A sharp stability criterion for single well Duffing and Duffing-like equations

2019/06/04 by Alain Haraux, Haraux, Alain
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations

paper · doi:10.48550/arxiv.1906.01298

openalex publication_date 2019/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We refine some previous sufficient conditions for exponential stability of the linear ODE u''+ cu' + (b+a(t))u = 0 where b, c>0 and a is a bounded nonnegative time dependent coefficient. This allows to improve some results on uniqueness and asymptotic stability of periodic or almost periodic solutions of the equationu''+ cu' + g(u)=f(t)where c>0, f ∈ L^∞ (R) and g∈ C¹(R) satisfies some sign hypotheses. The typical case is g(u) = bu + a\vert u\vert^p u with a≥ 0 , b>0. Similar properties are valid for evolution equations of the form u''+ cu' + (B+A(t))u = 0 where A(t) and B are self-adjoint operators on a real Hilbert space H with B coercive and A(t) bounded in L(H) with a sufficiently small bound of its norm in L^∞(R+, L(H)) .

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