Detection Of Critical Transitions In Realistic Models

A critical transition represents an abrupt shift in the evolution of a dynamical system, where the system moves from one stable state to another. Understanding these sudden transitions allows us to understand many real life systems that undergo drastic changes over time. Typically, we consider systems that depend on numerous variables and parameters. These systems are highly complex to model because they often involve a noise factor from an external effect, which is why stochastic differential equations are utilized. One possible algorithm to detect critical transitions early on is called “Detrended Fluctuation Analysis,” i.e. DFA method. The DFA procedure involves interpolation of time series to obtain uniform spacing in time, detrending or removing the slow drift of the equilibrium by subtracting a slowly moving average, and calculating the autocorrelation and the variance of the resulting time series. From a theoretical perspective, an abrupt shift between two stable states takes place when the autocorrelation grows asymptotically to 1 and variance to infinity. We predict that the DFA algorithm can be used to predict the onset of critical transitions in realistic systems. We develop and improve existing algorithms for early detection of critical transitions by testing them on mathematical models based on stochastic differential equations. Our tests conclude that the DFA algorithm is a powerful tool for an early detection, provided it is fine tuned to the specifics of the underlying system. We apply these algorithms to analyze models of semiconductor lasers subject to optical injection and some other models.