Periodic solutions of discontinuous damped Duffing equations
In this paper, the authors study the following damped Duffing equation with a line of discontinuity \[ x''+cx'+g(x)=e(t),\quad x\in \mathbb{R},\tag{1} \] where \(x'\triangleq \frac{dx}{dt} \) and \(x''\triangleq \frac{d^2x}{dt^2},\) \(c>0\) is a constant, \(g(x)\) is given by \[ g(x)=\begin{cases} g_+(x), & x > 0, \\ g_-(x), & x < 0, \end{cases} \] \(e \in C^0(\mathbb{R},\mathbb{R})\) satisfying \(e(t + T) = e(t)\) and \(T > 0\) is the minimum period. Firstly, they prove that the periodic solutions of equation (1) can only be harmonic and subharmonic solutions. Secondly, the geometric properties of solutions for equation (1) are given. Finally, by using the two-dimension Poincaré-Bohl theorem, they obtain the existence and uniqueness of crossing harmonic and subharmonic solutions of equation (1).
scientific article; zbMATH DE number 7030198
scientific article