Numerical analysis of the initial conditions in fractional systems

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 adequate values for the initial conditions in numerical simulations.

Probability Density Evolution Algorithm for Stochastic Dynamical Systems Based on Fractional Calculus

“…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.…”

Fractal Dynamics and Control of the Fractional Potts Model on Diamond-Like Hierarchical Lattices

“…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.…”

“…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.…”

Fractional Order Bilingualism Model Without Conversion from Dominant Unilingual Group to Bilingual Group

“…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].…”