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Hopf bifurcation in Beck’s problem. (English) Zbl 0657.35011

A so-called Beck’s problem is to determine the stability of a column with one end welded to a rigid support and the other end subjected to a compressive follower force. This problem is reduced to the treatment of a quasi-linear parabolic system: \[ {\dot \nu}_ t=A{\dot \nu}_{xx}- A\theta^ 2_ x{\dot \nu}+f[\nu,\eta,\theta,{\dot \eta},{\dot \theta}] \]
\[ {\dot \eta}_ t=B{\dot \eta}_{xx}-B\theta^ 2_ x{\dot \eta}+g[\nu,\eta,\theta,{\dot \nu},{\dot \theta}] \]
\[ {\dot \theta}_ t=(C/I){\dot \theta}_{xx}+h[\nu,\eta,\theta,{\dot \nu},{\dot \eta}]. \] In this paper it is proved that the equilibrium can be lost by Hopf bifurcation and that viscosity can cause destabilization.
Reviewer: J.H.Tian

MSC:

35B32 Bifurcations in context of PDEs
35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
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