Complex analysis of a finance system and generalized synchronization for n-dimension

The article in hand is the study of complex features, such as zero Hopf bifurcation, chaos, and synchronization, of integer and fractional order versions of a new 3D finance system. Trusted tools of averaging theory and active control method are utilized for the investigation of zero Hopf bifurcation and synchronization, respectively, for both versions of the system. The novelty of this paper is to find the answer to the following question: is it possible to find a chaotic system that can be synchronized with any other of the same dimension? Based on different examples, we categorically develop a theory that if a couple of master and slave chaotic dynamical systems is synchronized by selecting a suitable gain matrix with special conditions, then the master system is synchronized with any chaotic dynamical system of the same dimension. With the help of this study, we developed generalized theorems for synchronization of n-dimension dynamical systems for integer and fractional versions. We proposed that this investigation will contribute a lot to control dynamical systems, will shorten the required criteria to synchronize the system under consideration with any other chaotic system of the same dimension, and ultimately, will minimize the cost. Chaotic properties of fractional version of the new finance system are also analyzed at fractional order q = 0.87. Simulations results, where required, are also provided to authenticate the analytical study.