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首页> 外文期刊>Physica Scripta: An International Journal for Experimental and Theoretical Physics >Bifurcation control for the Zakharov-Kusnetsov equation
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Bifurcation control for the Zakharov-Kusnetsov equation

机译：Zakharov-Kusnetsov方程的分叉控制

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摘要

We consider the bifurcation control for the forced Zakharov-Kusnetsov (ZK) equation by means of delay feedback linear control terms. Using a perturbation method, we obtain two slow flow equations on the amplitude and phase of the response as well as external force-response and frequency-response curves for the fundamental resonance. We observe in the resonance response for the uncontrolled system a saddle-center bifurcation, jumps and hysteresis phenomena and, using energy considerations, we show the existence of closed orbits of the slow flow equations. A limit cycle corresponds to a two-period quasi-periodic modulated motion for the ZK equation and we demonstrate that, in certain cases, a second low frequency appears in addition to the forcing frequency and then stable two-period quasi-periodic motions are present with amplitudes depending on the initial conditions. The value of the low frequency depends on the amplitude of the external excitation. Subsequently, we compare the uncontrolled and controlled systems and, to reduce the amplitude peak of the fundamental resonance and to remove saddle-center bifurcations and two-period quasi-periodic motions, we find appropriate choices of the feedback gains and time delay.
机译：我们通过延迟反馈线性控制项考虑强迫Zakharov-Kusnetsov（ZK）方程的分叉控制。使用摄动方法，我们获得了两个关于响应的振幅和相位以及基本共振的外力响应和频率响应曲线的慢流方程。我们在不受控制的系统的共振响应中观察到一个鞍状中心分叉，跳跃和滞后现象，并从能量的角度考虑，我们表明了慢流动方程的闭轨道的存在。极限周期对应于ZK方程的两周期准周期调制运动，并且我们证明，在某些情况下，除了强迫频率外，还会出现第二个低频，然后存在稳定的两周期准周期运动幅度取决于初始条件。低频值取决于外部激励的幅度。随后，我们比较了不受控制和受控制的系统，并且为了减小基本共振的振幅峰值并消除鞍形中心分叉和两周期准周期运动，我们找到了反馈增益和时间延迟的适当选择。

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