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Jonas Bergman Knots Lorenz ¢¡¤£¦¥¨§ © ¡ ¥ £¡ ¤¥© £!¡¤§ "$#&%¨'¨(*),+.-0/213'4% 7CBD;E.>F687*987=:?@ ;GH@ IJB ;K L8>A@ ;M8BN>F;O>A687PQ;:<7=M8RS7=T8L8UV>F@ ;M8BDUW:<7*@ M8XV7=B?>F@ YVUW>A7=GUVM8G Teknisk- naturvetenskaplig fakultet 52687*98:<;987=:?>A@ UTH-enheten B L8ZZUW:<@ RW7CGH[8\¨]JX^UV:?]^@ M8Y_>A687`,UW]^K 7=@ YV6M8L8Za87=:b:?c8Ud98UW:<UWZ7C>A7=:D@ M>F687*7=T8L8UW>A@ ;M8B c8a8;>F6 IdUVM8Ge987=:?@ ;GH@ IJa87=68UVXV@ ;:b@ BN;a8B?7=:?X^7=GH[852687*f=M8;>FBNI^;:<:?7=B?98;M8GH@ M8Y_>A;O>F687*GH@ EgEg7C:<7=M8> Besöksadress: I^68UW;¨>F@ ;¨GH@ IdB?;K L8>F@ ;M8BN;E.>A687B?]^B >F7=ZhUV:?7*GH7=>F7=:<Z@ M87=GUVM8GeI^K UWB?B @ Eg@ 7=GH[8igM98UV:<>F@ I^L8K UW:b>F687 Ångströmlaboratoriet 987=:<@ @ IJEg@ YVL8:?7=jk7=@ YV68>,f=M8;>,@ BNB?7=UW:<I^687=GEg;¨:D>F;OI^;K K 7=I^>,@ M8Eg;:?ZUW>A@ ;¨MUVa8;L8>,lJ687=MUVM8G Lägerhyddsvägen 1 68]^987=:?a8;K Hus 4, Plan 0 68;lm@ >,UV98987CUV:<B [8\¨]JEg@ M8GH@ M8Y_>F68@ BDfCM8;¨>,>F687*B L8B?98@ IV@ ;M>A68UW>,>A687=:?7*7=n8@ B >FBDU_M87=lm:?;¨L8>F7*>F; Gea87B?>F:<7=M8YW>A687=M87CGH[8odM8Eg;:<>FL8M8UV>F7=K ]d>A68@ BNf=M8;>,lJUVBNM8;>,Eg;L8M8GH[8p¤;lJ7=X^7=:?c8>FlJ; Postadress: I^68UW;¨BNlJ;L8K @ I^UV>F7=GHc868]V987=:<a8;K @ Idf=M8;¨>FBNlJ7=:?7*Eg;L8M8GH[85268@ BDEgUWI^>,B?>F:<7CM8YV>F687=M8BD>F687 Box 536 ;>F687=:<c8Z;:<7I^;Z98K I^@ ;M>F68UV>,>F687*Eg@ YWL8:<7=jk7=@ YV68>,f=M8;>,7=n8@ B >FBDUWBDU_B?;K L8>F@ ;M>F;O>F687*PQ;:<7=M8RS7=T8L8UV>F@ ;M8B?[Q5,68@ B c 751 21 Uppsala B L8B?98@ @ M>FL8:<M8cQI^;L8K Gea87*U_B?@ YWM;E.>F687*7=n8@ B >A7CM8I^7*;E.]^7=>,UWM8;>A687=:D:<;L8>F7*>F;OI^68UV;BN>F68UVM>F687*;M87=B Telefon: B?;OEg;L8M8GeUVM8GI^K UVB B?@ Eg@ 7=GH[ 018 – 471 30 03 f=M8;lJM>F;GHUV]^[Qqr7=X^7=:?UVK8>F;:<L8BNf=M8;>ABNlJ7=:?7*UVK Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student p¤UVM8GHK 7=GHUV:<7=tQ DM8>F>A@Qs@ 7=Z@ ZM87=B YV:?UVM8B?f=UW:<7=tQ DM8>F>A@Qs@ 7=Z@ {8n8UVZ@ M8UW>A;:?t8K ;¨XdDYV:?7=M igqrqrstQu8v8w8u8jFx8y8x8y8cQodzH5,{8|~}Cw8vw88 Sammanfattning Beteendet hos de periodiska lÄosningarna till Lorenz-ekvationerna har undersÄokts och summerats. Genom att variera Rayleigh-numret r, en parameter i ekvation- erna, observeras antingen kaotiska eller periodiska lÄosningar till systemet. Knu- tarna, som motsvaras av de periodiska lÄosningarna, klassi¯ceras och bestÄams. FramfÄorallt Äonskar vi ¯nna, som lÄosning, den hyperboliska ºatta-knuten fÄor att samla information om hur och nÄar den uppkommer. Genom att ¯nna denna knut skulle misstanken om att en ny vÄag till kaos existerar stÄarkas. TyvÄarr kan den knuten, som lÄosning, ej hittas. DÄaremot hittas tvºa andra, mer komplicerade, hy- perboliska knutar med 12 korsningar vardera. Detta faktum stÄarker misstanken om att den hyperboliska ºatta-knuten ¯nns som lÄosning till Lorenz-ekvationerna och i fÄorlÄangningen att ytterligare en vÄag till kaos skulle kunna existera, utÄover de idag kÄanda. Dessutom lokaliserades och klassi¯cerades flertalet torusknutar. Contents 1 Introduction 1 1.1 Scope of thesis . 1 1.2 Structure of thesis . 2 2 The Lorenz equations 3 3 Knot theory 5 4 Methods and equipment 8 4.1 Methods . 8 4.2 Equipment . 9 5 Experimental 10 5.1 The x2y window (99:524 < r < 100:795) . 10 5.2 The x2y2xy window (126:4 < r < 126:55) . 10 5.3 The x2y2 window (145:96 < r < 166:06) . 10 5.4 The xy window (214:364 < r < 1) . 11 5.5 Other periodic orbits listed by Sparrow . 11 5.6 Orbits not listed by Sparrow or others . 11 5.6.1 Orbits corresponding to torus knots . 12 5.6.2 Orbits corresponding to hyperbolic knots . 12 5.7 Knot plots . 13 6 Results and discussion 14 6.1 Results . 14 6.2 Discussion . 14 6.3 Conclusions . 15 6.4 Recommendations . 15 Appendix A - Maple°c code 17 Appendix B - The template approach 18 Appendix C - Knot popular 22 References 23 1 Introduction The Lorenz equations are a set of three ordinary di®erential equations model- ing convection rolls in the atmosphere that appear as warm air rises into cooler layers of air at higher altitude. This set of equations was originally proposed by Edward Lorenz in 1963 [1]. Lorenz, a meteorologist as well as a mathemati- cian, derived them by introducing a truncation to the Navier-Stokes equation e®ectively reducing it to a set of three ordinary di®erential equations in 3-space. These equations soon started to puzzle Lorenz since they seemed to behave in a random way depending on their initial conditions. If one parameter was changed ever so little, the result turned out to be completely di®erent: this is the main characteristic of chaotic systems which implies a practical impossibility of long term prediction. But for Lorenz, and the entire scienti¯c world, this concept was not yet known. Lorenz began an investigation of the equations which re- sulted in the paper [1] published in Journal of atmospheric sciences, a paper not commonly read by mathematicians. This was the main reason why the re- search on chaos theory progressed slowly the ¯rst years: mathematicians simply did not know that the chaos theory existed. Chaos is today a well known phe- nomenon occurring in most dynamical systems. The most common formulation of the de¯nition of chaos states that: Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions. \Aperiodic long-term behavior" means that there are trajectories which do not settle down to ¯xed points, periodic orbits, or quasiperiodic orbits as t ! 1. \Deterministic" means that there are no random or noisy inputs or parameters. \Sensitive dependence on initial conditions" means that nearby trajectories sep- arate exponentially fast resulting in completely di®erent outputs. This feature is also known as the butterfly e®ect, since a butterfly, in principle, only by flapping her wings, may cause a hurricane on the other side of the planet. The Lorenz equations, which this project is devoted to, have later also been found to model other dynamical systems such as lasers, dynamos and a mechanical waterwheel invented by Willem Malkus and Lou Howard at MIT in 1994 [2]. 1.1 Scope of thesis The main goal of this project is to ¯nd a solution to the Lorenz equations cor- responding to the hyperbolic ¯gure-eight knot with four crossings. This search is motivated by a suspected existence of hyperbolic knots, such as the ¯gure- eight knot, as solutions to the Lorenz equations. Especially if the ¯gure-eight knot was found, it would strengthen the suspicion that there is yet another route to chaos in the Lorenz equations than the ones known today: the Feigenbaum route, also known as the period doubling route, and the intermittency route. If the ¯gure-eight knot is found, the close neighborhood of the parameter space will be investigated to learn more about when and how it occurs. The project is a 1 preparatory work that hopefully can deliver some interesting answers about the properties of the ¯gure-eight knot. We will also collect and summarize results and methods used to study periodic orbits, or knots, in the Lorenz equations. 1.2 Structure of thesis The outline of this work is as follows: In chapter 1, which is the present chapter, a short background of the Lorenz equations is given as well as a motivation and scope of the project. Chapter 2 then deals, in a more mathematical way, with the Lorenz equations and the behavior of the corresponding solutions. In chapter 3 some basic and necessary de¯nitions connected to knot theory are given which will be used throughout the report. Chapter 4 describes the methods and equipment used for solving the set of equations, analyzing the solutions and plotting the solutions and knots. In chapter 5 the experimentally obtained data are listed. Chapter 6 contains a summary of the main results as well as a discussion, a conclusion and a recommendation regarding them. Last in the report one can ¯nd three appendices containing additional, but not for the report necessary, information. The report is concluded with bibliography and acknowledgements. 2 2 The Lorenz equations This chapter will present the Lorenz equations in a more detailed way. The Lorenz equations are: x_ = σ(y ¡ x) y_ = rx ¡ y ¡ xz (1) z_ = xy ¡ bz Here σ is the Prandtl number, r is the Rayleigh number and b is an arbitrary parameter. These are all real parameters. By convention the σ and b are held constant while r is varied to change the behavior of the system. Usually σ is 8 set to 10 and b to 3 . We stick to these choices. Research has been performed on other parameter values, e.g. by Shimizu and Morioka [3, 4, 5] as well as by Shimada and Nagashima [6]. Roughly speaking, in the convective and original formulation of the Lorenz equations, one can say that the variable x measures the rate of convective over- turning and y and z the horizontal and vertical temperature variations respec- tively. However, the main interest in the set of equations nowadays is not their applicability to fluid convection or atmospheric prediction, but in their mathe- matical properties. The Lorenz equations are one of the the best studied set of equations in the ¯eld of dynamical systems and chaos. This is probably due to their simplicity and the great variety of behavior of the solutions, that is, both chaotic and periodic behavior as displayed in ¯gure 1.
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