DCDS
A new method for the boundedness of semilinear Duffing equations at resonance
Zhiguo Wang Yiqian Wang Daxiong Piao
We introduce a new method for the boundedness problem of semilinear Duffing equations at resonance. In particular, it can be used to study a class of semilinear equations at resonance without the polynomial-like growth condition. As an application, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^2x+g(x)+\psi(x)=p(t)$ under the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $\psi(x)$ are periodic and $g(x)$ is bounded.
keywords: periodic nonlinearity Moser's theorem. boundedness Hamiltonian system at resonance
DCDS-B
Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations
Qihuai Liu Dingbian Qian Zhiguo Wang
In this paper, we prove the existence of positive quasi-periodic solutions for the Lotka-Volterra competition systems with quasi-periodic coefficients by KAM technique. The result shows that, in most case, quasi-periodic solutions exist for sufficiently small quasi-periodic perturbations of the autonomous Lotka-Volterra systems. Moreover, these quasi-periodic solutions will tend to an equilibrium of the autonomous Lotka-Volterra systems.
keywords: KAM iteration quasi-periodic solution small denominator. Lotka-Volterra competition system

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