a 2d or two variable system it is a linear system it is autonomous and it is first-order next we want to look at some initial conditions now up until this point now of course we've always picked a specific value for our initial condition and we've run the simulation in this script we want to actually look at what happens for a range of different initial conditions like if we start X naught negative 2 and Y naught at negative 2 what happens what happens if we start these initial values at 1 or point 5 all right so we want X naught and y naught our initial conditions to range over are negative 2 to 2 in increments of 0.5 so we're going to run this for lots of different initial conditions we'll do this for 30 time steps and again this is a model that's been modified by syama okay so let's start setting up what we need equals 30 this is our number of time steps what else do we need well we'll need an array to hold the values of x and y so let's say x equals zeros 1t let's say y equals zeros 1 T and I'm suppressing the output at the end of those lines now I need to pick some initial conditions and to do this I am going to let a range from negative 2 to 0.5 up to 2 similarly I'm going to let B go from negative 2 in increments of 0.5 up to 2 now if you're wondering what the heck does that for a loop do let's go down here and take a look suppose I have a for loop that says a negative 2 to 0.5 up to 2 okay let's just display a and end the for loop let's see what this does ok what that did it was a for loop it started our index variable at a and it's going to march up until it gets to 2 and it's hopping up in increments of 0.5 so anytime you encounter code that you're a little bit unsure about just come out to the command window and play with it a little bit so you see what the command is doing okay so let's go back to our script we've got 4 equals negative 2 up to 2 and we're moving in increments of 0.5 same thing for B so we're going to initialize x and y so we start indexing at 1 the first position in our array is going to have the value a and for y we'll set that to be B so the first time we come through this double for loop a is negative 2 B is negative 2 and so that means the initial value for X will be negative - and the initial value for Y will be negative - and it will run around in this inner little for loop for B and that'll keep changing the initial conditions for Y value and then we'll hop out to the outer layer and repeat that as we change the values for X okay next thing we want to do is we want to run the simulation so at this stage we're going to have another for loop and let's index this one with n for N equals two up through our number of time steps here's where we need to run the model so I'll just go up and grab X sub n and I'll go up and grab the code for Y sub n we'll end our for loop 4n and I'm just going to make a note we've got several four loops here so this is which one are we ending we're ending the one that has it index over n at this point we want to plot X versus Y because this is a phase portrait and we're going to hold on to that plot we'll end our for loop for B and then we'll end our for loop for a let save and run I'm just going to go up here copy paste the name so I don't have to type that and our command window got a little bit messy so let's let's clean that up and I'm going to clear all variables because I have some variable a over here and I certainly don't want that to be messing up my code okay so let's run and we get a beautiful phase portrait now you notice there it looks like there's a fixed point at 0 0 and you could verify that analytically
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