Data arising from hyperchaotic financial systems. Control through Koopman operators and EDMD

Abstract

We present a method for linearizing control and stabilization of chaotic systems in finance. This method considers the deviation of some trajectory of the system from an ideal or desirable orbit. Using Koopman operators and EDMD, we model this deviation as a linear dynamical system. The linear system is necessarily defined in some augmented state space whose dimension is bigger than the dimension of the original state space. The linear system can then be used for control and stabilization properties. Namely, one may apply feedback control to drive the deviation to zero, which means that the trajectory is close to the desired one. This approach can also be applied to more than one trajectories. However, in order to maintain good approximation properties, the more trajectories we consider the larger the dimensions of the linear system will become and at some stage the method will not be computationally effective. For this reason, we do not take into consideration the whole set of trajectories, but we start with a smaller set of orbits. This is a realistic scenario, since in economic studies the macroeconomic variables (such as the gross domestic product) are not arbitrary numbers but depend on the data of the economy.