[MUSIC] Welcome to this third week of the MOOC simulation and modeling of natural processes. My name is Orestis Malaspinas, and today I will talk to you about dynamical systems and their numerical integration. Okay so, in this first module, the idea is to make you a very generic introduction to dynamical systems and to introduce you with notations we will use throughout this week of the course. So, what is a dynamical system? So generally speaking, the dynamical system is any system that varies through time. We used to describe such a system by state, which is noted s, and which will depend on t. So, s of t, is a function. The state can be a vector, of course, also, so here we noted it with s going from 1 to n. But to keep things simple, we'll only consider scalar as functions. So as examples for dynamical systems you can think of any system that is evolving in time. For example, the pendulum, or whether evolution, or the evolution of population of bacterias or any kind of season that evolves through time. So we will talk about two classes of description of dynamical system. The first one is discrete dynamical systems. These systems evolve in time with finite time steps. So, here you see on this first equation that the evolution of s at time t plus delta t, where delta t is a finite number Is equal to some function of s. And f is any function. Then we also have the containers dynamical systems which are described through differential equations. So we see in this second equation that the description is that the time derivative of s is given by a function f(s). So usually, you can make a really close link between those two kind of descriptions. And as we will see later in the course, the first version of the data, because system is something close to what you will usually implement in a computer. Whereas, in the second case, it is usually more handy to work with them. So, if the description of the dynamic system given only in terms of first derivative enough to describe all kinds of systems including those with the second derivatives, for example? So, in the first equation of the slide, you see that on the left side, we have the first derivative of s plus the second derivative of s which is equal to some function of s. Is this equation equivalent to what we saw in the previous slide? I think we can make it look exactly the same by just adding an extra variable. So we see that we can define s dot, which is equal to y, and y dot, which is equal to f(s)- y, and then we define a vector which is composed of s and y. And we have again a dynamical system which has exactly the same form as the previous one. So if we name u equal s and t, and y sorry. A vector composed of these two components, we can construct a dynamical system which is u dot, which is equal to some function of j of u. In the district case now, the second derivative is equivalent to add more than one time step. So here you see on the left side of the first equation we have s of t plus identity, but this time instead of having a dependence only of f of s(t). We also add a second time which is relevant, which is t minus delta t. Similarly, to what we did for the continuous case, we simply define an extra variable, which is y again. And then very similarly again, we can define a vector u which is composed of s and y and we can construct dynamical system which has the usual form u of t plus delta t which is equal to some function j of u. Now what happens if our function depends, or f depends on s n on t. Is this equivalent to just having the notation as dot equal f of s? As we see on the, and in this slide, we can do exactly the same trick as we just did for the second derivatives. We just add a variable to describe the state of resistance, here again y, and we can build new vector of u. Which is equal to a vector made of s and y, and we can reconstruct a dynamical system which is u dot = g(u). The only thing we did, basically, is instead of having a scalar s, now we have a vector u describing the state of our system. So with this I end this very generic introduction to dynamical systems, and in the next module I will talk to you about practical example of discrete dynamical system. Thank you for your attention. [MUSIC]
