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Stochastically perturbed dynamics of Hamiltonian systems near 1:1-resonance.

機(jī)譯：哈密??頓系統(tǒng)在1：1共振附近的隨機(jī)擾動(dòng)動(dòng)力學(xué)。

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This study is concerned with the dynamical behavior of stochastically perturbed Hamiltonian systems that are in 1:1-resonance and are weakly dissipative.; The main part of this investigation considers the near-resonant motion of integrable systems that are parametrically excited by noise. The emphasis of the study is on the effect of resonances on the averaged dynamics of the perturbed system. A framework for studying the motion of trajectories close to a resonance surface is developed. It is shown that the near-resonant motions can be approximated by an averaged system as in the non-resonant case. The effects of random perturbations on the near-resonant dynamics are analysed. Further, it is shown for 1:1-resonant systems having SO(2)-symmetry that the random perturbations result in passage of trajectories through a resonance zone in finite time.; The second important aspect of the present study is to obtain reduction of the dynamics of two-degrees of freedom perturbed integrable systems. This is achieved by making use of the integrable structure of the unperturbed motion and the separation of time scales of the perturbed dynamics. Reduction of the perturbed dynamics is obtained using the idea of stochastic averaging. Using this methodology the perturbed dynamics of a thin circular spinning disc subject to random fluctuations of its spin-rate is analysed. The stationary behavior of the randomly perturbed dynamics is analysed and stochastic bifurcation scenarios investigated for this system. The case where the unperturbed dynamics consists of multiple fixed points with a heteroclinic trajectory connecting two saddle points however, requires a different treatment. This scenario is considered in the present study in the context of stochastically perturbed weakly nonlinear Hamiltonian systems with 1:1-resonant semisimple linear form. By suitably extending the method of stochastic averaging on graphs to four-dimensional systems a reduced representation of the perturbed dynamics is obtained and the stationary behavior analysed. Various resonant motions are also considered. This part of the study is motivated by the problem of a nearly-square plate subject to random displacements of its edges.; A method to determine the moment-stability of stochastically perturbed two-degrees of freedom coupled linear systems in 1:1-resonance, is developed. The effect of the resonant coupling on the stochastic stability of the linear system is analysed. These results can be applied to examine the stability of stationary solutions of stochastically perturbed two-degrees of freedom nonlinear systems.; The importance of resonances and the effect of near-resonant motion of trajectories on the averaged dynamics of stochastically perturbed two-degrees of freedom integrable systems is emphasized throughout the study. The extension of the method of stochastic averaging on graphs to 1:1-resonant Hamiltonian systems forms the other major aspect of this work.
機(jī)譯：這項(xiàng)研究關(guān)注的是具有1：1共振和弱耗散性的隨機(jī)擾動(dòng)哈密頓系統(tǒng)的動(dòng)力學(xué)行為。本研究的主要部分考慮了可參數(shù)化系統(tǒng)的近共振運(yùn)動(dòng)，該系統(tǒng)被噪聲參數(shù)激發(fā)。研究的重點(diǎn)是共振對(duì)被攝動(dòng)系統(tǒng)平均動(dòng)力學(xué)的影響。建立了研究共振表面附近軌跡運(yùn)動(dòng)的框架。結(jié)果表明，與非共振情況一樣，可以通過(guò)平均系統(tǒng)來(lái)近似近共振運(yùn)動(dòng)。分析了隨機(jī)擾動(dòng)對(duì)近共振動(dòng)力學(xué)的影響。此外，對(duì)于具有 SO （2）對(duì)稱性的1：1共振系統(tǒng)，表明隨機(jī)擾動(dòng)會(huì)導(dǎo)致軌跡在有限時(shí)間內(nèi)通過(guò)共振區(qū)域。本研究的第二個(gè)重要方面是降低兩自由度擾動(dòng)可積系統(tǒng)的動(dòng)力學(xué)。這是通過(guò)利用不受干擾的運(yùn)動(dòng)的可積分結(jié)構(gòu)和受干擾的動(dòng)力學(xué)的時(shí)間尺度的分離來(lái)實(shí)現(xiàn)的。使用隨機(jī)平均的思想可以減少擾動(dòng)動(dòng)力學(xué)。使用這種方法，分析了一個(gè)薄的圓形紡絲盤(pán)的自旋速度隨機(jī)波動(dòng)的擾動(dòng)動(dòng)力學(xué)。分析了隨機(jī)擾動(dòng)動(dòng)力學(xué)的平穩(wěn)行為，并研究了該系統(tǒng)的隨機(jī)分叉情況。但是，如果擾動(dòng)動(dòng)力學(xué)由多個(gè)固定點(diǎn)組成，而該固定點(diǎn)具有連接兩個(gè)鞍點(diǎn)的異斜線軌跡，則需要進(jìn)行不同的處理。在本研究中，考慮了具有1：1共振半簡(jiǎn)單線性形式的隨機(jī)攝動(dòng)的弱非線性哈密頓系統(tǒng)的情況。通過(guò)將圖上的隨機(jī)平均方法適當(dāng)?shù)財(cái)U(kuò)展到四維系統(tǒng)，可以獲得擾動(dòng)動(dòng)力學(xué)的簡(jiǎn)化表示，并分析了穩(wěn)態(tài)行為。還考慮了各種共振運(yùn)動(dòng)。研究的這一部分是受一個(gè)接近正方形的板的邊緣隨機(jī)位移的問(wèn)題所驅(qū)使的。開(kāi)發(fā)了一種確定1：1共振隨機(jī)擾動(dòng)的兩自由度耦合線性系統(tǒng)的矩穩(wěn)定性的方法。分析了共振耦合對(duì)線性系統(tǒng)隨機(jī)穩(wěn)定性的影響。這些結(jié)果可用于檢驗(yàn)隨機(jī)擾動(dòng)的兩自由度非線性系統(tǒng)的平穩(wěn)解的穩(wěn)定性。在整個(gè)研究中，都強(qiáng)調(diào)了共振的重要性以及軌跡的近共振運(yùn)動(dòng)對(duì)隨機(jī)擾動(dòng)的兩個(gè)自由度可積系統(tǒng)的平均動(dòng)力學(xué)的影響。將圖上的隨機(jī)平均方法擴(kuò)展到1：1共振哈密頓量系統(tǒng)構(gòu)成了這項(xiàng)工作的另一個(gè)主要方面。

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