 So, positive feedback systems or we can define it as exponential growth equation, let us try to compute that. So, the initial system we started with is ds by dt with g into s that is what we start with. So, let us just d into g, integrate both sides, s at time 0 to s at time t, d into dt, it is integrate integral of 1 by s s logarithm of s. So, s at time t, s at time 0 equal to g into t means that I can raise it to the power e and I solve it, then I can get s at time t is s at time s naught into e power g t. So, I have just raised it to the power exponential and then moved the denominator to the right side. So, I get a stock at time t is nothing but initial value of stock into e power g t and as you can see these are positive values g is greater than 0. So, g is greater than 0 then it is going to exhibit exponential growth just by defined by this term here. Roughly s naught is a constant value I am just multiplying by a constant and given the initial value of stock and if I know g and for whatever value of t of course, I can directly compute what is going to be the stock value at time t. So, this is the stock time t, this is your initial stock. Of course, if you want to get any different point in time you just take that as the initial value and then take the time difference. So, it is you know it has equal to 1, bad thing about putting these things it means that there are many more equations coming. Let us say this is 1, now let us see what can I do that. Let me introduce this one. You can remember this s at time t is s naught into e power g t. It is not that we always use this fractional growth rate of g, we also define something called as a time constant. Let us define it as capital T which is nothing but a reciprocal of the fractional growth rate, growth rate g that is t is just defined as 1 over g. So, since the time units of the units of g is 1 over time, so capital T units is of course, time where it is useful is it is not that wherever we are looking at interest rates sometime it could be the delay. So, instead of multiplying g by the stock value you can take stock divided by time value it is going to mean the same thing. So, the equations stock at time t is equal to s naught into e power g t instead of g I can always use 1 by T 1 by capital T. So, suppose an interval of time capital T passes the stock at capital T time will be stock at time 0 into e power g t is s naught into e power g itself is 1 by capital T, capital T time units passes over small t is also capital T. So, that means, s at time t is s naught into e power 1 which is about 2.72 into s naught. So, after passage of 1 time constant the stock value will be approximately 2.72 times to the initial value of stock. And if say 2 time unit 2 T time passes if 1 T time passes 2.72 if 2 T time units pass then stock value will be 2.72 into 2.72 times that will be the stock at time 2 T. So, this is defined as the time constant. Now, let us come to our one question that we ask what is the doubling time let us see whether we can compute that let us look at doubling time let us say T d doubling time it is a time after which after time T d stock value doubles we need to compute this T d let us see. So, one states that s at time t is s naught into e power g t. So, I know the stock value doubles. So, let us say 2 into s naught is equal to s naught into e power g times T d already have it I am canceling s naught taking the logarithm on both sides v into T d which means that T d is equal to 0.69 divided by g or 0.69 into time constant capital T. So, this 0.69 will keep appearing because linear systems and we are looking at even role seeking etcetera or the work computation or a thumb rule purposes people sometimes refer to it as 70 percent rule that after with every d t time units 70 percent or is nothing but 0.7 times your time constant or nothing but 0.69 divided by g. So, for example, in our interest rate example T d is 0.69 divided by 0.15 g is 0.15 which is about 4.6 since our time resolution was time interval of integration was 1. So, only in the fifth period we are able to observe that it has double the value. If we had used a smaller time step then you would have observed more closer to 4.6. So, this is say every 4.6 years or 5 periods or 5 years. So, every 4.6 years the value of stock is going to keep doubling. So, and that can be computed just by given the g values. So, if you know what is the fractional growth rate then all you have to do is 0.7 by this for approximately 0.69 by 0.15 4.6 in this case here this is the doubling time. So, there is nothing much to do with the experiences systems other than just calculating what is going to be doubling time and identifying it. So, now, let us go back to our slides. We have seen this, we have seen a time constant for passage of one time constant stock value will be 2.72 times initial value, largerity or smaller g produces flatter growth curve. Now that we know all these equations it is very easy to compute that is not the intent as we are looking at a more modeling course we need to understand what will happen when based on values of g if you have large value of g then the growth has to be going to be steeper if it is a smaller value of g growth is going to be more what is it is flat up that is you are going to be much more time before you actually perceive the exponential growths. Doubling time, time interval required for an exponentially growing variable to double in its value T d is 0.69 times your time constant. So, now, if we start plotting the level and the rate that is you plot the rate for every value of the level that we just saw for the example for same example you will find a linear line with the slope of g it can be expected because this linear equation. So, for any positive value of g you will still produce exponential growth. So, even if this curve even if this line even if this line is like this I am going to still produce exponential growth or the line is going to be like this because I am going to produce exponential growth. We call these linear systems coming from this diagram right here where you can see that this line itself is linear which is why the entire system becomes a linear system right there any question so far not it fully away. Now, let us go back and spend some time on looking at some graph. So, time horizon over which exponential growth occurs alters the perception of growth even though the underlying system remains the same. So, it depends on how much data that we want to look at from the past or the time series data we are looking at alters are perception. So, let us see what we mean by that and let us relate it with our variables of t d or time constant t let us see. Suppose the time horizon taken happens to be just 0.1 times your doubling time then it looks like there is absolutely no growth it is very easy to just dismissing now system is fine it is just flat there is no growth in the system it looks very flat. This time horizon is just 0.1 times your doubling time. Your time horizon is 1 into doubling time if you if you squint your eyes or something then you will kind of start to perceive some exponential growth, but for practical purpose it looks like a linear growth it just goes from whatever in this example as 1 to 2 in the time constant. So, it looks like a linear growth to us. So, you do not start to perceive exponential growth at all even if you observe system only until it is doubling time. Doubling time is depends on 0.69 by g. So, if j is going to be very small then I am going to get a much longer doubling time. Suppose it is 10 times your t d then you can start to perceive some exponential growth which is shown here, but even then observe what is happening x axis x axis also changes with that time horizon 0 to 10 then 0 to 100 time units now 0 to 1000 time units when you start to perceive it, but the difficulty comes in is when you look at data from say 0 to 400 it is almost invisible to us and it seems to you say that nothing is affecting the system there and suddenly things are active only after time 400 which is not the case underlying system is the same. But if you again take too much data then you do not see anything you just see that something is just piking all of a sudden as an exponential growth. What is super exponential growth? Many positive feedback system is characterized by doubling time that decrease rather than remain fixed as a value or system level increases. Doubling time is constant that means you are having an exponential growth after every 5 time unit passes your level value the stock value became 100 then 200 then 400 then 800 after every 5 years the doubling time is constant, but system such as unfortunately population systems the doubling time is not constant the doubling time keeps reducing whereas the time it took take 5 years to reach the double the value next time it may take just 4 years to double in its value and following time it may take just 3 and a half years to double in its value. So, as doubling time reduces that means the value of G is increasing that is what it means. So, if you plot as the stock value increases the value of G changes. So, if you want to plot it the world population you will get a chart looking at a non-linear system because the value of G is also now getting affected by some other factors it is no more a external variable as we assume as long as G is constant does not change then you got a linear system as well as a exponential growth. So, system is linear growth is exponential, but if G itself starts to increase as we go along as the stock level increases then you start to get in what we call as a non-linear system because level and rate relation now became non-linear and the growth is going to be more pronounced. Look at world population example if you assume exponential growth then probably you will get the blue line, super exponential growth you are going to get this black line that you see here suddenly exponential growth does not seem so bad. You may be happy to have exponential growth then exponential growth is bad then super exponential growth is. So, you want to move to exponential growth then come to a world growth system.