# Numerical scheme and dynamic analysis for variable-order fractional van der Pol model of nonlinear economic cycle

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## Abstract

Considering the fact that the memory in economic series changes with dynamic economic environment, this paper is devoted to the proposal of a variable-order fractional van der Pol model (VOFVDPM), where the order of the derivative is replaced by a time-dependent function. A numeric scheme for this model is designed by the Adams-Bashforth-Moulton method. The dynamic behaviors of the VOFVDPM with linear and periodic variable-order functions are investigated through numerical experiment. Some dynamic characteristics of the VOFVDPM that do not exist in a fractional order van der Pol model are discovered in the numerical simulation, such as existing limit point when the linear order functions have the same ranges and opposite slopes.

### Keywords

van der Pol model variable-order fractional derivative dynamic behavior### MSC

26A33 34A08## 1 Introduction

As a classical nonlinear dynamic model, the van der Pol model has attracted scholars’ attention from many fields since it was proposed by the Dutch electrical engineer and physicist Balthasar van der Pol. It is extensively applied in neurology, physics, sociology, and economics. Based on Kaldor’s model of business cycles, the van der Pol model of nonlinear business cycles is proposed by postulating symmetric shapes of the investment and savings functions [1]. Assuming that the investment outlay is periodic and continuous function of time, the forced van der Pol model of nonlinear economic cycles is presented in [2], which is the generalization of Goodwin’s nonlinear accelerator-multiplier model. When the investment outlay is neglected, the forced van der Pol model of nonlinear economic cycles is the van der Pol model of nonlinear business cycles. In addition to Kaldor’s model and Goodwin’s nonlinear accelerator-multiplier model, the van der Pol model has been applied in other relevant economical problems; see [3] for more details.

The dynamic of the van der Pol model of nonlinear economic cycles is extensively studied by researchers. The bifurcation diagrams of the forced van der Pol model on varying the driver’s frequency and amplitude are analyzed in [4]. It points out the coexistence of asymmetric attractors ascribed to the system symmetry of the van der Pol oscillator, and it finds that periodic, quasiperiodic, and chaotic attractors coexist. In [5], the dynamic behaviors are investigated by considering the change of control parameters, especially the pattern of bifurcation with damping parameter changing. The number of coexisting attractors in overlaps of mode-locking subzones is an important dynamic behavior, and it is analyzed in [6]. For the dynamics structure of the forced van der Pol model of nonlinear economic cycles it is found that the chaotic attractor is composed of chaotic saddles and unstable periodic orbits located in the gap regions of chaotic saddles for the model [7]. For existing chaos in the forced van der Pol model with certain parameters, the chaos control of chaotic unstable limit cycles is discussed in [8].

Inspired by the development of fractional calculus, the fractional order derivatives are applied in the forced van der Pol model of nonlinear economic cycles, for which fractional order derivatives can depict a ‘memory principle’ in economic dynamic. Chaos in a fractional order modified van der Pol system is studied by [9], where chaos exists in the fractional order system with the order both less than and higher than the number of the states of the integer order generalized van der Pol system. And chaos is also discovered in non-autonomous and autonomous generalized van der Pol systems excited by a sinusoidal time function with fractional orders [10]. In [11], one presents a periodically excited van der Pol system with fractional damping, and one finds that the response of the system is very sensitive to changes in the order of fractional damping. Several other papers dealing with van der Pol oscillators could be found in the literature (see [12, 13, 14, 15, 16, 17, 18] and the references therein).

Compared to the usual forced van der Pol model of nonlinear economic cycles, the contribution of the fractional forced van der Pol model of nonlinear economic cycles is that it can describe memory in economic dynamics. However, the characteristic of this memory in economic time series will change with time, for which it is affected by the macroeconomic environment and government policy at different times. Therefore, this paper will propose a variable-order fractional forced van der Pol model of nonlinear economic cycles, which can describe the fact that memory in economic series is varying with time. The orders in the fractional forced van der Pol model of nonlinear economic cycles are replaced by positive bounded continuous functions. In order to obtain an effective and applicable numerical technique for solving the variable-order fractional van der Pol model, the Adams-Bashforth-Moulton method is adopted. Then we employ the technique to obtain the numerical results for the dynamics characteristics of the variable-order fractional van der Pol model with different parameters.

The remainder of this article is organized as follows: In Section 2, we review the mathematical preliminaries as regards the variable-order fractional derivative. In Section 3, we propose a variable-order fractional forced van der Pol model of nonlinear economic cycles. In Section 4, we present the numerical technique for solving the variable-order fractional forced van der Pol model by the Adams-Bashforth-Moulton method. In Section 5, numerical experiments are performed to analyze the dynamic characteristics. The conclusions are drawn in Section 6.

## 2 Variable-order fractional derivative

In this section, we introduce some preliminaries of variable-order fractional derivative. The variable-order fractional derivative is defined by replacing the order of the fractional derivatives with a continuous bounded function in the counterparts [19, 20].There exist three most frequently used definitions for the general fractional differ-integral, which are the Grünwald-Letnikov (GL) definition, the Riemann-Liouville (RL), and the Caputo definitions. According to these definitions, their corresponding variable-order fractional derivative can be defined as follows.

### Definition 1

([19])

### Definition 2

### Definition 3

### Remark 1

For the above definitions of variable-order fractional derivative, when \(\alpha(t)=q>0\), (1), (2), (3) describe the fractional derivative. If \(\alpha(t)=m-1\), (1), (2), (3) describe the classical (\(m-1\))th order derivative. If \(\alpha(t)=0\), \(D_{GL,RL,C}^{\alpha(t)}f(t)=f(t)\).

The main advantage of the Caputo derivative is that the initial conditions for the fractional differential equations are of the same form as that of the integer order differential equations [21]. Therefore, we study the VOFVDPM with the Caputo derivative in this paper.

## 3 Variable-order fractional van der Pol model of nonlinear business cycles

*x*is economic variable, such as national income;

*μ*(>0) is the coefficient on nonlinear damping term and determines the speed of adjustment;

*a*is the amplitude of the exogenous force, and

*ω*is the frequency of the exogenous force; \(a\sin(\omega t)\) is called the force function, and it can be explained as the amount of autonomous investment outlays at

*t*[2]. When \(a=0\), (4) is the unforced van der Pol model of nonlinear business cycles.

### Remark 2

If \(\alpha(t)=\alpha\), \(\beta(t)=\beta\), (5) is a forced variable-order fractional van der Pol model of nonlinear business cycles. If \(\alpha(t)=1\), and \(\beta(t)=1\), the system (5) is a forced van der Pol model of nonlinear business cycles.

### Remark 3

If \(a=0\), (5) is an unforced variable-order fractional van der Pol model of nonlinear business cycles.

For the VOFVDPM, *x* is the macroeconomic variable of a certain economy, such as national income, GDP, and so on. \(a\sin(\omega t)\) is for the periodical investment outlays, which is a controllable variable and has impacts on the macroeconomic variable *x*. The variable-order fractional derivative of VOFVDPM reflects the varying memory in the macroeconomic series *x*. So the order functions \(q_{1}(t),q_{2}(t)\) depict the evolution path of the memory in the macroeconomic series *x*. Therefore, VOFVDPM with a certain variable-order fractional derivative not only depicts the complex dynamics of national income, but also reflects the varying memory in the economical series of the national income. The application of a variable-order fractional derivative in the van der Pol model can be extended to other dynamic economic models.

## 4 Numerical scheme

The Adams-Bashforth-Moulton method is a type of predictor-corrector methods and a relatively new approach to provide a numerical approximation to solve the fractional order differential equations. It is studied and discussed thoroughly in [22, 23]. In this paper, we adapt the Adams-Bashforth-Moulton method to solve the variable-order fractional van der Pol model (VOFVDPM).

## 5 Numerical results

The above numerical experiments investigate the dynamic behaviors of the VOFVDPM with different linear and periodical functions in the case \(a=0\). It shows that some novel dynamic characteristics of the model have been found in the numerical results, such as the existence of a limit point \((0,0)\) when linear \(q_{1},q_{2}\) are in the same ranges and have opposite slopes. These novel dynamic characteristics indicate that the limit cycle is not the only characteristic of the FVDP model when the order of the derivative is a time-dependent function.

## 6 Conclusion

In this paper, we successfully extend the forced van der Pol model with nonlinear economic cycle to the VOFVDPM, which considers the memory in an economic series varying with time. The numeric scheme based on the Adams-Bashforth-Moulton method for the VOFVDPM has been designed. By the scheme, the dynamic behaviors of the model with linear or periodic variable-order function have been analyzed. The results of the analysis suggest that the VOFVDPM presents some novel dynamic characteristics, which cannot be found in an integral and constant fractional van der Pol model.

## Notes

### Acknowledgements

The project supported by National Natural Science Foundation of China (71501070) and Education Department foundation of Hunan Province (15B150).

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