Boundedness Of Solutions Of Duffing Equations With Asymmetric Terms

Duffing equations are important mathematical models in celestial mechanics,nonlinear oscillation and Hamiltonian system.The Duffing equation with asymmetric terms simulates the movement of particles subjected to asymmetric restoring forces after separation,which is a simplification of the suspension bridge model.It is of great theoretical significance and application value to study this kind of equation.In this paper,we investigated the boundedness of solutions for Duffing equations with asymmetric terms,that was,the following results in details:1.We proved that all solutions of the semilinear Duffing equation x"+ax+-bx-+?(x)=p(t)are bounded in resonance case,where the potential term ?(x)satisfies certain asymptotic limit conditions at infinity,and the perturbation term p(t)is periodic in time t and satisfies the critical condition.Firstly,we transformed the semilinear system into a sublinear one by eliminating the linear term in the Hamiltoian function through a rotational transformation.Then we constructed some canonical transformations and obtained the normal form of the sublinear system,for which a weak twist conditon hold true.Finally,we proved the boundedness of solutions of equations by Moser's twist theorem under reasonable conditions.2.We proved boundedness of all solutions of the p-laplace equation with asymmetric terms(?p(x'))'+a?p(x+)-b?p(x-)=g(x,t)+f(t),where the perturbation term f(t)and the potential function g(x,t)are sufficiently smooth and periodic in time t.The potential function satisfies the asymptotic limit conditon at infinity.We constructed canonical transformations such that the new Hamiltonian system was closed to a nearly integrable one.Then,we proved the boundedness of solutions of equations by Moser's twist theorem under the critical condition.3.We proved the Lagrangian stability of quasi-periodic super-linear asymmetric oscillators with impact (?)where f(t)is quasi-periodic in t.In order to cope with the nonsmoothness due to the existence of impact,we exchanged the role of time variables and angle variables.Then,We constructed canonical transformations such that the new Hamiltonian system is closed to a nearly integrable one.At last,we proved the Lag rangian stability by the twist theorem of quasi-periodic mappings.