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

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    • Keywords

       

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

    • Abstract

       

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

    • Author Affiliations

       

      B K Goswami1

      1. Laser and Plasma Technology Division, Bhabha Atomic Research Centre, Mumbai 400 085, India
    • Dates

       
  • Pramana – Journal of Physics | News

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