Exact computation of Lyapunov exponents via system parameters in multi-triangle chaotic maps: Bifurcation analysis and circuit realization

This study investigates chaotic maps based on multiple triangular functions with methods of summation and cyclic coupling, rigorously proving that their Lyapunov exponents depend solely on system parameters. As an illustrative example, a one-dimensional chaotic map, namely, the golden-ratio double-triangle chaotic map, is examined in detail. We theoretically derive the conditions for its boundedness and linear stability and identify the critical parameter threshold for chaos generation. Numerical simulations demonstrate that the system exhibits rich dynamic behaviors. Furthermore, an approximate arcsine function module is designed via Taylor expansion, leading to the construction of an alternative system that is easier to implement. The feasibility of both the original and the alternative systems is verified using PSIM. Simulation results confirm that the system employing the approximate function can effectively generate chaotic signals, highlighting its significant value for engineering applications. In conclusion, this work provides a theoretical foundation and a practical implementation framework for constructing highly configurable and realizable chaotic systems in various engineering contexts.