Quasi-Periodic, Phase-Locked and Chaotic Solutions in a Financial System with Two Feedback Delays

The complex dynamics observed in the financial system are key issues in the economic and social fields. Time delays also exist in lots of financial services and have some marked effects on the economy. Therefore, it is significant to study the intricate and abundant dynamical behaviors of a financial system with delays. In this paper, rich dynamics such as quasi-periodic, phase-locked and chaotic solutions, are worth to paying more attention to utilizing results of the time-delayed financial system to avoid disorder in the financial community. Firstly, double Hopf bifurcation points of the equilibrium are obtained by using DDE-BIFTOOL to draw critical curves of Hopf bifurcation. Furthermore, the method of multiple scales (MMS), a powerful analysis tool, is employed to derive the normal form, enabling their unfolding and classification. Two kinds of double Hopf bifurcation have been found. Finally, there are plentiful dynamical behaviors found in this system, such as quasi-periodic, phase locked and chaotic solutions, as well as bistability where two co-existing periodic solutions with different frequencies and amplitudes occur. It is worth mentioning that this paper presents two types of routes to chaos: torus breakdown and period-doubling bifurcation routes to chaos. Bifurcation diagrams and recurrent plots serve as tools for identifying quasi-periodic, phase-locked, and chaotic solutions within the system. These intricate dynamical phenomena offer valuable insights for researchers seeking a deeper understanding of the system's underlying mechanisms. For achieving optimal performance in the financial system, it becomes imperative to carefully select appropriate delay values.