Mathematical research for dependence of structure of dynamical system on insertion of random errors

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In the present study, the stochastic difference equations are considered and the relation between the size of noise and characteristics of obtained numerical solutions is discussed. On the stochastic discrete logistic equation, it is clarified that intensively inserted randomness induces the inverse period doubling bifurcation in the averaged dynamical system. Dependence of the unsteady structure of asymptotic numerical solutions of the incompressible Navier-Stokes equations on the randomness is also studied by comparing the dynamical structure of asymptotic numerical solutions. It is clarified that weak noises make the system change and the effects of the noises are similar to those of the fourth viscosity terms.