Universal quantification for deterministic chaos in dynamical systems

A cell dynamical system model for deterministic chaos enables precise quantification of the round-off error growth, i.e., deterministic chaos in digital computer realizations of mathematical models of continuum dynamical systems. The model predicts the following: (a) The phase space trajectory (strange attractor) when resolved as a function of the computer accuracy has intrinsic logarithmic spiral curvature with the quasiperiodic Penrose tiling pattern for the internal structure. (b) The universal constant for deterministic chaos is identified as the steady-state fractional round-off error k for each computational step and is equal to 1/τ² ( = 0.382) where τ is the golden mean. k being less than half accounts for the fractal (broken) Euclidean geometry of the strange attractor. (c) The Feigenbaum's universal constantsa and d are functions of k and, further, the expression 2a² = πd quantifies the steady-state ordered emergence of the fractal geometry of the strange attractor. (d) The power spectra of chaotic dynamical systems follow the universal and unique inverse power law form of the statistical normal distribution. The model prediction of (d) is verified for the Lorenz attractor and for the computable chaotic orbits of Bernoulli shifts, pseudorandom number generators, and cat maps.