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

Pramana

B K Goswami1¹ 

[1] Laser and Plasma Technology Division, Bhabha Atomic Research Centre, Mumbai 400 085, India$$

关键词: Toda oscillator;    Duffing oscillator;    soft-spring and hard-spring bistabilities;    principle of harmonic balance;    Hamiltonian chaos;    integrability;    Toda lattice.;   

DOI  :  

学科分类：物理（综合）

来源: Indian Academy of Sciences

【 摘 要 】

We study theoretically the effect of truncating the nonlinear restoring force (exp $(𝛷)−1 = Sum^{∞}_{n=1} 𝛷^n/n!$) 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).

【 授权许可】

Unknown