Math 19. Lecture 33Causes of ChaosT. JudsonFall 20041 The Lor enz Equ atio nsWe first look at a what seems a simple system called the Lorenz equations.This set of equations was devised to model certain weather-related phenom-ena. The system can be written asdxdt= −σx + σy,dydt= rx − y − xz,dzdt= −bz + xy.It was discovered that for certain parameters, the trajectories of the solutionswere incredibly convoluted and effectively unpredictable. Here, σ, r, and bare constants. For certain values of these constants, the traj ectories are bothcrazy and extremely sensitive to their starting positions.In general, a systemdxdt= f(x, y, z),dydt= g(x, y, z),dzdt= h(x, y, z),or in vector formdvdt= f (v).1has a unique solution for each initial condition.• The solution can stay in a bounded region of the three dimensional ver-sion of the phase plane and wind through the region along an incrediblyconvoluted path.• The solution may be very sensitive t o initial data. Since real data al-ways has some inherent uncertainty, starting values are never preciselyknown.• No matter how long you watch a trajectory, you may not be able topredict future behavior.• There is still value, but you must take care.2 Equilibrium Points When v Has Two Com-ponentsThe situation is fairly straightforward.3 Equilibrium Points Whe n v Has Three Com-ponentsThe situation is much more complicated in three dimensions.4 Throwing the DiceTwo trajectories that start close ending up far away.25 Unpredict abil i ty for Two- Compo nent Sys-temsThis sort of unpredictability can only occur once along a trajectory (once fo reach hyperbolic equilibrium point).Readings and References• C. Taubes. Modeling Differential Equations in Biology. Prentice Hall,Upper Sa ddle River, NJ, 2001. Chapter 2
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