Quantization and Turbulization of Deterministic Chaos of the Exponential Oscillons and Pulsons

An exact nonstationary three-dimensional (3-d) solution for deterministic chaos of exponential oscillons and pulsons, which are governed by the Navier-Stokes equations, has been derived in [1] using the theoretical Deterministic Scalar Kinematic (tDSK) structures, the theoretical Deterministic Vector Kinematic (tDVK) structures, the theoretical Deterministic Scalar Dynamic (tDSD) structures, and the theoretical Deterministic Vector Dynamic (tDVD) structures. In the current work, an alternative derivation of deterministic chaos is provided with the help of the experimental Deterministic Scalar Kinematic (eDSK) structures, the experimental Deterministic Vector Kinematic (eDVK) structures, the experimental Deterministic Scalar Dynamic (eDSD) structures, and the experimental Deterministic Vector Dynamic (eDVD) structures that significantly simplify and clarify derivation. Symbolic computations of exact expansions in these invariant structures have been performed using the experimental and theoretical programming in Maple. Previously [2], quantization of the kinetic energy of deterministic chaos was developed and displayed for the Fourier set of x- and y-periods. In the present work, we explore an effect of the Bernoulli set of x- and y-periods, which grow as prime numbers, on the rate of turbulization of the exact solution for deterministic chaos. The method of inhomogeneous Fourier expansions through eigenfunctions in coordinates and time is used to study topology, periodicity, and integral properties of the exponential pulsons and oscillons. Computed results show a significant growth of the rate of turbulization of the deterministic chaos of the exponential oscillons and pulsons compared with those for the Fourier periods. Contrary to [2], probe visualizations of the kinetic energy qualitatively emulate experimental results on wave turbulence.