The Model of Multi Limits of the Quantity of Electrons in Central Coulomb force Field

Based on the theory of periodic bifurcation of iterative equation, a conjectural model of periodic bifurcation of number of electrons in a central Coulomb force field is proposed. After that with the help of the methods Zeng’s The Course of Quantum Mechanics and Wu’s Methods of Mathematical Physics, [1], [2] the wave function of the electrons under the approximate state is solved in the central Coulomb force field. By using the method of separating variables for solving partial differential equations and some transformation and construction techniques, the strict mathematical solution of the Schrödinger equation for the electron in the field of central Coulomb force is obtained, and the iterative formula of the level of electron number is given theoretically. And using MATLAB, the multi-limit model of electron number is simulated under different initial value problems, to explore the change of the limit with the initial value and the factors affecting the limit number to a certain extent. Some potential research value of this model is also proposed.

On Time-Periodic Bifurcation of a Sphere Moving under Gravity in a Navier-Stokes Liquid

We provide sufficient conditions for the occurrence of time-periodic Hopf bifurcation for the coupled system constituted by a rigid sphere, S, freely moving under gravity in a Navier-Stokes liquid. Since the region of flow is unbounded (namely, the whole space outside S), the main difficulty consists in finding the appropriate functional setting where general theory may apply. In this regard, we are able to show that the problem can be formulated as a suitable system of coupled operator equations in Banach spaces, where the relevant operators are Fredholm of index 0. In such a way, we can use the theory recently introduced by the author and give sufficient conditions for time-periodic bifurcation to take place.

Nonlinear analysis of the locomotive traction system with rub-impact and nonlinear stiffness

Considering rub-impact forces, eccentricity of rotor, and nonlinear stiffness of armature shaft, a dynamic differential equation is built to investigate the bifurcation and chaos behavior of the locomotive traction system. The nonlinear analysis methods are inclusive of bifurcation diagrams, phase plane portraits, Poincaré maps, displacement–time curves, and spectrograms. The simulation results reveal the complex dynamic comprising period-3 to quasi-periodic bifurcation and intermittent bifurcation. Some research results are references for dynamic design and rub-impact fault diagnosis of the locomotive traction system.

Research on the Numerical Simulation of the Nonlinear Dynamics of a Supercavitating Vehicle

Little is known about the movement characteristics of the supercavitating vehicle navigating underwater. In this paper, based on a four-dimensional dynamical system of this vehicle, its complicated dynamical behaviors were analyzed in detail by numerical simulation, according to the phase trajectory diagram, the bifurcation diagram, and the Lyapunov exponential spectrum. The influence of control parameters (such as various cavitation numbers and fin deflection angles) on the movement characteristics of the supercavitating vehicle was mainly studied. When the system parameters vary, various complicated physical phenomena, such as Hopf bifurcation, periodic bifurcation, or chaos, can be observed. Most importantly, it was found that the parameter range of the vehicle in a stable movement state can be effectively determined by a two-dimensional bifurcation diagram and that the behavior of the vehicle in the supercavity can be controlled by selecting appropriate control parameters to ensure stable navigation.

DYNAMIC ANALYSIS OF THE CAROTID–KUNDALINI MAP

The nature of the fixed points of the Carotid–Kundalini (C–K) map was studied and the boundary equation of the first bifurcation of the C–K map in the parameter plane is presented. Using the quantitative criterion and rule of chaotic system, the paper reveals the general features of the C–K Map transforming from regularity to chaos. The following conclusions are obtained: (i) chaotic patterns of the C–K map may emerge out of double-periodic bifurcation; (ii) the chaotic crisis phenomena are found. At the same time, the authors analyzed the orbit of critical point of the complex C–K Map and put forward the definition of Mandelbrot–Julia set of the complex C–K Map. The authors generalized the Welstead and Cromer's periodic scanning technique and using this technology constructed a series of the Mandelbrot–Julia sets of the complex C–K Map. Based on the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, we investigated the symmetry of the Mandelbrot–Julia set and studied the topological inflexibility of distribution of the periodic region in the Mandelbrot set, and found that the Mandelbrot set contains abundant information of the structure of Julia sets by finding the whole portray of Julia sets based on Mandelbrot set qualitatively.

Periodic Bifurcation For Semilinear Differential Equations With Lipschitzian Perturbations in Banach Spaces

Let A : D(A) → E, D(A) ⊂ E; be an infinitesimal generator either of an analytic compact semigroup or of a contractive C

Experimental Study on Nonlinear Dynamics Characteristics of High-Speed Rotor-Gas Lubrication Bearing System

An experimental study was done with the nonlinear dynamics characteristics of high-speed rotor-gas lubrication bearing system analyzed using the orbits of shaft center, 3D frequency coupling diagram, frequency spectrum and bifurcation diagram. Experimental results indicate that gas whirl is the main cause for the nonlinear instability of the high-speed rotor-gas lubrication bearing system, and the double periodic bifurcation resulting from gas whirl leads to the chaos vibration of the rotor-bearing system. It is therefore concluded that it is a new way to improve the nonlinear stability of a rotor-bearing system to keep the instability rotation speed far away from the design rotation speed by modulating the coupling between the natural frequency of the system and the gas whirl frequency, and to use the boundary nature of chaos verified through experiments to control the amplitude of vibration.