Bifurcation analysis and periodic motions embedded in chaos in a DC-modulated Shinriki circuit

Bifurcation structures and periodic motions embedded in chaos play an important role in organizing complex oscillatory dynamics in nonlinear autonomous circuits. However, rigorous bifurcation analysis and the characterization of unstable periodic orbits remain insufficiently explored. This paper studies a DC-modulated Shinriki circuit, in which the introduced DC-offset acts as a control parameter that modifies circuit symmetry and reorganizes the associated dynamical states. The resulting behaviors are highlighted via equilibrium analysis, bifurcation diagrams, and Lyapunov exponents. Moreover, a finite Fourier series is employed to quantify the amplitude evolution of the decomposed harmonic components, providing a frequency-domain interpretation of the corresponding periodic and chaotic behaviors. The associated stable and unstable periodic motions are further discussed and validated experimentally on a field programmable gate array (FPGA) platform with oscilloscope observations. In addition, a pseudo random number generator (PRNG) is realized by exploiting the unstable oscillations hidden in chaos, and the generated bit sequences successfully pass the NIST tests under different output scales and long-term iteration intervals. These results provide meaningful insights into how symmetry variation reshapes the bifurcation dynamics and demonstrate the potential for related chaos-based applications.