Bifurcation Analysis of the Fractional Duffing System Based on the Improved Short Memory Principle Method

In this study, the improved short memory principle method is introduced to the analysis of the dynamic characteristics of the fractional Duffing system, and the basis for the improvement of the short memory principle method is provided. The influence of frequency change on the dynamic performance of the fractional Duffing system is studied using nonlinear dynamic analysis methods, such as phase portrait, Poincare map and bifurcation diagram. Moreover, the dynamic behaviour of the fractional Duffing system when the fractional order and excitation amplitude change is investigated. The analysis shows that when the excitation frequency changes from 0.43 to 1.22, the bifurcation diagram contains four periodic and three chaotic motion regions. Periodic motion windows are found in the three chaotic motion regions. Results confirm that the frequency and amplitude of the external excitation and the fractional order of damping have a greater impact on system dynamics. Thus, attention should be paid to the design and analysis of system dynamics.

Improved Short Memory Principle Method for Solving Fractional Damped Vibration Equations

In this paper, an improved short memory principle based on the Grünwald–Letnikov definition is proposed and applied in solving fractional vibration differential equations. The improved idea is to adjust the truncation of memory time in short memory principle (SMP) to the truncation of binomial coefficient terms, and the finite coefficients are repeatedly applied to the step size gradually enlarged. In this method, a very small initial step size is used to meet the accuracy requirements, and then the step size is gradually enlarged to prolong the memory time and reduce the amount of calculation. Examples of free vibration, forced vibration with a single-degree-of-freedom and a vehicle suspension two-degree-of-freedom vibration reduction model verify the method’s accuracy and effectiveness.

A note on short memory principle of fractional calculus

In this paper, from the classical short memory principle under Grünwald-Letnikov definition, several novel short memory principles are presented and investigated. On one hand, the classical principle is extended to Riemann-Liouville and Caputo cases. On the other hand, a special kind of principles are formulated by introducing a discrete argument instead of the continuous time, resulting in principles with fixed memory length or fixed memory step. Apart from these, several interesting properties of the proposed principles are revealed profoundly.