Analysis of Steady-State Solutions and Bifurcations in Higher-Dimensional Piecewise-Linear Systems

Vibration systems with multiple potential wells can often be approximated by higher-dimensional piecewise-linear nonsmooth systems, whose responses typically consist of complex frequency components. Existing studies provide limited analytical insights into the steady-state responses of such systems, incompletely revealing the response characteristics. This work investigates the bifurcation behavior of a higher-dimensional piecewise-linear system subjected to low-frequency excitation. The system consists of a coupled linear oscillator and a piecewise-linear quadruple-well structure. The Euler–Lagrange equation is employed to establish the dynamic model, from which a corresponding dimensionless form is derived. The phase space of the piecewise-linear system is divided into seven regions by switching manifolds, within which the equations of motion are linear. On this basis, an analytical method is developed by constructing mappings across the switching manifolds. Numerical simulations are then performed to examine the superharmonic responses of the system under low-frequency sinusoidal excitation. The bifurcation behavior is analyzed using phase portraits, time histories, bifurcation diagrams, power spectral density, frequency spectra, and Lyapunov exponent spectra.