Homoclinic Bifurcations and Chaotic Dynamics in a Bistable Vibro-Impact SD Oscillator Subject to Gaussian White Noise

IF 1.9 4区 数学 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Lele Jia, Shuangbao Li, Liying Kou, Kongran Wu
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Abstract

This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures.
受高斯白噪声影响的双稳态振荡器中的同链分岔和混沌动力学
本文研究了高斯白噪声对双稳态振动冲击平滑连续(SD)振荡器的同室分岔和混沌动力学的影响。首先,通过在固定杆上安装一个滑块来再现和泛化 SD 振荡器,因此滑块由一对在垂直方向上初始预压缩的线性弹簧连接,以实现双稳态振动特性,并在杆上安装两个螺母作为两个可调节的双边刚性约束,以产生振动冲击。通过研究独特的、未受扰动的、非光滑的同室结构的持续性,推导出了一个不连续动力学方程,该方程在切换边界上定义了一个映射,以表示每次碰撞过程中的速度损失,从而描述了双稳态振动撞击 SD 振荡器的振动模式。其次,推导出了一类受高斯白噪声污染的双稳态振冲系统的随机梅利尼科夫过程的一般框架,并通过相应的梅利尼科夫函数得到了双稳态振冲自标振荡器同切线和可能混沌的必要参数阈值。第三,利用最大李雅普诺夫指数、分岔序列和 0-1 检验验证了梅尔尼科夫函数半分析预测的有效性。最后,通过分形吸引盆验证了混沌初始值的敏感性,并通过波恩卡雷映射研究了高斯白噪声对周期性结构和混沌结构的影响,展示了丰富的动力学几何结构。
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来源期刊
International Journal of Bifurcation and Chaos
International Journal of Bifurcation and Chaos 数学-数学跨学科应用
CiteScore
4.10
自引率
13.60%
发文量
237
审稿时长
2-4 weeks
期刊介绍: The International Journal of Bifurcation and Chaos is widely regarded as a leading journal in the exciting fields of chaos theory and nonlinear science. Represented by an international editorial board comprising top researchers from a wide variety of disciplines, it is setting high standards in scientific and production quality. The journal has been reputedly acclaimed by the scientific community around the world, and has featured many important papers by leading researchers from various areas of applied sciences and engineering. The discipline of chaos theory has created a universal paradigm, a scientific parlance, and a mathematical tool for grappling with complex dynamical phenomena. In every field of applied sciences (astronomy, atmospheric sciences, biology, chemistry, economics, geophysics, life and medical sciences, physics, social sciences, ecology, etc.) and engineering (aerospace, chemical, electronic, civil, computer, information, mechanical, software, telecommunication, etc.), the local and global manifestations of chaos and bifurcation have burst forth in an unprecedented universality, linking scientists heretofore unfamiliar with one another''s fields, and offering an opportunity to reshape our grasp of reality.
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