Performance tests for a Micro-Integrator algorithm which reduces the numerical butterfly effect in time evolving nonlinear systems

In computing solutions to chaotic systems, slight deviations in step size could lead to completely diverging trajectories as the system´s 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 `η´ 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 10,000; most of them were above 1,000. 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 led to higher performance factors.