 So, we have nicely seen for the model for Romeo and Juliet how things exponential growth because of positive feedback system, both like each other and the reinforce of each other's behavior. If one partner allows the other, more than other person reciprocates the love, so we model that. First of all, we found that some initial settings resulted in asymptotic behavior. Now, let us take up another setting, love-hate relationship, things are not as it seems. Perhaps when the boy is disinterested or doesn't know, doesn't pay attention, maybe she loves her, I mean she loves him. When the boy starts to reciprocate his love, she starts to spurn him or doesn't like him and the same way, more she spurns, he loves her more but then when she starts to reciprocate her love, he walks away, it's all very complicated, so let's call them Julian and Clare. After the protagonist from the red and the black by Stendhal, it's a French book translated in English and setting of many Hollywood movies or any language movies. So let us take this scenario, Clare is attracted to Julian when he shows no interest in her. When Julian begins to love her, he becomes pathetic in her eyes and develops disgust for him. Julian is like this, attracted to Clare only if she gives him his cold shoulder, then she becomes a prize for his honor mandates that he catch. What will happen when these two characters meet, so how do we model this? Similarly we will define stock of love for each other and we will see if we can just capture the personalities using the parameter settings. So Julian, Clare model will look very similar to Romeo and Juliet. Now let us observe, see the stock values and the flow equations doesn't change. In initial love, we can as we told we can set some values, let's try to capture their personalities by using the terms on Julian's reaction and Clare's reaction. In Julian's reaction is opposite of what Clare demonstrates. If Clare dislikes him, he likes her. If Clare likes him, he dislikes her, so it is the opposite direction. So let us put Julian's reaction is minus 1. Now looking to Clare's reaction, if Julian likes her, likes her, then she is annoyed, so she dislikes him. If Julian ignores her, then she likes him, so it is also in opposite direction. So let us put Clare's reaction is minus 1. So I just going to put Julian's reaction is minus 1, Clare's reaction is minus 1 because both Julian's reaction to Clare's love and Clare's reaction to Julian's love are negative and every unit of love that Julian displays to Clare, decreases Clare's love for Julian by one unit and vice versa. Now let's go ahead and try to simulate this. Let us assume the initial love of Julian, initial love of Clare is 1, that is they both meet each other, but instantly they like each other. But let us see what happens to the relationship based on their personalities. Is both love each other, what do we expect? Both will start to push each other away, right? Julian initial likes Clare, but Clare dislikes him, let Julian's love be plus 1 love unit and Clare's love be minus 1 love unit. She likes her, she does not like him, let us see where it goes, okay. We are also going to get a exponential behavior. The green one is Julian's love for Clare and red one is Clare's love for Julian. As you can see, more Clare dislikes Julian, Julian's love is going to keep increasing and as Julian's love keeps increasing, Clare's love, Clare's love or order, dislike for him increases and again we are getting exponential behavior. So if we trace the loops, you will see that it is actually a positive feedback loop. So say Clare's love increases for Clare, I mean Julian's love for Clare increases, suppose. Then Clare dislikes, starts to dislike him, as more Clare dislikes the more Julian is going to love her. So it becomes a positive feedback system because there are two negatives, there are two negatives in the loop which resulted in a positive feedback system which is caused this behavior. I think there are quite a few movie plotlines based on this, maybe I do not know if I am pronouncing Hindi correctly, Ranjana, similar plotline, right. He loves her, she does not like him, she goes to kill him, etc., etc., all the things happen because a dislike for him keeps growing. So let us try out other such fun things. Suppose initially Julian loves Clare, little more than Clare loves Julian, both love each other, but let us assume Julian initially feels plus 1 love unit, where Clare only feels 0.9 love units, she likes him 10% less than he likes her, what do you expect will happen? Let us see what happens. Actually we got a surprising the same behavior that is happening, right, but it is not always so. We need to zoom into period 0 and 2, so I cannot do it here. Let me just, so we call the initial love for Julian for Clare is 1, initial love of Clare for Julian is 0.9, Julian's reaction, Clare's reaction is minus 1, let us simulate it. This graph here shows again exponential behavior which is fine, but I told we need to zoom into period 0 and 2, already zoom graph I have. Actually if you zoom in, you get some interesting crazy dynamics initial 4 periods. So the red one is Clare's love for Julian and blue is Julian's love for Clare. So as Clare finds out that Julian loves him, her love decreases and as her love decreases, his love is also decreases slightly, because when she loves him, he starts to dislike her a bit. So her love, since it is positive, she or he also likes her lesser. But at around period 1.4, around here, her love actually becomes dislike from positive it goes to negative, starts to dislike. As soon as she starts disliking him, his love starts to grow exponentially. So the change in loop dominance is what you just saw here, but it is at such a small timescale, you may not have realized it. So you just saw that, that is an interesting dynamics which happened, but still positive negative feedback loops is what was kept dominating within the system. Now suppose both Julian and Clare initially like each other, both feel plus 1 love units towards each other, then what will happen, oscillations, no oscillations. She both like each other, so then since they both like each other, they are going to like each other even less as time goes on and we are going to get a goal seeking behavior to a point of kind of indifference within this model where, because each are going to like each other even lesser and lesser as time goes on and we will have exponential. Since the first order system, we are getting a, or rather it is a positive feedback system, but in this special case, it is showing asymptotic behavior. You can expect similar behavior when say both Clare and Julian dislike each other initially. When they dislike each other, their dislike reduces with time and they results in again a point of indifference and eventually suppose Julian and Clare were initially indifferent to each other that both felt 0 love units, that means the stock initially 0, so there is no dynamics that can be observed within the system, right. So, though the personalities are opposite when they like, when one person likes, the partner does not like them, when the partner likes, the other person does not like them, even then the system becomes a positive feedback system where, one's love is going to grow exponentially and others dislike is going to grow exponentially, there is a dynamics that we just saw. So, the personalities does not end there, we have many other personalities also. We will take up one more scenario based on Gone With The Wind, I do not know how many saw the movie or read the book or read some synopsis or some adaptation of it in some version somewhere. So, when he is initially not interested in her, she loves him and when she, when he reciprocates the love, she starts to spurn him, but then when he, when she loves him is when he loves her, he likes to his love to be reciprocated and when she is, she spurns him, then he also wants to walk away, he does not want to hang around, right. So, let us call them Scarlett and Rhett, when Rhett is indifferent to Scarlett, she is attracted to him, when Rhett is attracted to her, she scones him, Rhett however becomes frustrated and Scarlett treats him with contempt, he is attracted to her when she is attracted to him, but he loses interest when she snubs him, what will happen when these two meet, yeah. So, our model setting is going to be similar, we are going to have exact same model setting, same stock of love, so Rhett's love for Scarlett and Scarlett's love for Rhett and we are going to use the parameters, how can we set the parameters for say Rhett's reaction, how will Rhett's reaction be? So, Scarlett loves him, Rhett is more attracted to her, right. So, it is in the same direction, so Rhett's reaction is plus 1, positive, but when Rhett loves Scarlett, then Scarlett's reaction is to scorn him, it is in opposite direction, so it is in, we can model it as minus 1, let us say, Scarlett's reaction is minus 1, so Rhett's reaction, Scarlett's love is positive, as Scarlett's reaction, Rhett's love is the negative opposite direction. Now, let us suppose initially Scarlett and Rhett feel one unit of love for each other, that is their initial attracted to each other, then what kind of dynamics can we expect? So, what feedback loop do we have? The previous two cases are positive feedback systems, what feedback system is this? It is a negative feedback system, right, then what kind of dynamic can we expect? What you are going to get is system is going to oscillate, if you have seen the movie, this is the plot line. So, both love each other, but as Scarlett is going to dislike him as he loves her, but as long as love is somewhat positive, his love increases, but then when she starts to actively scorn him and dislike him, his love falls, continues to fall and then he starts to ignore her and when he starts ignoring her, her dislike reduces and then slowly again she falls back, falls back in love with him. Once he starts loving him, he also reciprocates the love and the game continues. So, you get a nice oscillating systems of love for each other, Rhett's love for Scarlett and Scarlett's love for Rhett, the red and green lines. This is because we are having a negative feedback within the system. So, as Rhett's love increases and Scarlett dislikes him, as Scarlett dislikes him, his love reduces. So, the direction of change is negative. So, it is a negative feedback system is what we have just observed, which is causing an oscillations. That is the first order negative feedback system. Again, we can have some interesting settings. Suppose initially Rhett's love for Scarlett, but Scarlett is indifferent to Rhett, that is Scarlett initial value is 0, where Rhett's initial value is 1. So, she does not dislike him or like him. So, since she and then he loves her. So, immediately we can expect that Scarlett will start to dislike him and as soon as she starts disliking, Rhett's love is going to fall, but we can expect a similar oscillating systems. So, instead of initial increase in love for Rhett, he will start to it will start to fall as Scarlett's dislike for him increases and system oscillates. Initially, Rhett likes Scarlett, but Scarlett actively dislikes Rhett. They are in opposite sides, right? As Scarlett, as Scarlett actually dislikes Rhett, Rhett's love has to fall and as initially Rhett likes Scarlett, then she should actively dislike him more, which is what happens. Initially, since he likes her, she likes him even less. Again, system will oscillate because of this second order negative feedback system that is present in our model. Suppose they are indifferent to each other, what can we expect? Will system oscillate? No, your stocks are 0. So, you do not expect it to oscillate. So, that is what you get. System does not oscillate. I think I effectively ruined the books for you. So, that is the conclusion. We saw three scenarios, Romeo, Juliet, Julian, Claire and Scarlett, Rhett. But interesting is all had the same generic structure and this is called a second order system that we have. We had a stock, two stocks, each individual changing, but the change in stock 2 is affected by stock 1 and change in stock 1 is affected by stock 2. This is generic structure. This is a generic structure for actually any system which is going to exhibit a sustained oscillation, this is a generic structure. When you have this generic structure, it is going to cause sustained oscillations. But what else is needed to ensure sustained oscillation? We need negative feedbacks. So, second order system with negative feedbacks is the only thing that is going to cause oscillations. So, oscillations are a feature for this. Negative feedback loops in higher order, two stocks are more systems, similar to the gone with the wind model. So, you have multiple stocks with negative feedbacks, that is when you are going to see oscillation within the system. Even if you have two stocks, but with the positive feedback system, no set of initial conditions can make them oscillate. So, Romeo and Juliet and Clare and Julian Clare model are driven by positive feedback loops. So, they are going to generate exponential behavior. In some exceptional cases, it does produce asymptotic behavior. So, for us to get sustained oscillations, we need a second order negative feedback system is what is going to cause it. So, there is a learning in this class. So, I suggest you think of this typical movie plot line and see how we can model it. Boy meets girl, he likes her on first sight, but she dislikes him. He woos her, but her dislike towards him grows, then suddenly she starts to reciprocate his love. As of now, I cannot, we cannot change anything in the parameter setting to model this. There is a non-linear shift in feelings for each other, their reactions. Try to model this, how we are going to do it, what kind of stock, is the reaction even governed by the others, their response or not and see how it works. So, some more descriptions on that is also available online. Thank you.