I am trying to make some simulations of chaotic systems, for instance X(k) 4 X(k) (1 - X(k-1)) but I noticed that for all these systems, the loss of precision propagates exponentially, to the point that after 50 iterations, all values generated are completely wrong. I wrote some code in Perl using the BigNum library (providing hundreds of decimals accuracy) and it shows how dramatic standard arithmetic fails in this context.

In this course, you will skill-up with techniques related to various aspects of concurrent programming in Python, including common thread programming techniques and approaches to parallel processing. Filled with examples, this course will show you all you need to know to start using concurrency in Python. You will learn about the principal approaches to concurrency that Python has to offer, including libraries and tools needed to exploit the performance of your processor. Learn the basic theory and history of parallelism and choose the best approach when it comes to parallel processing. After taking this course you will have gained an in-depth knowledge of using threads and processes with the help of real-world examples.

Here we describe well-known chaotic sequences, including new generalizations, with application to random number generation, highly non-linear auto-regressive models for times series, simulation, random permutations, and the use of big numbers (libraries available in programming languages to work with numbers with hundreds of decimals) as standard computer precision almost always produces completely erroneous results after a few iterations -- a fact rarely if ever mentioned in the scientific literature, but illustrated here, together with a solution. It is possible that all scientists who published on chaotic processes, used faulty numbers because of this issue. This article is accessible to non-experts, even though we solve a special stochastic equation for the first time, providing an unexpected exact solution, for a new chaotic process that generalizes the logistic map. We also describe a general framework for continuous random number generators, and investigate the interesting auto-correlation structure associated with some of these sequences. References are provided, as well as fast source code to process big numbers accurately, and even an elegant mathematical proof in the last section.