Abstract
In this paper, the dynamic characteristics of fractional Duffing system are analyzed and studied by using the improved short memory principle method. This method has small amount of calculation and high precision, and can effectively improve the problem of large amount of calculation caused by the memory of fractional order. 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 changes are 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. It is confirmed that the frequency and amplitude of the external excitation and the fractional order of damping have a greater impact on system dynamics. Thus, attention shall be paid to the design and analysis of system dynamics.
1. Introduction
Fractional calculus was proposed almost at the same time as integer calculus. Its development was relatively slow for nearly 300 years due to the lack of a clear physical meaning of fractional calculus, and it was mainly studied in the field of pure mathematics. In 1974, Oldham and Spanier [1] coauthored the first monograph on fractional calculus titled ‘Applied Fractional Calculus’, which paved its way towards the application of fractional calculus. In the 1970s, Mandelbrot [2] pointed out that many fractional dimensions exist in nature. Since then, the theory of fractional calculus developed rapidly, and numerous applications of fractional calculus emerged [3][9]. The application of fractional calculus in mechanical engineering gradually increased in recent years mainly because many physical objects had fractional characteristics, such as viscoelasticity, damping, mechanical friction and impact [10][17].
Many popular fractional systems, such as the Lorenz system [5], the Rossler system [13], Chua’s circuit [6] and the Duffing system [9], were used for research. The Duffing system was applied in many fields, including fluidinduced vibration, largeamplitude oscillation of centrifugal speed control systems and mathematical modeling [9], [18], [19]. The Duffing system that considers fractional differential damping is equivalent to introducing polymer damping into the classic Duffing system. Compared with the traditional integerorder damping model, the fractionalorder damping model can more accurately describe the dynamic behavior of the system [20][22]. In Reference [23], the nonlinear dynamic behavior of the fractionalorder Duffing system was studied using the AdamsBashforthMoulton predictorcorrector method. The Adomian decomposition method was utilized to find the numerical solution to the combined fractional differential equation of the Duffing equation and the Van der Pol equation in Reference [24]. The Melnikov method transforms a fractionalorder system into an equivalent integerorder system, which can effectively detect chaos in the fractionalorder Duffing system [25]. Based on the definition formula with an amount of calculations, reference [26] studies the bifurcation and vibration resonance of fractional doubledamping Duffing time delay system driven by external excitation signal with two wildly different frequencies. In Ref. [27], phase diagrams, bifurcation diagrams and Lyapunov exponents are used to determine the presence of chaos over a wide range of fractional orders, based on the Caputo fractional difference form of Duffing maps.
In this study, the improved short memory principle method is used to study the dynamic characteristics of the fractional Duffing system. Firstly, the GrünwaldLetnikov definition of fractional calculus is adopted. Secondly, the classical short memory principle is improved, and the numerical solution of the system is obtained. Lastly, the influence of excitation frequency 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 behavior of the fractional Duffing system when the external excitation amplitude and fractional order change are investigated.
2. Fractional duffing system
2.1. Fractional differential definition
The three most frequently used definitions for fractional derivatives are GrünwaldLetnikov, RiemmanLiouville and Caputo definitions. The GrünwaldLetnikov definition is used in this work, as it is expressed as:
where $\alpha $ is the fractional order, $h$ is the time step and [∙] means rounding. If the time step $h$ is small enough, the abovementioned formula can be written as:
where ${w}_{j}$ is the coefficient of binomial ${\left(1z\right)}^{\alpha}$:
Eq. (3) is changed to the recursive form:
Its first item ${w}_{0}=1$.
2.2. Improved short memory principle method
The classical short memory principle truncates the memory time. In Eq. (2), the upper limit of summation becomes $\left[L/h\right]$, where $L$ is the fixed memory time length, and the part exceeding $L$ is discarded. The discarded part often brings errors that cannot be ignored. In particular, when fractional order $\alpha $ approaches zero, the binomial coefficient changes slowly.
The improved short memory principle method changes the truncation of the memory time in the classic short memory principle to the number of terms of the binomial coefficient. Then, the finite binomial coefficient is repeatedly applied to gradually enlarging time steps for error reduction. The expression of a time step being enlarged for the first time is:
The definition of GrünwaldLetnikov is used in this study, and the formulas are no longer marked. In Eq. (5), $N$ is the number of binomial truncated terms and $m$ is the magnification of the step length. Similarly, the expression of the second enlargement of step size is:
$+\frac{1}{{\left({m}^{2}h\right)}^{\alpha}}{\sum}_{j=\left[\frac{N}{m}\right]+1}^{\left[t/\left({m}^{2}h\right)\right]}{w}_{j}x\left(tj{m}^{2}h\right).$
In the same manner, other expressions of step size can be derived.
2.3. Basis for step enlargement
Step enlargement is based on the initial time step. After the step size is enlarged, although the interval for taking the function value increases, the binomial coefficient of the function value also increases, and the magnification is almost the same as the step size magnification. For example, Eq. (5) can be rewritten as:
The second term at the right end of the formula above is the result of the step size being enlarged for the first time. The binomial coefficient before the function value $x\left(tjmh\right)$ is ${w}_{j}/{m}^{\alpha}$. If the step size is not enlarged, the binomial coefficient in front of the function value shall be ${w}_{jm}$. Table 1 shows the ratio of the corresponding binomial coefficients when $N=$ 100, $\alpha =$ 0.5 and $m$ are 5, 10 and 20, respectively. $A$ is the ratio of binomial coefficient ${w}_{j}/{m}^{\alpha}$ after the first step is enlarged to binomial coefficient ${w}_{jm}$. $B$ is the ratio of binomial coefficient ${w}_{j}/{m}^{2\alpha}$ after the second step is enlarged to binomial coefficient ${w}_{j{m}^{2}}$. The ratio is almost equal to step magnification $m$ and ${m}^{2}$, and the ratio increases with the increase in step magnification. After this increase, the first binomial coefficient increases; in particular, when $n=$20, it is larger than the number of initial steps in the enlarged step. A large magnification causes a large error. In addition, whether the adopted function value is representative in the interval, that is, whether the adopted function value is close to the average value of the function value in the interval, which is also a question to be considered. The error in the longterm calculation is limited because expanding the step size to derive the function value is a dynamic ergodic process. In summary, it is required to try to intercept as many binomial coefficients as possible if the calculation conditions permit, and a small step size magnification shall be selected to obtain a highly accurate numerical solution.
Table 1Partial results of wj/mαwjm and wj/m2αwjm2 when α= 0.5
$m=\text{5}$  $m=\text{10}$  $m=\text{20}$  
$j$  $\frac{{w}_{j}/{m}^{\alpha}}{{w}_{jm}}$  $\frac{{w}_{j}/{m}^{2\alpha}}{{w}_{j{m}^{2}}}$  $j$  $\frac{{w}_{j}/{m}^{\alpha}}{{w}_{jm}}$  $\frac{{w}_{j}/{m}^{2\alpha}}{{w}_{j{m}^{2}}}$  $j$  $\frac{{w}_{j}/{m}^{\alpha}}{{w}_{jm}}$  $\frac{{w}_{j}/{m}^{2\alpha}}{{w}_{j{m}^{2}}}$ 
6  21.302  427.31  
11  10.3225  103.543  11  20.6804  414.279  
21  5.07335  25.4396  21  10.1649  101.813  21  20.348  407.306 
31  5.04926  25.2953  31  10.1108  101.218  31  20.2338  404.909 
41  5.03708  25.2224  41  10.0834  100.917  41  20.176  403.696 
51  5.02973  25.1783  51  10.0669  100.735  51  20.1412  402.964 
61  5.02481  25.1488  61  10.0558  100.614  61  20.1178  402.474 
71  5.02129  25.1277  71  10.0479  100.527  71  20.1011  402.123 
81  5.01865  25.1118  81  10.0419  100.461  81  20.0885  401.859 
91  5.01658  25.0995  91  10.0373  100.41  91  20.0788  401.654 
100  5.01508  25.0905  100  10.0339  100.373  100  20.0716  401.504 
2.4. Fractional duffing system
The form of the classic integerorder Duffing system is:
where $D$ is the firstorder differential operator, ${D}^{2}$ is the secondorder differential operator, $m$ is the mass, $c$ is the damping coefficient, $k$ is the linear stiffness coefficient, $a$ is the nonlinear stiffness coefficient and $f$ and $\omega $ are the amplitude and frequency of external excitation, respectively. If the damping term is changed to the fractional order, it becomes a fractionalorder Duffing system:
where $\alpha $ is the fractional order of the damping of the fractional Duffing system. In this study, the improved short memory principle is used to solve the fractional Duffing equation.
3. Nonlinear dynamic analysis of fractional duffing system
In this section, several fixed parameters in Eq. (9) are taken as $m=$ 1, $c=$ 0.9, $k=$ –1 and $a=$ 1. The initial conditions are $x\left(0\right)=0$ and $\dot{x}\left(0\right)=0$.
3.1. Fractional order duffing system with $\mathit{\alpha}=1.2$ and $\mathit{f}=0.6$
Fractional order $\alpha =$ 1.2 and excitation amplitude $f=$ 0.6 are adopted to analyze the influence of excitation frequency on the system. The calculation parameters are as follows: the basic step size is $h=$ 0.001, the number of truncated items is $Nt=$ 10,000, and the step length magnification is $m=$ 5. Excitation frequency $\omega $ is changed from 0.43 to 1.23 in the experiment to analyze the influence of excitation frequency on the fractional Duffing system, and the change step is 0.001. A total of 250 cycles is calculated for each frequency point, and the first 150 cycles are discarded. The bifurcation diagram of the system obtained with $\omega $ as the control parameter is shown in Fig. 1. The figure indicates that excitation frequency $\omega $ has a relatively large impact on the dynamic behavior of the system. Three chaotic and four periodic motion regions exist during the change of $\omega $ from 0.43 to 1.22.
The system performance in response to stable period1 motion occurs at $0.43<\omega \le 0.487$. The phase portrait and Poincaré map when $\omega =$ 0.45 are shown in Figs. 2(a1) and 2(a2), respectively. The phase portrait is a universal limit cycle; correspondingly, the Poincaré map exhibits an isolated point. Notably, $0.488<\omega \le 0.518$ is the first chaotic motion region of the system.
Fig. 1Bifurcation diagram of fractionalorder Duffing system with α= 1.2, 0.43<ω≤1.23
Fig. 2Phase portraits and Poincaré maps of fractional Duffing system. a1) Phase portrait with ω=0.45, a2) Poincaré map with ω=0.45, b1) phase portrait with ω=0.509, b2) Poincaré map with ω=0.509, c1) phase portrait with ω=0.55, c2) Poincaré map with ω=0.55, d1) phase portrait with ω=0.6, d2) Poincaré map with ω=0.6, e1) phase portrait with ω=0.8, e2) Poincaré map with ω=0.8, f1) phase portrait with ω= 0.88, f2) Poincaré map with ω=0.88, g1) phase portrait with ω=1.02 and g2) Poincaré map with ω=1.02
a1)
a2)
b1)
b2)
c1)
c2)
d1)
d2)
e1)
e2)
f1)
f2)
g1)
g2)
The phase portrait and Poincaré map when $\omega =\text{0.509}$ are shown in Figs. 2(b1) and 2(b2), respectively. On the Poincaré map, many discrete points are concentrated in two areas. With the increase in excitation frequency $\omega $, the system enters the period3 motion region, and its frequency range is $0.519\le \omega \le 0.577$. The phase portrait and Poincaré map when $\omega =$ 0.55 are shown in Figs. 2(c1) and 2(c2), respectively. The Poincaré map appears as three isolated points, and $0.578\le \omega \le 0.657$ is the second chaotic region of the system. The phase portrait when $\omega =$ 0.6 is shown as Fig. 2(d1), and scattered points can be seen on the corresponding Poincaré map in Fig. 2(d2). When $\omega \ge 0.658$, the system enters a stable period5 motion region, and its interval is $0.658\le \omega \le 0.848$. The phase portrait and Poincaré map when $\omega =$ 0.8 are shown in Figs. 2(e1) and 2(e2), respectively. The Poincaré map has five scattered points, and $0.849\le \omega \le 0.995$ is the third chaotic motion region. Fig. 2(f1) shows the phase portrait when $\omega =$ 0.88, and the corresponding Poincaré map is given in Fig. 2(f2). As $\omega $ further increases, the system enters periodic motion from chaotic motion. Fig. 1 indicates that when $\omega \ge $ 1.17, the system moves stably in period 1. A perioddoubling bifurcation transition region exists between the periodic motion region and the third chaotic motion region. The phase portrait and Poincaré map of the transition region when $\omega =$ 1.02 are shown in Figs. 2(g1) and 2(g2), respectively.
Notably, periodic motion windows are present in the three chaotic motion regions. The bifurcation diagram of the periodic motion window in the first chaotic motion region is shown in Fig. 3(a). The chaotic motion transits from perioddoubling motion to period2 motion. The phase portrait and Poincaré map when $\omega =$ 0.506 in the period window are shown in Figs. 3(b) and 3(c), respectively. The bifurcation diagram of the periodic motion window in the second chaotic motion region is shown in Fig. 4(a). In the period window, $\omega =$ 0.635 is period4 motion, and its phase portrait and Poincaré map are shown in Figs. 3(b) and 3(c), respectively. Fig. 5(a) presents the bifurcation diagram of the periodic window in the third chaotic region. Figs. 5(b) and 5(c) respectively show the phase portrait and corresponding Poincaré map when $\omega =$ 0.904 in the window.
Fig. 3a) Bifurcation diagram of periodic window in chaotic motion region 1, b) phase portrait when ω= 0.506 and c) Poincaré map when ω= 0.506
a)
b)
c)
Fig. 4a) Bifurcation diagram of periodic window in chaotic motion region 2, b) phase portrait when ω= 0.635 and c) Poincaré map when ω= 0.635
a)
b)
c)
Fig. 5a) Bifurcation diagram of periodic window in chaotic motion region 3, b) phase portrait when ω= 0.904 and c) Poincaré map when ω= 0.904
a)
b)
c)
Notably, an interesting change occurs in the periodic window of the second chaotic motion region. Although the system moves in period 4 of the window, the bifurcation diagram has eight dotted lines instead of four solid lines. Although the system exhibits period4 motion in the period window, the bifurcation diagram has eight dotted lines instead of four solid lines. This observation shows that the motion of the system jumps with the change in frequency in the period window, but it is still period4 motion. For example, the phase portraits when $\omega =$ 0.6352 and $\omega =$ 0.6353 are shown in Figs. 6(a) and 6(b), respectively. The phase portraits of two adjacent frequency points are rotated by exactly 180 degrees. When the frequency is further refined, this jumping property remains. Figs. 6(c) and 6(d) present phase portraits with $\omega $ of 0.63280 and 0.63281, respectively.
Fig. 6a) Phase portrait with ω= 0.6352, b) phase portrait with ω= 0.6353, c) phase portrait with ω= 0.63280 and d) phase portrait with ω= 0.63281
a)
b)
c)
d)
3.2. Fractional order duffing system with fractional order $\mathit{\alpha}$ or excitation amplitude $\mathit{f}$ fluctuations
This paper also examines the dynamic characteristics of the system when the fractional order changes. The excitation amplitude is $f=$ 1, the excitation frequency is $\omega =$ 1, the basic step length is $h=$ 0.001, the basic truncation term is $Nt=$ 10,000, and the step length magnification is $m=$ 5. its memory weakens as the fractional order increases. Thus, the number of truncated items ${N}_{t}$ is set to decrease by 1000 items with each increase in $\alpha $ by 0.2. The bifurcation diagram is shown in Fig. 7(a). In the process of changing fractional order $\alpha $ from 0.2 to 2, three periodic and three chaotic motion regions appear. The transition is from the first chaotic zone to a stable period1 motion region after perioddoubling bifurcation. As $\alpha $ continues to increase, it enters the second chaotic motion region through perioddoubling bifurcation.
Fig. 7(b) presents the bifurcation diagram of the system when excitation amplitude $f$ fluctuates from 0.1 to 2.1. The fractional order is $\alpha =$ 0.8, the excitation frequency is $\omega =$ 1, the basic step length is $h=$ 0.001, the basic truncation item number is $Nt=$ 10,000, and the step length magnification is $m=$ 5. The figure shows that three periodic and two chaotic motion regions appear during the fluctuation of excitation amplitude. The first is a stable period1 motion; the first chaotic motion region is entered through perioddoubling bifurcation, followed by entrance to the period3 motion region. A part of the period6 movement is contained in the period3 motion region, and the second chaotic motion region is entered from period3 motion. As the excitation amplitude continues to increase, the system transitions from the second chaotic region to the period1 motion region through the perioddoubling bifurcation.
Fig. 7a) Bifurcation diagram of x versus α, 0.2<α<2 and b) bifurcation diagram of x versus f, 0.1<f<2.1
a)
b)
4. Conclusions
An improved short memory principle method is introduced to solve the fractional Duffing system numerically. And based on high computational efficiency, some previously unobserved phenomena are obtained. The influence of excitation frequency on the dynamic performance of the system is studied using nonlinear dynamic analysis methods, such as phase portrait, Poincaré map and bifurcation diagram. The bifurcation diagram has four periodic and three chaotic motion regions. The third chaotic motion region exhibits an obvious transition from perioddoubling motion to stable period1 motion. In addition, periodic motion windows are found in the three chaotic motion regions. The periodic motion of the periodic window in the second chaotic motion region has a jumping property. The influence of excitation amplitude and fractional order on the dynamic system performance is also studied. The results show that a change in the system parameters of the fractional Duffing equation affects the dynamic system performance. Thus, such a change shall be considered during the dynamic system design and analysis.
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About this article
This work was supported by the National Natural Science Foundation of China [Grant No. 11472133].