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首页> 外文期刊>Acta Mechanica >Chaotic analyses of weakly damped parametrically excited cross waves with surface tension
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Chaotic analyses of weakly damped parametrically excited cross waves with surface tension

机译：具有表面张力的弱阻尼参激交叉波的混沌分析

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摘要

The Wiggins-Holmes extension of the Generalized Melnikov Method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to non-amonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to demonstrate that cross waves are chaotic. The Luke Lagrangian density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates (or, equivalently, three degrees of freedom) that are the time-dependent components of three velocity potentials that represent three standing waves. The generalized momenta are computed from the Lagrangian, and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains both autonomous and non-autonomous components that must be suspended by applying an extension of the Herglotz algorithm for non-autonomous transformations in order to apply the Kolmogorov-Arnold-Moser (KAM) averaging operation and the GMM. Three canonical transformations are applied to (i) eliminate cross product terms by a rotation of axes; (ii) to transform to action-angle canonical variables and to eliminate two degrees of freedom; and (iii) to suspend the non-autonomous terms and to apply the Hamilton-Jacobi transformation. The system of nonlinear non-autonomous evolution equations determined from Hamilton's equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic separatrices are computed from the unperturbed autonomous system. The non-dissipative perturbed Hamiltonian system with surface tension satisfies the KAM non-degeneracy requirements, and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic separatrix; consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents.
机译：广义梅尔尼科夫方法（GMM）的Wiggins-Holmes扩展到更高的维度以及广义Herglotz算法（GHA）到非自治系统的扩展被应用到在长矩形波通道中具有表面张力的弱阻尼参激交叉波为了证明横波是混沌的。具有表面张力和耗散的表面重力波的卢克拉格朗日密度函数用三个广义坐标（或等效地，三个自由度）表示，它们是代表三个驻波的三个速度势的时间相关分量。广义矩由拉格朗日计算得出，而哈密顿则由拉格朗日的勒让德变换计算得出。该哈密顿量包含自治和非自治分量，必须对非自治变换应用Herglotz算法的扩展才能将其暂停，以便应用Kolmogorov-Arnold-Moser（KAM）平均运算和GMM。应用了三种规范的转换，以（i）通过轴旋转消除叉积项; （ii）转换为作用角规范变量并消除两个自由度; （iii）暂停非自治条款并应用汉密尔顿-雅各比变换。必须对由第二类运动的汉密尔顿运动方程式确定的非线性非自治演化方程组进行平均，以获得可以由GMM分析的自治系统。由非斜向分离线连接的双曲鞍点是从不受干扰的自治系统中计算出来的。具有表面张力的非耗散摄动哈密顿系统满足KAM非简并性要求，并且通过计算梅尔尼科夫积分证明运动是混沌的。对于具有表面张力的耗散耗散系统，唯一在平均方程中幸存下来的双曲不动点是弱混沌的不动点，该不动点不由同斜向分离线连接。因此，梅尔尼科夫积分等于零。通过正Liapunov特征指数的数值计算证明了带有表面张力的耗散系统的混沌运动。

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