Multiple duffing problem in a folding structure with hill-top bifurcation

• We model a formula of the multiple Duffing oscillation with hill-top bifurcation.
• We extend to study theoretically mechanical model for the Duffing oscillation.
• It is recognized scientifically chaos phenomena with multiple-well potential system.
• It is more useful to analyze the multiple Duffing if there is bifurcation point.
• We will be able to get more knowledge about this Duffing oscillation.

This paper reviews the theoretical basis and its application for a multiple type of Duffing oscillation. This paper uses a suitable theoretical model to examine the structural instability of a folding truss which is limited so that only vertical displacements are possible for each nodal point supported by both sides. The equilibrium path in this ideal model has been found to have a type of “hill-top bifurcation” from the theoretical work of bifurcation analysis. Dynamic analysis allows for geometrical non-linearity based upon static bifurcation theory. We have found that a simple folding structure based on Multi-Folding-Microstructures theory is more interesting when there is a strange trajectory in multiple homo/hetero-clinic orbits than a well-known ordinary homoclinic orbit, as a model of an extended multiple degrees-of-freedom Duffing oscillation. We found that there are both globally and locally dynamic behaviours for a folding multi-layered truss which corresponds to the structure of the multiple homo/hetero-clinic orbits. This means the numerical solution depends on the dynamic behaviour of the system subjected to the forced cyclic loading such as folding or expanding action. The author suggests simplified theoretical models for hill-top bifurcation that help us to understand globally and locally dynamic behaviours, which depends on the static bifurcation problem. Such models are very useful for forecasting simulations of the extended Duffing oscillation model as essential and invariant nonlinear phenomena.