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DY: Dynamik und Statistische Physik

DY 46: Poster

DY 46.19: Poster

Donnerstag, 11. März 2004, 16:00–18:00, Poster D

Exact Analytical Solutions and Unpredictability of the Lorenz System — •Oliver Strebel — Handjerystr. 31, 12159 Berlin

For the Lorenz equations [1] exact analytical solutions are given. Special care was taken to construct graphs of solution curves by evaluating the analytical formula up to the accuracy of the floating point arithmetic of numbers in the used computer system. In addition to an analytical check the solution and its time derivative are evaluated on a computer and the difference between the left and right hand side of the Lorenz equations is calculated. This results in a tiny difference of several hundreds times the floating point precision constant stemming from different rounding errors during the evaluations of the formulas for the solution and its time derivative.

For the chaotic case of the Lorenz system it is demonstrated that for a given accuracy of the initial conditions it becomes undetermined after a few time units whether the system will reside in the left or right half space having a x coordinate less or greater than zero. This indicates fast transition to unpredictability of the system. The dynamics has for large regions of initial conditions a close resemblence to a Smale horseshoe [2], where adjacent initial conditions generate completely different trajectories.

[1] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos and Strange Attractors, Springer AMS 41, Berlin (1982).

[2] S. Wiggins, Global Bifurcations and Chaos, Springer AMS 73, Berlin (1988).