Abstract
Pairs of numerically computed trajectories of a chaotic system may coalesce because of finite arithmetic precision. We analyse an example of this phenomenon, showing that it occurs surprisingly frequently. We argue that our model belongs to a universality class of chaotic systems where this numerical coincidence effect can be described by mapping it to a first-passage process. Our results are applicable to aggregation of small particles in random flows, as well as to numerical investigation of chaotic systems.
Highlights
When we numerically compute two distinct trajectories of a deterministic system, after a certain number of iterations their coordinates may happen to be exactly equal, to machine precision
Be expected that it would be unusual to see this numerical coalescence in chaotic systems, where nearby trajectories have an exponentially growing separation
We argue that our results are representative of a physically significant universality class of chaotic systems
Summary
When we numerically compute two distinct trajectories of a deterministic system, after a certain number of iterations their coordinates may happen to be exactly equal, to machine precision. We might say that the trajectories have undergone numerical coalescence This coalescence effect is readily observed in systems with stable dynamics, where nearby trajectories approach each other, typically with an exponentially decreasing separation. In our numerical work we used variable precision arithmetic implemented in the mpmath package [3] for the Python programming language, and comparable results were obtained using the Maple mathematical software system [4]. These packages allow the number of decimal digits, M, to be set to an arbitrary integer value. We review the relationship between the results in this paper and those earlier works in our conclusions, section 5
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