In this study, we investigate the changes in the dynamics of generalized Chua's circuit through α-stable Lévy noise in the framework of stochastic bifurcation concept. By choosing the noise intensity, characteristic exponent (impulsiveness parameter) and the skewness parameter of the α-stable Lévy noise as the bifurcation parameters we have observed the qualitative changes in the stationary probability distributions and in the structures of the stochastic attractors.

In this paper, the probability density functions (PDFs) of the states of Generalized Chua's Circuit (GCC) have been modeled by Finite Mixture α-Stable (FMαS) distributions which is a Bayesian mixture model of α-stable distributions and it provides semiparametric characterization for the distributions of multiscroll chaotic attractors. Fully Bayesian approach has been applied to estimate the mixture parameters of multimodal distributions corresponding to the multiscroll chaotic attractors.

In this paper, the stochastic excitations in single machine infinite bus power systems have been modeled as alpha-stable Levy processes. Through the simulations of the corresponding stochastic differential equations, we have shown that the impulsiveness and/or asymmetry in the distributions of the load fluctuations can cause the instability of the rotor angle. Hence, the synchronism is lost and the rotor angle although it is stable in the sense of probability, it might not be stable in the mean square sense. However, by properly choosing the parameters of Levy type fluctuations the rotor angle stability can be improved in the sense of probability as the beneficience of noise.

In this study, the stochastic phenomenological bifurcations (P-bifurcations) of generalized Chua’s circuit (GCC) driven by skew-normal distributed noise have been investigated by numerically obtaining the stationary distributions of the stochastic responses. The noise intensity and/or skewness parameters of skew-normal distributed noise have been chosen as the bifurcation parameters to change the structure of the stochastic attractor. While the number of breakpoints in the piecewise-linear characteristics of the GCC are fixed, it has been observed that the number of scrolls have been changed by tuning the noise intensity and the skewness parameter of the skew-normal distributed noise.

The integration of renewable energy sources into the power systems and the growth of electricity consumption leads to a considerable increase in the power fluctuations. In the first part of this study, the control of the rotor angle stability of single machine infinite bus system in the presence of Wiener type power fluctuations has been achieved by minimizing the corresponding stochastic sensitivity function. In the second part, the power fluctuations have been modeled by alpha-stable Levy processes and since stochastic sensitivity function is not available for alpha-stable Levy processes, then the control of the rotor angle stability has been numerically achieved by minimizing the corresponding rotor angle dispersion for the first time in the literature.