[MUSIC] Hello. The atmosphere is a system in which the behavior of air masses is unpredictable and seems to happen by chance. But in fact it is governed by deterministic laws. It's chaos in the scientific sense of the word. In this sequence, you will learn about determinism and the concept of predictability that stems from chaos. Determinism is the philosophical principle that any event occurring in the future is the inevitable result of events and actions that have occurred before. Determinism was introduced into science in the 16th century with the idea that the principle of cause and effect completely governs the material world. It is the astronomer and mathematician Pierre Simon Laplace, who has affirmed universal determinism in all its rigour. According to this model, the evolution of the universe proceeds like a perfect machine. That is to say, without chance or deviation. Isaac Newton in the 18th century confirms this principle with the publication of his laws governing the movement of the planets. According to Newton, it is possible to predict with great precision, position and the speed of the planet. If we know the initial speed and position of the other stars in the region of influence of the planet of interest. The first triumph of mutant mechanics. The two bodies problem is entirely analytically soluble. The current position of planet is completely determined by its position, speed at a previous time. In other words, the position of planet is entirely dictated by the initial conditions of its position and its speed. It's pure determinism. At the top, the animation shows two bodies of equal mass gravitating around a common barry center with elliptical orbits. Below the animation shows two bodies with a slight difference in mass gravitating around a common barry center. Since the work of Henri Poincare, in particular the theorem, which published in 1890 in its article on the problem of three bodies and the equations of dynamics. We know that from the problem with three bodies, solutions are sensitive to initial conditions and analytical solution even approximate is illusory. Only statistical methods apply to this case. The animation shows the approximate trajectories of three identical bodies in a gravitational field and having zero initial velocities. We see that the center of mass, in accordance with the law of conservation of momentum, remains in place. Increasing the precision of the initial positions of the three bodies does not improve forecast of the time, but leads to very different results. It is a dynamic system very sensitive to initial conditions and the exact forecast becomes possible only if the exact initial conditions are known. However, in experimental science, no measurement is infinitely precise. This imprecision comes from the fact that the instruments used to make the measurements have final precision. For example, it is difficult to obtain accuracy greater than 1/10 of a kilogram for mass. The imagine an individual with a mass of 75 kg, 75 kg mass is therefore the target mass to be found by the measuring instruments. If we weigh with the scale, we get measurement of 75.1 kg. Due to the precision of the instrument this measurement is not false, it is just empresize. If we continue the wings, we describe probability density function around the target. This means that the initial conditions of a system cannot be determined exactly. Two sources of error must be considered, statistical dispersion referred as precision and systematic error referred as accuracy. If we continue the comparison with arrows that we shoot at target, the precision designates the fact that the arrows are close to each other or on the country's scattered on the target. The accuracy indicates whether the arrows were aimed at center or at another point on the target. These sources of error are reproduced on the probability density function. Let us consider a dynamic system describing the temporal evolution of variable which can be positioned, temperature, humidity at a point in space. This equation is of the form Dy dy/dt equals F of Y, where y is the variable. Dy/ DT it's derivative in time and F is a function describing the properties of y. What is Y worth to at an instant T knowing Y =Y0 at the initial instant T0? Let us illustrate this with a linear dynamic system like Newton's first law. Let a body of mass m subjected to its weight and speed, Y=V, then its acceleration DV / DT equals MG. Where m is its mass and g the acceleration of gravity, which we consider here constant. We measure a mass of 2kg with perfect balance. If m = 2kg and g = 10 m per second squared, then the force F that is the weight to which the body is subjected equals 2 * 10 = 20 N. We repeat the measurement with a device whose uncertainty is 0.1 kg. If m = 2.1 kg and g = 10 m per second squared, then this time F = 2.1 * 10 = 21 N. So in this case a 5% error results in a 5% error on the final results. This example shows that if we improve the accuracy of our measurement by five person, our results also improves by five persons. So in principle, the fact that we have an uncertain X% on the measurements, does not have much influence on our results. This is what Newton believed. But is it still true? The disk retired version of Newton's first law can be written in the form Y [t +1] is proportional to Y [t] where Y [t] is the speed of the body at 20. So I think previously, if we make 10% error on the initial conditions, we will have a 10% error on Y[t] at all times. Let us consider this time a nonlinear system of the form, Y[ t + 1] = 2y[t^2]. If we consider a perfectly known initial condition, Y[0] = 1. The values obtained at the first three times steps are shown below. If we make a 10% error on the initial conditions that Y[0] = 1.1. This time we see that the error increases from 21% at the first time step to 115% at the third time step. So when a process is normally near a small error in the initial conditions grows very quickly and degrades the final results significantly. This phenomenon is called dynamic and stability or chaos. In the atmosphere context, the mathematician and meteorologist Edward Lorenz, developed in 1963, a system of three first order differential equations as a simplified model for convection processes in the atmosphere, with the aim of studying meteorological phenomenon. The model consists of the differential equations visible on the screen. The variable X is the speed of rotation of the convection cell. The other two variables Y[t] and Z[t] represent the temperature distributions in the cell. The symbol sigma, R and B denote constant parameters. The values of which are specified according to certain atmospheric conditions. For example, R is called the railway number and is proportional to the temperature difference between the warm base of the convection cells and the cool top. This is the main parameter with the effect is of interest in the model. The dynamic system is nonlinear due to the presence of the multiplicative of terms XZ and XY. Lorenz observed extremely unusual solution behaviors in this model. For example, the trajectories of simulated solutions with very slightly different initial conditions are executed in this animation. But simulations predict completely different trajectories after a certain period of time called predictability. This led to the discovery of major attributes of Celtic dynamic systems dependent sensitive to initial conditions. So very small changes in the initial conditions can make very big differences in long term behavior. This phenomenon has been nicknamed the Butterfly Effect. In this sequence we discussed the concept of predictability. We have shown that a non linear system is very sensitive to initial conditions. There's a small error in the initial conditions grows very quickly and degrades the final results significantly. This phenomenon is called dynamic instability, or chaos. It was popularized under the name Butterfly Effect from the dynamic system developed by Edward Lorenz, whose mathematical formulation you have learned and which represents a simplified model of atmospheric content action. Thank you for your attention.