Extreme Events and Markov Model

 Despite that the border between chaotic and stochastic systems is exactly defined, scientists, use high dimensional chaotic dynamics to model numerous stochastic models and sometimes use stochastic models to study chaotic systems. Indeed, the border between these two is usually much vaguer in high dimensional systems. With the outstanding breakthrough in data science, eliciting the statistical properties of the complex dynamical system has become easier and more beneficiary.

In this work, we have investigated chaotic systems with a stochastic approach and proposed an estimator for the chaotic system which is used to present different algorithms for chaos control, extreme event prediction and extreme event mitigation.

The stochastic estimator is constructed by meshing the phase space and applying the cell mapping method (with some considerations) which provides us with a model-free approximation of the systems. The technique used here is unique between many of the chaos control approaches and almost all of the extreme events prediction and mitigation algorithms. The algorithms are ideal for real-world applications where there are many noises and the system is high-dimensional with limited access to state variables. Besides, the proposed methods are adopted for when the control signal is limited to a few specific values. We have tested these algorithms on Logestic, Henon and a physiological control system and the results showed that the algorithms work very well and can be used on other systems.