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Homoclinic orbits in the dynamic phase-space analogy of an elastic strut. (English) Zbl 0755.73023
Summary: The equation (*) u IV (x)+Pu '' (x)+u(x)-u 2 (x)=0, is a possible dimensionless version of a model for the configuration of a very long strut resting on a nonlinear elastic foundation with axial loading P. By seeking to establish the existence of homoclinic orbits connecting the zero equilibrium of (*), now regarded as defining a four dimensional dynamical system, to itself, one is pursuing the so-called ‘dynamical phase-space analogy’ for the spatial configuration suggested by the form of the equation. The existence of homoclinic solutions is then interpreted as indicating the presence of spatially localized buckling of the deformed strut.
MSC:
74B20Nonlinear elasticity
74G60Bifurcation and buckling
34C37Homoclinic and heteroclinic solutions of ODE