Abstract
This study explores the impact of force nature (rational, algebraic, irrational) on solution dynamics in mass-spring systems, with a focus on a given square root nonlinearity. We establish a connection between force nature and solution nature, highlighting a hierarchy of solutions, stability control, and predictability. For the system with the chosen square root nonlinearity, we uncover irrational dynamics, superpitchfork bifurcation, and a hierarchy of equilibrium points with complex behaviors. As the bifurcation parameter increases, the system transitions from a stable equilibrium to a saddle point and symmetric stable equilibria. This reveals novel phenomena such as equilibrium cross-sensitivity to eigenvalues, intricated equilibrium complexity, equilibrium points' asymmetrical influence, and mutual influence symmetry breaking. Under sinusoidal excitation, ALTC (Amplitude, Lyapunov exponents, Time scale ratio magnitude, and frequency Coefficient of variation) maps identify diverse dynamics, including chaotic and non-chaotic bursting, with chaos sensitivity to truncation or rounding errors. These findings highlight the interplay between irrational algebraic and transcendental dynamics, raising new questions about the theoretical foundations of such systems.