Flip-flop between soft-spring and hard-spring bistabilities in the approximated Toda oscillator analysis

We study theoretically the effect of truncating the nonlinear restoring force (exp(F)-1=?_n=1^￥Fⁿ/n!\exp (\Phi)-1={\sum}_{n=1}^{\infty}\Phi^{n}/n!) in the bistability pattern of the periodically driven, damped one-degree-of-freedom Toda oscillator that originally exhibits soft-spring bistability with counterclockwise hysteresis cycle. We observe that if the truncation is made third order, the harmonic bistability changes to hard-spring type with a clockwise hysteresis cycle. In contrast, for the fourth-order truncation, the bistability again becomes soft-spring type, overriding the effect of third-order nonlinearity. Furthermore, each higher odd-order truncation attempts to introduce hard-spring nature while each even-order truncation turns to soft-spring type of bistability. Overall, the hard-spring effect of every odd-order nonlinear term is weaker in comparison to the soft-spring effect of the next even-order nonlinear term. As a consequence, higher-order approximations ultimately converge to the soft-spring nature. Similar approximate analysis of Toda lattice has in recent past revealed remarkably similar flip-flop pattern between stochasticity (chaotic behaviour) and regularity (integrability).