How Far Can We Trust Chaos? Extending the Horizon of Predictability

Abstract: Chaos reveals a fundamental paradox in the scientific understanding of Complex Systems. Although chaotic models may be mathematically deterministic, they are practically non-determinable due to the finite precision, which is inherent in all computational machines. Beyond the horizon of predictability, numerical computations accumulate errors, often undetectable. We investigate the possibility of reliable (error-free) time series of chaos. We prove that this is feasible for two well-studied isomorphic chaotic maps, namely the Tent map and the Logistic map. The generated chaotic time series have unlimited horizon of predictability. A new linear formula for the horizon of predictability of the Analytic Computation of the Logistic map, for any given precision and acceptable error, is obtained. Reliable (error-free) time series of chaos serve as gold standard for chaos applications. The practical significance of our findings include: (i) the ability to compare the performance of neural networks that predict chaotic time series, (ii) the reliability and numerical accuracy of chaotic orbit computations in encryption, maintaining high cryptographic strength, and (iii) the reliable forecasting of future prices in chaotic economic and financial models.