Numerical analysis of spectra of the Frobenius-Perron operator of a noisy one-dimensional mapping: toward a theory of stochastic bifurcations.

A different method to detect the stochastic bifurcation point of a one-dimensional mapping in the presence of noise is proposed. This method analyzes the eigenvalues and eigenfunctions of the noisy Frobenius-Perron operator. The invariant density or the eigenfunction of the eigenvalue 1 of the operator possesses "static" information of the noisy one-dimensional dynamics while the other eigenvalues and eigenfunctions have "dynamic" information. Clear bifurcation phenomena have been observed in a noisy sine-circle map and both stochastic saddle-node and period-doubling bifurcation points have been successfully defined in terms of the eigenvalues.