Energy control and chaos prediction of a fractional-order financial risk contagion system

Abstract This paper delves into the dynamic characteristics and behaviors of a fractional -order financial risk contagion (FOFRC) system. Through systematic dynamic analysis incorporating stability, Lyapunov exponents spectrum, bifurcation diagrams, 0–1 test and complexity algorithms, the intrinsic nonlinear characteristics of the proposed model are analyzed. To improve system stability, a Hamilton energy feedback control strategy is further developed, which demonstrates effective chaotic behavior suppression through energy regulation principles. Furthermore, a PSR-ESN framework, which combines Echo State Networks with Phase Space Reconstruction, is employed to forecast chaotic time series. Comparative evaluations reveal that the PSR-ESN framework is superior to conventional ESN, RNN and LSTM architectures’ prediction accuracy and computational efficiency. These theoretical and computational advancements substantiate the unique advantages of fractional calculus in financial system modeling and chaotic behavior control, offering new perspectives for advancing risk management strategies.