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Nonlinear oscillations: One hundred years after Lyapunov and Poincaré. (English) Zbl 0801.34037
This paper shows on examples the present influence of Lyapunov’s and Poincaré’s concepts, methods and ideas especially when they are combined with new mathematical techniques. The first example deals with the converse of the Lagrange-Dirichlet stability theorem. A proof is given of a sufficient condition for instability of the equilibrium combining a new variational principle and the fundamental theory of optimization in a Hilbert space. So it is proved that any local maximum of the potential energy of the considered mechanical system is an unstable equilibrium position. The second example is in connection with a differential equation with periodic nonlinearities, which is a perturbation of the equation of the mathematical pendulum. Mainly existence and multiplicity results for periodic solutions are considered. Systems with \(n\)-degrees of freedom having certain Lagrangians are subject of the review too. Also heteroclinic and rotation type solutions for autonomous systems with periodic nonlinearities are examined.

MSC:
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
01A55 History of mathematics in the 19th century
34-03 History of ordinary differential equations
70K20 Stability for nonlinear problems in mechanics
70H05 Hamilton’s equations
70H03 Lagrange’s equations
70-03 History of mechanics of particles and systems
74H45 Vibrations in dynamical problems in solid mechanics
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
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