Arnold tongues and mode-jumping in the supercritical post-buckling of an archetypal elastic structure

The buckling and post-buckling of a classical simply supported elastic strut on a stiffening elastic foundation is treated as a nonlinear dynamical system. For super critical loads this fourth-order system has a pair of imaginary eigenvalues, indicating the possibility of either a double or a quasi-periodic response. Finite lengths impose constraints that restrict such responses to the rational positions on the real line of the ratio of competing wavelengths: for irrational positions the response locks onto a single periodic wave. The extent of the mode-locking is described in parameter space by the geometry of Arnold tongues. Critical positions on the tongues are identified which. mark conditions for mode-jumping from one wavelength to another.