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In Order to Derive His Equations, Lorenz (1963)

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In Order to Derive His Equations, Lorenz (1963) THE LORENZ EQUATIONS MARIANNA KATERINA PELEKANOS MSc MATHEMATICS SCHOOL OF MATHEMATICAL SCIENCES QUEEN MARY UNIVERSITY September 10, 2009 1 ABSTRACT The project aims to analyse the work of Edward Lorenz. The way dynamical systems with complex behaviour, such as the weather behave after a long period of time. The project will mention the Lorenz equations and what they represent. What happens to the system if we vary one parameter having the other parameters fixed will also be discussed here. In addition it will be discussed about what happens to the dynamical system if we slightly change the initial conditions. Finally it will talk about what is meant by sensitive dependence on initial conditions and the butterfly effect and how these to concepts are related to chaos theory. 2 TABLE OF CONTENTS Acknowledgements 4 1. Introduction 5 2. Derivation of the Lorenz Equations 7 2.1 The Bernard Rayleigh Experiment 7 2.2 Derivation of the Lorenz Equations 9 3. The Parameter r 18 3.1 Solutions of the Lorenz Equations 18 3.2 Different Values of r 19 4. Nontrivial Fixed Points 21 4.1 The Region r>1 21 4.2 Time Dependent non Stationary Solutions 23 4.3 Period Doubling Scenarios 24 5. The Butterfly Effect 27 6. Conclusion 29 Bibliography 30 3 ACKNOWLEDGEMENTS This dissertation was an interesting experience for me, but would not have been possible without the support of my supervisor Doctor Wolfram Just which I thank very much for all the help and support he have given me. My special thanks to my parents, who made this Masters possible and also to my friends for their support. 4 CHAPTER 1 INTRODUCTION Could anyone think that a tiny error in an input number of a computer program can have huge consequences in the long run behaviour of a complex dynamical system? Edward Lorenz was a meteorologist and he devoted his work to try to give reasons why it is difficult to give accurate weather forecasting for more than two or three weeks. Lorenz studied Mathematics at Dartmouth Collage and then had a Masters degree in Harvard University also in Mathematics. He, as well received a SM and a ScD in Meteorology from MIT. Furthermore, he served for the US Army Air Corps in World War II as a weather forecaster and it is then when he chose to do a graduate job in Meteorology. Soon after he took his doctorate in Meteorology. Lorenz was also awarded with many prized such as the Crafoord prize and the Kyoto prize. (MIT, 2008, April 16). The atmosphere can be modelled by mathematical equations. A small incident during a survey to model the atmosphere led Lorenz to discover chaos. After this event, Lorenz was searching for complex mathematical systems. Such system is the Lorenz attractor or Lorenz equations which will be studied in the project. At the beginning the Lorenz equations were derived to model the atmosphere mathematically. Firstly, the project will concentrate on deriving the Lorenz equations. What Lorenz considered before the derivation, what mathematical and physical notations were used is something it will also be mentioned in this chapter. It will be explained what the variables and what the parameters of these equations represent. The next two chapters will look at the different parameter range of these equations. Basically we will adjust one parameter having the other parameters fixed and observe the different behaviours the system takes. If it has stable or unstable points, if it has periodic orbits and even if the system has chaotic behaviour. It will also be discussed 5 why the invariant set of the Lorenz equations is also called Lorenz attractor and about the nontrivial fixed points of the attractor. After that, the project will talk about the butterfly effect, a term first conceived by Lorenz himself in a paper he wrote in 1972. This chapter will also mention sensitive dependence on initial conditions and how this is related to chaos theory. This will be also explained by the means of some plots. Finally, chapter 6 presents the conclusions of the project. Who else is involved with chaos theory and where the Lorenz equations are used. 6 CHAPTER 2 DERIVATION OF THE LORENZ EQUATIONS Before deriving Lorenz equations, let us first say a few things about what Lorenz did, some mathematical details he considered and about the small experiment he prepared that led him to find the equations the project will talk about. 2.1 The Benard Rayleigh experiment Lorenz (1963) considered a two-dimensional fluid element, such as water, placed between two plates. The plates are separated by a distance H . The lower plate is then heated uniformly and the upper plate was cooled. The difference of the temperature between the two plates is ΔΤand this difference is a fixed parameter. The x and y coordinates are parallel to the two plates and the z coordinate is perpendicular to the plates. The equations of motion are given by the Navier Stokes equations for an incompressible fluid (⋅∇ u = 0), and the equation of the temperature is given by the heat transfer equation. It is assumed that motion variations are in one direction and are parallel to the two plates. This means that the temperature and the velocity of the fluid depend only on x and z and independent of the y direction. So there are no variations in the y direction. In his paper, Lorenz (1963), considered systems of deterministic equations used in hydrodynamics. By saying deterministic equation we mean an equation that has the power over a dynamical system without arbitrary forces. He was mostly interested in periodic solutions and we will discuss this later on in the project. To derive his equations, he coped with a phase space Γ in which there was only one trajectory that passed through every point and where time was continuous. In addition the trajectory was bounded with a uniform way. This means that there is a bounded area, say R in which all trajectories eventually stayed in this area. Lorenz organised the trajectories in three ways. The first way was according to the existence of transient properties. Regarding the stability of the trajectories was the second way and the third way was according to the periodicity of the trajectories. 7 Lorenz (1963) used Saltzman’s (1962) equations of free convection to derive his system of three ordinary differential equations. When we say convection we mean the heat flow of a fluid from a hot area to a cold area. Saltzman (1962) rewrote the equations of motion in terms of θ and ψ .He derived his equations using double Fourier transforms in x and z , but we will not discuss this here. Saltzman’s two equations are: 2 ∂ 2 ∂(ψ, ∇ ψ ) 4 ∂θ ∇ψ = − +ν ∇ ψ + g α , ∂t ∂()x, z ∂x (1) ∂θ∂() ψ, θΔΤ ∂ ψ = + +κ ∇ 2 θ , (2) ∂t∂() x, zΗ ∂ x where ψ is a stream function for a two-dimensional motion, θ is the leaving temperature in the state of no convection, g is a constant representing the acceleration of gravity, α is the coefficient of thermal expansion, ν is the kinematic viscosity, κ ∂ψ, ∇ 2 ψ∂ ψ ∂ ∇ 2ψ ∂∇ 2ψ ∂ ψ is the thermal conductivity and ( ) = ( ) − ( ) . ∂()x, z∂ x ∂z ∂x ∂z These equations are introduced, taking into account that motion is parallel to the x − z plane and there is no motion in the y direction. At both the boundaries, the vertical velocity is equal to zero. Furthermore at a free boundary the tangential stress is also zero and at a rigid boundary the tangential velocity is zero. At the boundaries this means that ψ = ,0 ∇2ψ = 0, ∂ψ = 0 . ∂z For the derivation of the Lorenz equations we will only useψ = ∇2ψ = 0 . Here, we note that these boundary conditions apply to z = 0 and z = H , where H is the height of the fluid. 8 2.2 Derivation of Lorenz Equations The nonlinear system of the partial differential equations (1) and (2) with the boundary conditions mentioned above is difficult to solve. To make things easier, one should choose some approximations to reduce the partial differential equations approximately to nonlinear ordinary differentia equations. For this reason Lorenz (1963) introduced solutions of the following form, as these expressions satisfy the boundary conditions. −1 a()1+ a 2 κ−1 ψ = Χ 2 sin()πaH−1 xsin(π H−1 z) , (3), (4) −1 −1 −1 −1 −1 πRRc a ΔΤθ = Y 2 cos())((πaH xsin π H z− Zsin 2π H z) . Here, XYZ, , are only functions of timet , Ra is the Rayleigh number, after the one 3 −1 − 1 4− 1 2 3 who invented it. This is equal to Ra = gαH ΔΤν κ and Rc =π a(1 + a ) is the critical value of the Rayleigh number. Lorenz (1963) substituted equations (3) and (4) into equations (1) and (2) and with some calculations he ended up with the following set of ordinary differential equations, which are known as the Lorenz equations. ⋅ XXY= −σ + σ . Y= − XZ + rX − Y . Z= XY − bZ. σ ,,r b are constants to be determined and the dot represents derivative with respect to time. It will now be discussed in some detail how the Lorenz equations are obtained from Saltzman’s (1962) partial differential equations. Now we will substitute (3) into (1). First we will compute the LHS of (1). 9 ∂ 2ψ∂ 2 ψ∂ 2 ψ ∇2ψ = + + ∂x2 ∂ y2 ∂ z 2 ∂ 2ψ From equation (3) we see that nothing depends on y so = 0 and hence ∂y 2 ∂ 2ψ∂ 2 ψ ∇2ψ = + .
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