Multi-degenerate hill-top bifurcation of Fermi-Pasta-Ulam softening chains: Exact and asymptotic solutions

A one-dimensional nonlinear elastic chain, known as Fermi-Pasta-Ulam system, is analyzed in the static field. The chain is made of elements admitting a quartic potential, with softening nonlinear behavior. When the chain is subject to pure tension, it exhibits a multi-degenerate hill-top bifurcation, from which several softening branches bifurcate. On each path, the springs either behave softening or hardening, in all the possible combinations, making the response non-unique. Both exact and asymptotic solutions are pursued, and the multitude of the bifurcated paths is illustrated by bifurcation diagrams. A proof of their instability is given. The role of the imperfections is commented, either in modifying the equilibrium paths and in unfolding the degenerate bifurcation.

Automated summary

Analyzed the behavior of a one-dimensional nonlinear elastic chain, known as the Fermi-Pasta-Ulam system, in a static field. The chain consists of elements with a quartic potential and softening nonlinear behavior. When subjected to pure tension, the chain exhibits a multi-degenerate hill-top bifurcation, resulting in several softening branches. The behavior of the springs on each path can either soften or harden, leading to non-unique responses. Bifurcation diagrams illustrate the multitude of bifurcated paths, and their instability is proven. The role of imperfections in modifying equilibrium paths and unfolding degenerate bifurcation is discussed.