Mathematicians crack formula for predicting chaotic patterns in complex systems
Researchers have solved a long-standing problem in chaos theory by deriving an exact formula for how often points return to their starting location in certain mathematical systems. The breakthrough could improve models used in physics, engineering, and data analysis where systems exhibit unpredictable but measurable behavior.
Originaltitel: On recurrence sets for toral endomorphisms
Let A be a 2\times 2 integral matrix with an eigenvalue of modulus strictly less than 1. Let T be the natural endomorphism on the torus \mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2} , induced by A . Given \tau>0 , let R_{\tau} be the set of points x\in \mathbb{T}^{2} such that T^{n}x\in B(x,e^{-n\tau}) for infinitely many n\in\mathbb{N} . We obtain a formula for the Hausdorff dimension of R_{\tau} , and also prove that R_{\tau} has a large intersection property.