Comparative Analysis of Numerically Computed Chaos Diagrams in Duffing Oscillator

T. Salau, O. Ajide
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引用次数: 5

Abstract

This study utilised optimum fractal disk dimension algorithms to characterize the evolved strange attractor (Poincare section) when adaptive time steps Runge-Kutta fourth and fifth order algorithms are employed to compute simultaneously multiple trajectories of a harmonically excited Duffing oscillator from very close initial conditions. The challenges of insufficient literature that explore chaos diagrams as visual aids in dynamics characterization strongly motivate this study. The object of this study is to enable visual comparison of the chaos diagrams in the excitation amplitude versus frequency plane. The chaos diagrams obtained at two different damp coefficient levels conforms generally in trend to literature results[1] and qualitatively the same for all algorithms. The chances of chaotic behaviour are higher for combined higher excitation frequencies and amplitudes in addition to smaller damp coefficient. Fourth and fifth order Runge-Kutta algorithms indicates respectively 62.3% and 53.3% probability of chaotic behaviour at 0.168 damp coefficient and respectively 77.9% and 78.9% at 0.0168 damp coefficient. The chaos diagrams obtained by fourth order algorithms is accepted to be more reliable than its fifth order counterpart, its utility as tool for searching possible regions of parameter space where chaotic behaviour/motion exist may require additional dynamic behaviour tests.
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Duffing振荡器数值计算混沌图的比较分析
当采用自适应时间步长龙格-库塔四阶和五阶算法在非常接近的初始条件下同时计算谐波激发Duffing振荡器的多重轨迹时,本研究利用最佳分形盘维数算法来表征进化的奇异吸引子(庞加莱截面)。研究混沌图作为动力学表征的视觉辅助的文献不足的挑战强烈地激发了这项研究。本研究的目的是使在激励幅值与频率平面的混沌图的视觉比较。两种不同阻尼系数水平下得到的混沌图在趋势上与文献结果[1]基本一致,各算法的混沌图在质量上是一致的。在较高的激励频率和幅值以及较小的阻尼系数的组合下,混沌行为的可能性更高。四阶和五阶龙格-库塔算法在0.168阻尼系数下的混沌概率分别为62.3%和53.3%,在0.0168阻尼系数下的混沌概率分别为77.9%和78.9%。通过四阶算法获得的混沌图被认为比其五阶对应的混沌图更可靠,其作为搜索参数空间中存在混沌行为/运动的可能区域的工具可能需要额外的动态行为测试。
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