Abstract:a b s t r a c tFractional dynamics is a growing topic in theoretical and experimental scientific research. A classical problem is the initialization required by fractional operators. While the problem is clear from the mathematical point of view, it constitutes a challenge in applied sciences. This paper addresses the problem of initialization and its effect upon dynamical system simulation when adopting numerical approximations. The results are compatible with system dynamics and clarify the formulation of ad… Show more
“…By solving the dynamic response of the simple vehicle model and the vehicle-track coupled system model, Chen Guo et al analyzed their power spectra and found that the agreement is better in the low-frequency region, but there will be a large gap in the high-frequency region. Lin Jiahao proposed the virtual excitation method to transform the complex random track upset into simple harmonic virtual track upset, which can improve the calculation accuracy and efficiency, gave a new method to evaluate the train operation comfort based on the elastic vehicle model [15], combined the virtual excitation method and the fine integration method to analyze the nonsmooth stochastic response of a bridge system under the action of random track upset, and concluded that the root mean square of the system response would be doubled for each level of track upset. Zeng et al used the virtual excitation method to calculate the vehicle-bridge coupled system under the assumption of wheel-rail close connection and the wheel-rail transverse creep rate; considering the vehicle moving load, the random dynamic response of the coupled vehicle-bridge system under the combined effect of vehicle moving load, random uneven excitation of the bridge deck and ground vibration load was calculated using the virtual excitation method and its statistical law was analyzed.…”
With the increasing load and speed of trains, the problems caused by various random excitations (such as safety and passenger comfort) have become more prominent and thus arises the necessity to analyze stochastic dynamical systems, which is important in both academic and engineering circles. The existing analysis methods are inadequate in terms of computational accuracy, computational efficiency, and applicability in solving complex problems. For that, a new efficient and accurate method is used in this paper, suitable for linear and nonlinear random vibration analysis of large structures as well as static and dynamic reliability assessment. It is the direct probability integration method, which is extended and applied to the random vibration reliability analysis of dynamical systems. Dynamical models of the dynamic system and coupled system “three-car vehicle-rail-bridge” are established, the time-varying differential equations of motion are derived in detail, and the dynamic response of the system is calculated using the explicit Newmark algorithm. The simulation results show the influence of the number of representative points on the smoothness of the image of the probability density function and the accuracy of the calculation results.
“…By solving the dynamic response of the simple vehicle model and the vehicle-track coupled system model, Chen Guo et al analyzed their power spectra and found that the agreement is better in the low-frequency region, but there will be a large gap in the high-frequency region. Lin Jiahao proposed the virtual excitation method to transform the complex random track upset into simple harmonic virtual track upset, which can improve the calculation accuracy and efficiency, gave a new method to evaluate the train operation comfort based on the elastic vehicle model [15], combined the virtual excitation method and the fine integration method to analyze the nonsmooth stochastic response of a bridge system under the action of random track upset, and concluded that the root mean square of the system response would be doubled for each level of track upset. Zeng et al used the virtual excitation method to calculate the vehicle-bridge coupled system under the assumption of wheel-rail close connection and the wheel-rail transverse creep rate; considering the vehicle moving load, the random dynamic response of the coupled vehicle-bridge system under the combined effect of vehicle moving load, random uneven excitation of the bridge deck and ground vibration load was calculated using the virtual excitation method and its statistical law was analyzed.…”
With the increasing load and speed of trains, the problems caused by various random excitations (such as safety and passenger comfort) have become more prominent and thus arises the necessity to analyze stochastic dynamical systems, which is important in both academic and engineering circles. The existing analysis methods are inadequate in terms of computational accuracy, computational efficiency, and applicability in solving complex problems. For that, a new efficient and accurate method is used in this paper, suitable for linear and nonlinear random vibration analysis of large structures as well as static and dynamic reliability assessment. It is the direct probability integration method, which is extended and applied to the random vibration reliability analysis of dynamical systems. Dynamical models of the dynamic system and coupled system “three-car vehicle-rail-bridge” are established, the time-varying differential equations of motion are derived in detail, and the dynamic response of the system is calculated using the explicit Newmark algorithm. The simulation results show the influence of the number of representative points on the smoothness of the image of the probability density function and the accuracy of the calculation results.
“…where k is the control parameter. Since 1 is the fixed point for any ], μ, p in system (15), we take z * � 1 in controlled system (17). So, the controlled system (17) becomes Figures 2 and 3, we can see that Julia sets of the controlled system (19) are shrinking with the increasing of control parameters k.…”
Section: Control Of Julia Sets Of Discrete Fractional System (15)mentioning
confidence: 99%
“…Fractional calculus is a generalization of the ordinary differential and integral to an arbitrary order. e fractional dynamical systems are related with the past status and can reflect the situation of the system more realistically [16][17][18]. And, the fractional difference provides us a new powerful tool to depict the dynamics of discrete complex systems.…”
The fractional Potts model on diamond-like hierarchical lattices is introduced in this manuscript, which is a fractional rational system in the complex plane. Then, the fractal dynamics of this model is discussed from the fractal viewpoint. Julia set of the fractional Potts model is given, and control items of this fractional model are designed to control the Julia set. To associate two different Julia sets of the fractional model with different parameters and fractional orders, nonlinear coupling items are taken to make one Julia set change to another. The simulations are provided to illustrate the efficacy of these methods.
“…In population models, the future state of the population depends on the past state which is called memory effect. By including a delay term or using fractional derivative in the model, one can handle the memory effect of the population [9,10].…”
A fractional order model of a population with one bilingual and two unilingual components, in which conversion from dominant unilingual to bilingual doesn't exist is studied. Equilibrium points are found, criteria for the existence and the stability of the positive equilibrium are then investigated. Also, numerical solutions for an example of the fractional order system are obtained by transforming the fractional system to the corresponding integer order one.
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