Chaos theory is a relatively new scientific paradigm for the analysis, simulation and prediction of non-linear phenomena whose initial conditions determine the behavior of their entire time series representation. It finds many applications in mathematics, science, and engineering. These include, but are not limited, to data encryption and decryption, designing secure communication systems, predicting weather patterns, noise fluctuations on data lines, understanding turbulence in fluid flow, and analyzing quantum wells. Systems that exhibit chaos are called chaotic systems. In computing solutions to nonlinear chaotic partial differential equation sets, slight deviations in step size could lead to completely diverging trajectories as the systems time series progresses. This is called the numerical butterfly effect. Smaller step sizes produce arrays closer to the desired continuous time solution, but they require more sampling points and as a result more memory. The Micro-Integrator produces results with a high level of accuracy while using only a fraction of the amount of memory required by conventional numerical integration methods. The reduction in memory requirements by the Micro-Integrator was quantified by introducing a performance factor 'Î·' that was mathematically equal to the ratio of the amount of memory required for computing without the Micro-Integrator to that required for computing with it. Recorded values of the performance factor from the tests ranged from 5 to 4 10 , out of which 75% were above 3 10 . The performance factor was also found to depend on the type of chaotic system, the numerical method, and the time window for computation. Less computationally efficient numerical methods resulted in higher performance factors than the more efficient ones.
Date of publication
Jemegbe, Joshua, "Using a Micro-Integrator to Eliminate the Numerical Butterfly Effect in Non-Linear Chaotic Partial Differential Equations" (2011). Electrical Engineering Theses. Paper 2.