Complementarity, duality and symmetry in nonlinear mechanics: DUALITY, COMPLEMENTARITY, AND POLARITY IN NONSMOOTH/NONCONVEX DYNAMICS

摘要

This paper presents a brief survey and a unified theory in nonsmooth and nonconvex dynamical systems. The canonical dual/polar transformation methods and the associated bi-duality and triality theories proposed recently in nonconvex variational problems are generalized into fully nonlinear dissipative dynamical systems governed by nonsmooth constitutive laws and boundary conditions. It is shown that by this method, nonsmooth and nonconvex Hamilton systems can be reformulated into certain smooth dual-complementary variational problems. An bi-polarity variational principle is established for 3-D nonsmooth elastodynamical systems, and a potentially powerful complementary variational principle can be used for solving unilateral variational inequality problems governed by nonsmooth boundary conditions. Applications are illustrated by dynamically post-buckling analysis of extended beam model. A very interesting new phenomenon: meta-buckling transition period in chaotic vibration is revealed and a bifurcation criterion for dissipative Duffing system is obtained by the dual analysis. It is shown that the chaotical solutions form an invariant set in canonical dual phase space. This invariant set and the bifurcation criterion play key roles in feedback controlling against chaos.