New exact soliton solutions, bifurcation and multistability behaviors of traveling waves for the (3+1)-dimensional modified Zakharov-Kuznetsov equation with higher order dispersion

The goal of the present paper is to obtain and analyze new exact travelling wave solutions and bifurcation behavior of modified Zakharov-Kuznetsov (mZK) equation with higher order dispersion term. For this purpose, first and second simple methods are used to build soliton solutions of travelling wave solutions. Furthermore, bifurcation behavior of traveling waves including new type of quasiperiodic and multi-periodic traveling wave motions have been examined depending on the physical parameters. Multistability for the nonlinear mZK equation has been investigated depending on fixed values of physical parameters with various initial conditions. The suggested methods for the analytical solutions are powerful and benefical tools to obtain the exact travelling wave solutions of nonlinear evolution equations (NLEEs). Two and three-dimensional plots are also provided to illustrate the new solutions. Bifurcation and multistability behaviors of traveling wave solution of the nonlinear mZK equation with higher order dispersion will add some value in the literature of mathematical and plasma physics.

Formal Chaos Existing Conditions on a Transmission Line Circuit with a Piecewise Linear Resistor

By using one-dimensional (1-D) map methods, some lossless transmission line circuits with a short at one side terminal have been actively studied. Bifurcation results or chaotic states in the circuits have been reported. On the other hand, many weak or strong definitions such that a 1-D map is mathematically chaotic are still being studied. In such definitions, the definition of formal chaos is well known as being the most traditional and most definite. However, formal chaos existences have not been rigorously proven in such circuits. In this paper, a general lossless transmission circuit is considered first with a dc bias voltage source in series with a load resistor at one side terminal and with a three-segment piecewise linear resistor at another side terminal. Secondly, the method for deriving a 1-D map describing the behavior of the circuit is summarized. Thirdly, to provide a basis of chaotic application for the 1-D map, the mathematical definition of formal chaos and the sufficient conditions of the existence of formal chaos are discussed. Furthermore, by using Maple, formal chaos existences and bifurcation behavior of 1-D maps are presented. By using the Lyapunov exponent, the observability of formal chaos in such bifurcation processes is outlined. Finally, the principal results and the future works are summarized.

Assessing the Non-Linear Dynamics of a Hopf–Langford Type System

In this paper, the non-linear dynamical behavior of a 3D autonomous dissipative system of Hopf–Langford type is investigated. Through the help of a mode transformation (as the system’s energy is included) it is shown that the 3D nonlinear system can be separated of two coupled subsystems in the master (drive)-slave (response) synchronization type. After that, based on the computing first and second Lyapunov values for master system, we have attempted to give a general framework (from bifurcation theory point of view) for understanding the structural stability and bifurcation behavior of original system. Moreover, a family of exact solutions of the master system is obtained and discussed. The effect of synchronization on the dynamic behavior of original system is also studied by numerical simulations.

Influence of Design Parameters on Static Bifurcation Behavior of Magnetic Liquid Double Suspension Bearing

Magnetic liquid double suspension bearing (MLDSB) includes electromagnetic system and hydrostatic system, and the bearing capacity and stiffness can be greatly improved. It is very suitable for the occasions of medium speed, heavy load, and starting frequently. Due to the mutual coupling and interaction between electromagnetic system and hydrostatic system, the probability and degree of static bifurcation are greatly increased and the operation stability is reduced. And flow of bearing cavity, coil current, oil film thickness, and galvanized layer thickness are the key parameters to ensure operation safe and stable, which has an important influence on the static bifurcation behavior. So this article intends to establish the coupling model of MLDSB to reveal the range of parameter combination in the case of static bifurcation. The influences of different parameter groups on the singularity characteristics, phase trajectory, x − t curves, and suction basin of the single DOF bearing system are analyzed. The result shows that there are nonzero singularities and static bifurcation occurs when ε 2 > 0 or δ 2 > 0 . As the flow of bearing cavity, coil current, oil film thickness, and galvanized layer thickness changes in turn, the singularities will convert between stable focus, unstable focus, stable node, and saddle point, and then the stable limit cycle may be generated. The attractiveness of singularity will change greatly with the flow of the bearing cavity and coil current changes slightly in the case of small current or large flow. The minimal change of galvanized layer thickness will lead to the fundamental change of the final stable equilibrium point of the rotor, while the final equilibrium point is slightly affected by the oil film thickness. This study can provide a reference for the supporting stability of MLDSB.

The Calculation Process of the Limit Cycle for Lorenz System

This work focuses on the bifurcation behavior before chaos phenomenon happens. Traditional numerical method is unable to solve the unstable limit cycle of nonlinear system. One algorithm is introduced to solve the unstable one, which is based one of the continuation method is called DEPAR approach. Combined with analytic and numerical method, the two stable and symmetrical equilibrium solutions exist through Fork bifurcation and the unstable and symmetrical limit cycles exist through Hopf bifurcation of Lorenz system. With the continuation algorithm, the bifurcation behavior and its phase diagram is solved. The results demonstrate the unstable periodical solution is around the equilibrium solution, besides the trajectory into the unstable area cannot escape but only converge to the equilibrium solution.

Analysis of nonlinear dynamic characteristic of a planetary gear system considering tooth surface friction

Based on the lumped mass method, a torsional vibration model of the planetary gear system is established considering the nonlinear factors such as friction, time-varying meshing stiffness, backlash, and comprehensive error. The Runge–Kutta numerical method is used to analyze the motion characteristics of the system with various parameters and the influence of tooth friction on the bifurcation and chaos characteristics of the system. The numerical simulation results show that the system has rich bifurcation behavior with the excitation frequency, damping ratio, comprehensive error amplitude, load and backlash, and experiences multiple periodic motion and chaotic motion. Tooth friction makes the bifurcation behavior of the system fuzzy in the high frequency and heavy load areas, makes the chaos of the system restrained in the low-damping ratio and light load areas, advances the bifurcation point of the system in the small comprehensive error amplitude area, and makes the period window of the chaos area larger in the large-backlash area, which makes the bifurcation behavior of the system more complex.