Random Oscillatory and Pulsatory Models in Elliptic Functions

Abstract

To explore experimental quantization of stochastic chaos and exact wave turbulence in exponential oscillons, it is necessary to construct smooth random functions of time. In the current paper, we develop a new method of modeling stochastic variables described by a closed system of ordinary differential and algebraic equations. Primarily, oscillatory and pulsatory dynamic models produced by the first triplet of copolar elliptic functions are studied from the viewpoint of the Hamiltonian and Newtonian dynamics. Secondly, the Hamiltonian systems of the first triplet and the first triplet squared are meticulously investigated in the hyperbolic limit that results in oscillations and pulsations with rectangular and point pulses and a variable period. Thirdly, the relative Hamiltonian systems are used to develop two stochastic models of a random oscillatory cn-noise and a random pulsatory cn 2 -noise. Numerical experiments show that for the Bernoulli frequencies the random oscillatory cn-noise approaches a smooth random oscillatory variable with an unbounded period and the Gaussian probability distribution and the random pulsatory cn 2 -noise tends to a smooth random pulsatory variable with an unbounded period and the truncated Gaussian probability distribution as the number of elliptic modes approaches infinity.