Science and Computers I
Science and Computers I PHY 307
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This 6 page Class Notes was uploaded by Ms. Bryce Wisoky on Wednesday October 21, 2015. The Class Notes belongs to PHY 307 at Syracuse University taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/225633/phy-307-syracuse-university in Physics 2 at Syracuse University.
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Date Created: 10/21/15
Lec5 Nonlinear systems chaos Phase space Poincare maps strange at tractors Period doubling Lorenz model balls in boxes Simple observations Initially transients seen remnant of decay ing natural oscillation Small driving force small amplitude mo tion in step with driving force like hamr moniC case Larger F apparently random or Chaotic behavior seen Windows of regular motion found at larger F Cannot be truly random motion deter ministic Something more subtle happen ing Sensitivity to initial conditions Two identical pendula with slightly different initial conditions In regular regime motions converge with time o In chaotic regime diverge o In first case poor knowledge of initial con ditions is irrelevant to predicting long time motion 0 In other case implies no predictability at long times eg weather Phase space Useful to examine motion not as t0 and tw but in phase space 9w 0 Regular non chaotic motion yields simple closed curve a Chaotic motion much structure Many nearly closed orbits sudden departures to new orbits never repeating Poincare plots Instead of plotting entire phase space trajec tory plot 9w only at multiples of time period of driving force a For regular motion single point seen a For chaotic motion non space filling struc ture seen Does not depend on initial con ditions o Predictable aspect of chaotic motion called a strange attractor All chaotic motions of system approach a motion on the attrac tor a Not a 1D curve in general fractal object later Lorenz model Another example of model showing chaos Very simplified model of convective fluid flow container containing fluid with bot tom and top surfaces held at different tem peratures Three variables 13 y z corresponding to tem perature density and fluid velocity Three parameters arb temperature dif ference and fluid parameters Full solution involves Navier Stokes and very many variables Weather simulations etc
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