Analytical period-3 motions to chaos in a hardening Duffing oscillator

In this paper, bifurcation trees of period-3 motions to chaos in the periodically forced, hardening Duffing oscillator are investigated analytically. Analytical solutions for period-3 and period-6 motions are used for the bifurcation trees of period-3 motions to chaos. Such bifurcation trees are based on the Hopf bifurcations of asymmetric period-3 motions. In addition, an independent symmetric period-3 motion without imbedding in chaos is discovered, and such a symmetric period-3 motion possesses saddle-node bifurcations only. The switching of symmetric to asymmetric period-3 motions is completed through saddle-node bifurcations, and the onset of asymmetric period-6 motions occurs at the Hopf bifurcations of asymmetric period-3 motions. Continuously, the onset of period-12 motions is at the Hopf bifurcation of asymmetric period-6 motions. With such bifurcation trees, the chaotic motions relative to asymmetric period-3 motions can be determined analytically. This investigation provides a systematic way to study analytical dynamics of chaos relative to period-m motions in nonlinear dynamical systems.     

Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).     

Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator

In this paper, the analytical dynamics of asymmetric periodic motions in the periodically forced, hardening Duffing oscillator is investigated via the generalized harmonic balance method. For the hardening Duffing oscillator, the symmetric periodic motions were extensively investigated with the aim of a good understanding of solutions with jumping phenomena. However, the asymmetric periodic motions for the hardening Duffing oscillators have not been obtained yet, and such asymmetric periodic motions are very important to find routes of periodic motions to chaos in the hardening Duffing oscillator analytically. Thus, the bifurcation trees from asymmetric period-1 motions to chaos are presented. The corresponding unstable periodic motions in the hardening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out as well. This investigation provides a comprehensive understanding of chaos mechanism in the hardening Duffing oscillator.     

Centered solutions for uncertain linear equations

Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we derive convex representations for united and tolerable solution sets. Secondly, to obtain centered solutions for uncertain linear equations, we develop a new method based on adjustable robust optimization (ARO) techniques to compute the maximum size inscribed convex body (MCB) of the set of the solutions. In general, the obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We use recent results from ARO to characterize for which convex bodies the obtained MCB is optimal. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input–output model, Colley’s Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method.     

Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

Effects of control measures on the dynamics of COVID-19 and double-peak behavior in Spain

Nonlinear Dynamics - The COVID-19 disease significantly has threatened the human lives and economy. It is a dynamic system with transmission and control as factors. Modeling the dynamics of the...     

High pressure X-ray diffraction study of potassium azide

Crystal structure and compressibility of potassium azide was investigated by in-situ synchrotron powder X-ray diffraction in a diamond anvil cell at room temperature up to 37.7 GPa. In the body-centered tetragonal (bct) phase, an anisotropic compressibility was observed with greater compressibility in the direction perpendicular to the plane containing N₃⁻ ions than directions within that plane. The bulk modulus of the bct phase was determined to be 18.6(7) GPa. A pressure-induced phase transition may occur at 15.5 GPa.     

基于甲氧基柱[5]芳烃的聚丙烯酸酯有机膜对对硝基苯衍生物的吸附行为

柱芳烃是一类有别于冠醚、杯芳烃、葫芦脲等的柱状大环分子,具有独特的富电子空腔和口腔的可修饰性。 它们可以包结多种有机污染物,对有机污染物的吸附和脱除具有广泛的应用前景。 本文通过核磁共振和紫外滴定的方法首先研究了对硝基苯衍生物与甲氧基柱[5]芳烃(MeP5A)的包结行为,测定了包结常数。 在此基础上,将MeP5A物理混合到聚丙烯酸酯(PA)乳液中,首先制备了甲氧基柱[5]芳烃/聚丙烯酸酯(MeP5A/PA)共混乳液,然后通过静电纺丝技术,得到了MeP5A/PA纳米纤维膜。 采用红外光谱、扫描电子显微镜对MeP5A/PA纳米纤维膜的结构和形貌进行了表征。 将MeP5A/PA纳米纤维膜用于4种对硝基苯衍生物的吸附实验。 结果表明,对硝基苯乙腈显示出与MeP5A最强的包结作用,其K_a=(6.0±0.3)×10² L/mol,PA膜中引入MeP5A,增大了吸附量,但并未改变纤维状形貌。 MeP5A/PA纳米纤维膜的最佳吸附平衡时间为2 h,且MeP5A/PA纳米纤维膜中MeP5A的含量越高,吸附量越大。 但当吸附溶液中MeP5A浓度达到4 mmol/L时(对应的膜中含有的MeP5A物质的量为1.4×10^-2 mmol),其吸附基本达到平衡,继续增加MeP5A物质的量,其吸附量变化不大。     

High pressure X-ray diffraction study on icosahedral boron arsenide (B12As2)

The high pressure properties of icosahedral boron arsenide (B₁₂As₂) were studied by in situ X-ray diffraction measurements at pressures up to 25.5 GPa at room temperature. B₁₂As₂ retains its rhombohedral structure; no phase transition was observed in the pressure range. The bulk modulus was determined to be 216 GPa with the pressure derivative 2.2. Anisotropy was observed in the compressibility of B₁₂As₂—c-axis was 16.2% more compressible than a-axis. The boron icosahedron plays a dominant role in the compressibility of boron-rich compounds.     

Robust convex optimization: A new perspective that unifies and extends

Mathematical Programming - Robust convex constraints are difficult to handle, since finding the worst-case scenario is equivalent to maximizing a convex function. In this paper, we propose a new...