0-1 Test for Chaos in a Fractional Order Financial System with Investment Incentive

Abstract

A new integer-order chaotic financial system is extended by introducing a simple investment incentive into a three-dimensional chaotic financial system. A four-dimensional fractional-order chaotic financial system is presented by bringing fractional calculus into the new integer-order financial system. By using weighted integral thought, the fractional order derivative's economics meaning is given. The 0-1 test algorithm and the improved Adams-Bashforth-Moulton predictor-corrector scheme are employed to detect numerically the chaos in the proposed fractional order financial system.

1. Introduction

In recent years there has been a high level of interest in the study of chaotic economic systems [1, 2]. Chaotic systems refer to nonlinear dynamical systems which are very sensitive to initial conditions, in a such way that a small perturbation of these could have unpredictable consequences on the evolution equations. As a complex system, the financial system shows a lot of nonlinear dynamical phenomena, such as chaos, fractals, and bifurcation. Reference [3] firstly presented a nonlinear financial model as follows: ̇̇𝑥=𝑧+𝑦−𝑎𝑥,𝑦=1−𝑏𝑦−𝑥²,𝑧̇=−𝑥−𝑐𝑧, (1) where 𝑥 denotes the interest rate, 𝑦 denotes the investment demand, 𝑧 denotes the price index, 𝑎 is the saving amount, 𝑏 is the cost per investment, 𝑐 is the demand elasticity of commercial markets, and all three constants 𝑎,𝑏,𝑐≥0.

References [4–7] analyzed some complex behavior of system (1), such as bifurcation and chaos. Reference [8] introduced fractional calculus into system (1) and studied its complex dynamics [9]. References [10, 11] studied chaos control of the fractional-order form of system (1). Reference [12] proposed an uncertain fractal-order form of system (1) and studied its chaos control via adaptive sliding mode. Reference [13] proposed an uncertain and stochastic form of system (1) with multiple delays. Reference [14] proposed a discrete form of system (1) and studied its Neimark-Sacker bifurcation. References [15, 16] presented a form of system (1) with time-delayed feedback and studied its dynamics and control. Reference [17] studied Hopf-pitchfork bifurcation and periodic phenomena in the system (1) with time delays. Reference [18] extended system (1) with the average profit margin.

Investment is an important economic activity [19]. In order to cope with high unemployment or backwardness, in many cases a government would use their policies to stimulate investment which will raise employment, exports, tax revenue, and so on. These incentives may take the form of investment grants or investment credit that reduces capital costs for investors. But these various forms of investment incentives also will arouse the investment demand of the financial system (1). In other words, there may be a positive feedback mechanism between investment command and investment incentive. Thus we can describe the above mechanism by using the following four-dimensional financial system with a simple investment incentive: ̇̇𝑥=𝑧+𝑦−𝑎𝑥,𝑦=1−𝑏𝑦−𝑥²̇̇+𝑢,𝑧=−𝑥−𝑐𝑧,𝑢=𝑑𝑦, (2) where 𝑥, 𝑦, 𝑧, 𝑎, 𝑏, and 𝑐 have the same meanings as those defined in the system (1), 𝑢 denotes the investment incentive, and 𝑑≥0 is the coefficient of investment incentive intensity.

Researchers’ ever-growing interests in fractional calculus have led the ever-broadening applications in a lot of fields [20–25]. When fractional calculus is introduced into the system (2), its fractional-order form can be described with fractional derivative as follows: 𝑑^𝑞1𝑥𝑑𝑡^𝑞1𝑑=𝑧+𝑦−𝑎𝑥,_𝑞2𝑦𝑑𝑡^𝑞2=1−𝑏𝑦−𝑥²𝑑+𝑢,_𝑞3𝑧𝑑𝑡^𝑞3𝑑=−𝑥−𝑐𝑧,_𝑞4𝑢𝑑𝑡^𝑞4=𝑑𝑦, (3) in which 𝑞=(𝑞₁,𝑞₂,𝑞₃,𝑞₄) is subject to 0<𝑞₁,𝑞₂,𝑞₃,𝑞₄<1.

Remark 1. When 𝑞=(1,1,1,1), the system (3) degenerates into the system (2).

The remainder of this paper is organized as follows. In Section 2, we give some preliminaries, such as Caputo fractional derivative’s economics meaning, numerical solutions of the system (3), and the 0-1 test algorithm. In Section 3, we numerically illustrate the chaos existence of the proposed financial system by using the improved Adams-Bashforth-Moulton predictor-corrector scheme and 0-1 test algorithm. Finally conclusions in Section 4 close the paper.

2. Preliminaries

2.1. Caputo Fractional Derivative and Its Economics Meaning

2.1.1. Weighted Integral

The integer integral and derivative are very helpful to understand fractional integral and derivative. An integral can be thought of as an area or a generalization of area.

Definition 2. Given a weighted factor 𝑓(𝑥) and a function 𝑔(𝑥) for 𝑎≤𝑥≤𝑏, the definite weighted integral ∫^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)𝑑𝑥 is a real number whose geometrical interpretation is the signed area under the graph of weighted function 𝑦=𝑓(𝑥)𝑔(𝑥) for 𝑎≤𝑥≤𝑏.

The Riemann integral is a simple and well-understood integral definition. If the continuous function 𝑔(𝑥) is defined on interval [𝑎,𝑏] and a partition 𝑃 of [𝑎,𝑏] can be divided into a collection of subintervals [𝑥₀,𝑥₁],[𝑥₁,𝑥₂],…,[𝑥_𝑛−1,𝑥_𝑛], for each 𝑖=1,2,…,𝑛, a point 𝑥^∗_𝑖 in [𝑥_𝑛−1,𝑥_𝑛], and its weighted factor 𝑓(𝑥^∗_𝑖), weighted Riemann integral ∫^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)𝑑𝑥 can be rewritten as ^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)𝑑𝑥=lim^𝑛_{𝑛→∞𝑖=1}𝑓𝑥^∗_𝑖𝑔𝑥^∗_𝑖Δ𝑥_𝑖 (4) for any choice of the 𝑥^∗_𝑖 in [𝑥_𝑛−1,𝑥_𝑛] with Δ𝑥=(𝑏−𝑎)/𝑛 and 𝑥_𝑖=𝑎+𝑖Δ𝑥.

When 𝑓(𝑥)=1, the weighted integral ∫^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)𝑑𝑥=∫^𝑏_𝑎𝑔(𝑥)𝑑𝑥 completely degenerates into a common integral.

2.1.2. Caputo Fractional Derivative

Loosely speaking, a derivative can be interpreted as a kind of change ratio. More specifically it is the measure of infinitesimal change ratio of the output over the infinitesimal change of the input. For example, consider the following.

Definition 3. The first order derivative of function 𝑓(𝑡) with respect to 𝑡 is given by 𝑓^′(𝑡)=lim_ℎ→0𝑓(𝑡+ℎ)−𝑓(𝑡)ℎ. (5)

Definition 4. The 2nd order derivative of function 𝑓(𝑡) with respect to 𝑡 is given by 𝑓^′′(𝑡)=lim_ℎ→0𝑓^′(𝑡+ℎ)−𝑓^′(𝑡)ℎ. (6)

Definition 5. The 𝑚th order derivative of function 𝑓(𝑡) with respect to 𝑡 is given by 𝑓^(𝑚)(𝑡)=lim_ℎ→0𝑓^(𝑚−1)(𝑡+ℎ)−𝑓^(𝑚−1)(𝑡)ℎ=lim_ℎ→01ℎ^𝑚𝑚_𝑛=0(−1)^𝑛𝑚𝑛𝑓(𝑡−𝑛ℎ). (7)

And there are several definitions of fractional derivatives, roughly speaking, which can be thought of as generalized above derivatives. The Caputo definition is a common notation of fractional derivative. Briefly the Caputo fractional derivative is a kind of weighted change ratio in a time interval. More specifically, the Caputo fractional derivative can be considered as a kind of weighted integral of the integer order derivative of a function on an interval. In this paper, we will adopt the following Caputo definition.

Definition 6. The 𝑞th order fractional derivative of function 𝑓(𝑡) with respect to 𝑡 is given by 𝐷^𝑞_𝑎𝑓(𝑡)=^𝑡_𝑎𝐾_𝑞(𝑡−𝜏)𝑓^(𝑚)(𝜏)𝑑𝜏, (8) where 𝐾_𝑞(𝑡−𝜏)=(𝑡−𝜏)^{𝑚−𝑞−1}/Γ(𝑚−𝑞), Γ(𝑚−𝑞)=∫^∞₀𝜏^{𝑚−𝑞−1}𝑒^−𝜏𝑑𝜏, 𝑡∈[𝑎,𝑏], 𝑚 is an integer and 𝑚−1<𝑞≤𝑚.

The functions 𝑓^(𝑚)(𝑡) in Definitions 5 and 6 are the same. When 𝐾_𝑞(𝑡−𝜏) is regarded as a kind of weight, then 𝐷^𝑞_𝑎𝑓(𝑡) can be regarded as a kind of weighted integral with respect to 𝑓^(𝑚)(𝑡). What is more, 𝑓^(𝑚)(𝑡) can be regarded as 𝑚th order change ratio of the function 𝑓(𝑡) at 𝑡; that is, there is no memory effect in 𝑓^(𝑚)(𝑡). However, 𝐷^𝑞_𝑎𝑓(𝑡) can be regarded as a weighted integral of the 𝑚th order derivative of the function 𝑓(𝑡) on the whole interval [𝑎,𝑏], and the weight 𝐾_𝑞(𝑡−𝜏) depends on the specific time; that is, there are memory effects in 𝐷^𝑞_𝑎𝑓(𝑡).

2.1.3. Caputo Fractional Derivative’s Economics Meaning

In economics, let me illustrate a concrete example like the system (3), for the investment demand function 𝑦(𝑡), its integer order derivative 𝑦^(𝑚)(𝑡), and its fractional order derivative 𝐷^𝑞₀𝑦(𝑡), 𝑦^(𝑚)(𝑡) can be regarded as 𝑚th order change ratio of the investment demand 𝑦(𝑡) at the time 𝑡; that is, it is unrelated to any time before the time 𝑡; that is, there is no memory effect in 𝑦^(𝑚)(𝑡). However, 𝐷^𝑞₀𝑦(𝑡) can be regarded as a weighted integral of all 𝑚th order derivative of the investment demand 𝑦(𝑡) on the whole time interval [0,𝑡]; that is, 𝐷^𝑞₀𝑦(𝑡) does not only depend on the time 𝑡, but also depends on any time before the time 𝑡; that is, there are memory effects in 𝐷^𝑞₀𝑦(𝑡). In the system (3), 𝑚=1 and its memory effects are easier to be understood, so we will not repeat them here. In fact, a lot of economics variables have memory effects; for example, our past economic behavior may affect our present and future ones. In other research fields, there are also a lot of memory effects, such as psychological shadow, escalation of commitment, sunk cost effect, lazy battery effect, and shape memory effect. As mentioned above, there are some memory effects in financial systems, so the fractional derivative may be more suitable to study them.

2.2. Numerical Solutions of Fractional Order System

Based on the improved Adams-Bashforth-Moulton predictor-corrector scheme [26–29], one can solve fractional-order ordinary differential equations. The numerical calculation formula of fractional-order system (3) can be described as follows.

With the initial value (𝑥₀^(𝑘),𝑦₀^(𝑘),𝑧₀^(𝑘),𝑢₀^(𝑘)), 𝑘=0,1,…,[𝑚]−1, and the fractional-order system (3) is equivalent to the Volterra integral equations as follows: 𝑥(𝑡)=^[𝑚]−1_𝑘=0𝑥₀^(𝑘)𝑡^𝑘+1𝑘!Γ𝑞₁×^𝑡₀(𝑡−𝜏)^𝑞1−1𝑧(𝜏)+𝑦(𝜏)𝑥(𝜏)𝑦(𝑡)=−𝑎𝑑𝜏,_{[𝑚]−1𝑘=0}𝑦₀^(𝑘)𝑡^𝑘+1𝑘!Γ𝑞₂×^𝑡₀(𝑡−𝜏)^𝑞2−11−𝑏𝑦(𝜏)−𝑥²(𝜏)(𝜏𝑧(𝑡)=+𝑢)𝑑𝜏,_{[𝑚]−1𝑘=0}𝑧₀^(𝑘)𝑡^𝑘+1𝑘!Γ𝑞₃×^𝑡₀(𝑡−𝜏)^𝑞3−1((𝜏)(𝜏𝑢(𝑡)=−𝑥−𝑐𝑧))𝑑𝜏,_{[𝑚]−1𝑘=0}𝑢₀^(𝑘)𝑡^𝑘+1𝑘!Γ𝑞₄×^𝑡₀(𝑡−𝜏)^𝑞4−1𝑑𝑦(𝜏)𝑑𝜏. (9)

Let ℎ=𝑇/𝑁, 𝑡_𝑛=𝑛ℎ, 𝑛=0,1,…,𝑁∈𝑍⁺, and the fractional-order system (3) can be approximated to the following difference equations: 𝑥_𝑛+1=𝑥₀+ℎ^𝑞1Γ𝑞₁×𝑧+2_𝑝𝑛+1+𝑦^𝑝_𝑛+1−𝑎𝑥^𝑝_𝑛+1+ℎ^𝑞1Γ𝑞₁×+2_𝑛𝑗=0𝛼_{1,𝑗,𝑛+1}𝑧_𝑗+𝑦_𝑗−𝑎𝑥_𝑗,𝑦_𝑛+1=𝑦₀+ℎ^𝑞2Γ𝑞₂×+21−𝑏𝑦^𝑝_𝑛+1−𝑥^𝑝_𝑛+1²+𝑢^𝑝_𝑛+1+ℎ^𝑞2Γ𝑞₂×+2_𝑛𝑗=0𝛼_{2,𝑗,𝑛+1}1−𝑏𝑦_𝑗−𝑥_𝑗²+𝑢_𝑗,𝑧_𝑛+1=𝑧₀+ℎ^𝑞3Γ𝑞₃×+2−𝑥^𝑝_𝑛+1−𝑐𝑧^𝑝_𝑛+1+ℎ^𝑞3Γ𝑞₃×+2_𝑛𝑗=0𝛼_{3,𝑗,𝑛+1}−𝑥_𝑗−𝑐𝑧_𝑗,𝑢_𝑛+1=𝑢₀+ℎ^𝑞4Γ𝑞₄+2𝑑𝑦^𝑝_𝑛+1+ℎ^𝑞4Γ𝑞₄×+2_𝑛𝑗=0𝛼_{4,𝑗,𝑛+1}𝑑𝑦_𝑗, (10) where 𝑥^𝑝_𝑛+1=𝑥₀+1Γ𝑞₁×^𝑛_𝑗=0𝛽_{1,𝑗,𝑛+1}𝑧_𝑗+𝑦_𝑗−𝑎𝑥_𝑗,𝑦^𝑝_𝑛+1=𝑦₀+1Γ𝑞₂×^𝑛_𝑗=0𝛽_{2,𝑗,𝑛+1}1−𝑏𝑦_𝑗−𝑥_𝑗²+𝑢_𝑗,𝑧^𝑝_𝑛+1=𝑧₀+1Γ𝑞₃^𝑛_𝑗=0𝛽_{3,𝑗,𝑛+1}−𝑥_𝑗−𝑐𝑧_𝑗,𝑢^𝑝_𝑛+1=𝑢₀+1Γ𝑞₄^𝑛_𝑗=0𝛽_{4,𝑗,𝑛+1}𝑑𝑦_𝑗,𝛼_{𝑖,𝑗,𝑛+1}=𝑛^𝑞𝑖+1−𝑛−𝑞_𝑖(𝑛+1)^𝑞𝑖,𝑗=0,𝑛−𝑗+2^𝑞𝑖+1+𝑛−𝑗^𝑞𝑖+1−2𝑛−𝑗+1^𝑞𝑖+1𝛽,1≤𝑗≤𝑛,𝑖=1,2,3,4,1,𝑗=𝑛+1,_{𝑖,𝑗,𝑛+1}=ℎ^𝑞𝑖𝑞_𝑖𝑛−𝑗+1^𝑞𝑖−𝑛−𝑗^𝑞𝑖,0≤𝑗≤𝑛,𝑖=1,2,3,4. (11) Errors of the above method are Δ𝑥=max_{𝑗=0,1,…,𝑁}𝑥𝑡_𝑗−𝑥_ℎ𝑡_𝑗=𝒪ℎ^𝑝1,Δ𝑦=max_{𝑗=0,1,…,𝑁}𝑦𝑡_𝑗−𝑦_ℎ𝑡_𝑗=𝒪ℎ^𝑝2,Δ𝑧=max_{𝑗=0,1,…,𝑁}𝑧𝑡_𝑗−𝑧_ℎ𝑡_𝑗=𝒪ℎ^𝑝3,Δ𝑢=max_{𝑗=0,1,…,𝑁}𝑢𝑡_𝑗−𝑢_ℎ𝑡_𝑗=𝒪ℎ^𝑝4, (12) where 𝑝_𝑖=min(2,1+𝑞_𝑖).

2.3. The 0-1 Test Algorithm

There are many approaches to detect chaos, but 0-1 test is one of the simplest and most effective ones, which was proposed by Gottwald and Melbourne [30] and also has already been successfully tested for various discrete or continuous systems, such as [31–45].

In this section, we introduce the detailed steps of the 0-1 test algorithm to robustly distinguish regular from chaotic dynamics in deterministic continuous dynamical systems. In fact, one can directly implement the 0-1 test algorithm if the given time series is generated from a discrete time system. But if the given time series is generated from a continuous time system, there may be a well-known oversampling issue that must overcome it. So the oversampling issue should be considered when we distinguish regular from chaotic dynamic system (3).

Consider a set of measurement data 𝜙₀(𝑖) obtained from a continuous time system, where 𝑖=1,2,…,𝑇, and 𝑇 is the amount of the data, the 0-1 test algorithm can be described as follows.

Step 1. Rearrange the initial measurement data with an optimal sampling time.
The average mutual information [46] between 𝜙₀(𝑖) and 𝜙₀(𝑖+𝜏) is defined as 𝐼(𝜏)=_{𝜙0(𝑖),𝜙0(𝑖+𝜏)}𝑃𝜙₀(𝑖),𝜙₀(𝑖+𝜏)×log₂𝑃𝜙₀(𝑖),𝜙₀(𝑖+𝜏)𝑃𝜙₀(𝑖)𝑃𝜙₀(),𝑖+𝜏 (13) where 𝜏 is the time delay, that is, sampling time.
The value 𝜏 corresponding to the first local minimum of the mutual information 𝐼(𝜏) is the optimal sampling time. Thus we can rearrange the initial measurement data 𝜙₀(𝑖) into the finely sampled measurement data 𝜙(𝑗) in the following way: 𝜙𝑗=𝜙₀𝑖+𝑗𝜏, (14) where  𝑖=1,2,…,𝑇, and  𝑗=1,2,3,…,𝑁.

Step 2. Choose a random number 𝑐∈(𝜋/5,4𝜋/5) and define the following new coordinates (𝑝_𝑐(𝑛),𝑠_𝑐(𝑛)): 𝑝_𝑐(𝑛)=^𝑛_𝑗=1𝜙𝑗𝜃𝑗,𝑠cos_𝑐(𝑛)=^𝑛_𝑗=1𝜙𝑗sin𝜃𝑗, (15) where 𝜃𝑗=𝑗𝑐+^𝑗_𝑖=1𝜙𝑗,𝑗=1,2,3,…,𝑛. (16)

Step 3. Define the mean square displacement 𝑀_𝑐(𝑛) as follows: 𝑀_𝑐(𝑛)=lim_𝑁→∞1𝑁×^𝑁_𝑗=1𝑝_𝑐(𝑗+𝑛)−𝑝_𝑐(𝑗)²+𝑠_𝑐𝑗+𝑛−𝑠_𝑐𝑗²𝑁,𝑛∈1,.10 (17)

Step 4. Define the modified mean square displacement 𝐷_𝑐(𝑛) as follows: 𝐷_𝑐(𝑛)=𝑀_𝑐(𝑛)−lim_𝑁→∞1𝑁^𝑁_𝑗=1𝜙𝑗²1−cos𝑛𝑐.1−cos𝑐 (18)

Step 5. Define the median value of correlation coefficient 𝐾 as follows: 𝐾=median𝐾_𝑐, (19) where 𝐾_𝑐=cov(𝜉,Δ)(𝜉)(Δ)∈[],varvar−1,1 (20) in which 𝜉=(1,2,3,…,𝑛_cut), Δ=(𝐷_𝑐(1),𝐷_𝑐(2),…,𝐷_𝑐(𝑛_cut)), 𝑛_cut=round(𝑁/10), and the covariance and variance are defined with vectors 𝑥,𝑦 of length 𝑞 as follows: =1cov𝑥,𝑦𝑞^𝑞_𝑗=1𝑥𝑗−𝑥𝑦𝑗−𝑦,1𝑥=𝑞^𝑞_𝑗=1𝑥𝑗,var(𝑥)=cov(𝑥,𝑥). (21)

Step 6. Interpret outputs as follows.(1)𝐾≈0 indicates the underlying dynamics is regular (i.e., periodic or quasiperiodic), whereas 𝐾≈1 indicates the underlying dynamics is chaotic.(2)Bounded trajectories in the (𝑝,𝑠) plane imply the underlying dynamics is regular (i.e., periodic or quasiperiodic), whereas Brownian-like (unbounded) trajectories imply the underlying dynamics is chaotic.

3. Some Basic Properties of the System (3)

3.1. Symmetry and Invariance

Obviously, the four-dimensional system (3) has a natural symmetry about the coordinate plane 𝑦-𝑢 since the coordinate transformation (𝑥,𝑦,𝑧,𝑢)→(−𝑥,𝑦,−𝑧,𝑢) satisfies the system is invariant for all values of the system parameters.

3.2. Dissipativity and the Existence of Attractor

From system (3), we obtain 𝜕𝑑∇𝑉=^𝑞1𝑥/𝑑𝑡^𝑞1+𝜕𝑑𝜕𝑥^𝑞2𝑦/𝑑𝑡^𝑞2+𝜕𝑑𝜕𝑦_𝑞3𝑧/𝑑𝑡^𝑞3+𝜕𝑑𝜕𝑧^𝑞4𝑢/𝑑𝑡^𝑞4(),𝜕𝑢=−𝑎+𝑏+𝑐 (22) that is, the system (3) is dissipative with an exponential contraction rate 𝑑𝑉/𝑑𝑡=𝑒^{−(𝑎+𝑏+𝑐)} when 𝑎+𝑏+𝑐>0. In other words, the volume element 𝑉(𝑡)=𝑉₀𝑒^{−(𝑎+𝑏+𝑐)𝑡} for the initial volume 𝑉₀ means the trajectories of chaotic system (3) would converge ultimately onto an attractor as 𝑡→∞ with an exponential rate (𝑎+𝑏+𝑐).

3.3. Equilibrium and Stability

Theorem 7 (see [47]). Consider a nonlinear autonomous fractional order system as follows: 𝐷^𝑞(𝑋(𝑡,𝑋(0)=𝑥𝑥=𝐹))₁(0),𝑥₂(0),…,𝑥_𝑛(0)^𝑇, (23) where 𝑋(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇, 𝐹(𝑋(𝑡))=(𝑓₁(𝑋(𝑡)),𝑓₂(𝑋(𝑡)),…,𝑓_𝑛(𝑋(𝑡)))^𝑇, 𝑞=(𝑞₁,𝑞₂,…,𝑞_𝑛)^𝑇, and 0<𝑞_𝑖<1 (𝑖=1,2,…,𝑛). If 𝐴=𝐷𝐹₍𝑋₎ is the jacobian matrix at its equilibrium 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇, then the point 𝑋 is asymptotically stable when |arg(eig(𝐴))|>𝑞_𝑚𝜋/2, where 𝑞_𝑚=max_{1≤𝑖≤𝑛}{𝑞_𝑖}.

The equilibrium of the system (3) satisfies the following equations: 𝑧+𝑦−𝑎𝑥=0,1−𝑏𝑦−𝑥²+𝑢=0,−𝑥−𝑐𝑧=0,𝑑𝑦=0. (24)

By simple computation, one can obtain that system (3) has only one equilibrium point 𝐸₀(0,0,0,−1).

Proposition 8. The equilibrium point 𝐸₀ is unstable.

Proof. For the equilibrium point 𝐸₀, the Jacobian matrix is 𝐽𝐸₀=−𝑎0100−𝑏01−10−𝑐00𝑑00. (25)
It is easy to get its eigenvalues corresponding to the equilibrium 𝐽(𝐸₀) as follows: 𝜆₁=−(1/2)𝑏+(1/2)𝑏²+4𝑑, 𝜆₂=−(1/2)𝑏−(1/2)𝑏²+4𝑑, 𝜆₃=−(1/2)(𝑐+𝑎)+(1/2)(𝑐−𝑎)²−4, and 𝜆₄=−(1/2)(𝑐+𝑎)−(1/2)(𝑐−𝑎)²−4.
Obviously 𝜆₁>0, that is, |arg(𝜆₁)|=0, thus the equilibrium point 𝐸₀ is unstable based on Theorem 7.
This proposition is proved.

4. 0-1 Test for Chaos in the System (3)

In the next two subsections, some properties of the system (3) will be, respectively, discussed by varying investment incentive intensity 𝑑 and investment demand derivative 𝑞₂. The abovementioned Adams-Bashforth-Moulton predictor-corrector scheme and 0-1 test algorithm can be employed to get the desired numerical solutions of system (3).

4.1. Varying 𝑑 and Fixing 𝑞₁=0.96, 𝑞₂=0.94, 𝑞₃=0.92,𝑞₄=0.78, 𝑎=3, 𝑏=0.4, and 𝑐=0.4

To simplify representation, one can fix 𝑞₁=0.96, 𝑞₂=0.94, 𝑞₃=0.92, 𝑞₄=0.78, 𝑎=3, 𝑏=0.4, and 𝑐=0.4 in this subsection. When the investment incentive intensity 𝑑 varies, the corresponding 𝐾 values are shown in Figure 1(a). The bifurcation diagram of interest rate 𝑥 with increasing the investment incentive intensity 𝑑 is presented in Figure 1(b). Obviously the bifurcation diagram well coincides with the 𝐾 diagram. The two abovementioned figures both show that system (3) changes gradually from chaos to stability as the investment incentive intensity 𝑑 grows larger.

(a) 0-1 test with 𝐾 versus 𝑑

(b) Bifurcation with 𝑑 versus 𝑥

Figure 1 

The evolvement of 𝐾 and 𝑥 versus 𝑑 with 𝑞1=0.96, 𝑞2=0.94, 𝑞3=0.92, 𝑞4=0.78, 𝑎=3, 𝑏=0.4, and 𝑐=0.4.

From Figure 1, the complex dynamical behavior of the system (3) can be clearly understood. When the investment incentive intensity 𝑑 is determined by its corresponding 𝐾≈1, the system (3) is chaotic. Whereas the investment incentive intensity 𝑑 is determined by its corresponding 𝐾≈0, the system (3) is regular.

Let me take two specific examples as illustrations in the system (3). When the investment incentive intensity 𝑑=0.1, its corresponding phase portraits are strange attractors as shown in Figure 2 and its trajectories in the new (𝑝,𝑠)-plane are Brownian-like as shown in Figure 3(a); that is, the system (3) is chaotic. Whereas the investment incentive intensity 𝑑=2.8, its corresponding phase portraits are quasiperiodic as shown in Figure 4 and its trajectories in the new (𝑝,𝑠) plane are bounded as shown in Figure 3(b); that is, the system (3) is regular (quasiperiodic).

(a) Attractor in 𝑥-𝑦-𝑧 space

(b) 𝑥-𝑦-𝑢 space

Figure 2 

Phase portraits of system (3) with 𝑞1=0.96, 𝑞2=0.94, 𝑞3=0.92, 𝑞4=0.78, 𝑎=3, 𝑏=0.4, 𝑐=0.4, and 𝑑=0.1.

(a) 𝑑=0.1

(b) 𝑑=2.8

Figure 3 

Plots in new coordinates (𝑝,𝑠) space with 𝑞1=0.96, 𝑞2=0.94, 𝑞3=0.92, 𝑞4=0.78, 𝑎=3, 𝑏=0.4, and 𝑐=0.4.

(a) 𝑥-𝑦-𝑧 space

(b) 𝑥-𝑦-𝑢 space

Figure 4 

Phase portraits of system (3) for 𝑞1=0.96, 𝑞2=0.94, 𝑞3=0.92, 𝑞4=0.78, 𝑎=3, 𝑏=0.4, 𝑐=0.4, and 𝑑=2.8.

4.2. Varying 𝑞₂ and Fixing 𝑞₁=0.98, 𝑞₃=0.97, 𝑞₄=0.78, 𝑎=3, 𝑏=0.4, 𝑐=0.4, 𝑑=0.1

In the following subsection, 𝑞₁=0.98, 𝑞₃=0.97, 𝑞₄=0.78, 𝑎=3, 𝑏=0.4, 𝑐=0.4, 𝑑=0.1 are fixed. When 𝑑 varies, the corresponding 𝐾 values are shown in Figure 5(a). The bifurcation diagram of interest rate 𝑥 with increasing the investment demand derivative 𝑞₂ is given in Figure 5(b). Obviously the bifurcation diagram well coincides with the 𝐾 diagram, too. Figure 5 shows that the system (3) changes gradually from stability to chaos as the investment demand derivative 𝑞₂ grows larger.

(a) 0-1 test with 𝐾 versus 𝑞2

(b) Bifurcation with 𝑞2 versus 𝑥

Figure 5 

The evolvement of 𝐾 and 𝑥 versus 𝑞2 with 𝑞1=0.98, 𝑞3=0.97, 𝑞4=0.78, 𝑎=3, 𝑏=0.4, 𝑐=0.4, and 𝑑=0.1.

From Figure 5 of the system (3), when the investment demand derivative 𝑞₂ is determined by its corresponding 𝐾≈1, the system (3) is chaotic. Whereas the investment demand derivative 𝑞₂ is determined by its corresponding 𝐾≈0, the system (3) is regular.

For example, in the system (3), when the investment demand derivative 𝑞₂=0.4, its corresponding phase portraits are quasiperiodic as shown in Figure 7 and its trajectories in the (𝑝,𝑠) plane are bounded as shown in Figure 6(a); that is, the system (3) is regular (quasiperiodic). Whereas the investment demand derivative 𝑞₂=0.89, its corresponding phase portraits are strange attractors as shown in Figure 8 and its trajectories in the (𝑝,𝑠) plane are Brownian-like as shown in Figure 6(b); that is, the system (3) is chaotic.

(a) 𝑞2=0.4

(b) 𝑞2=0.89

Figure 6 

Plots in new coordinates (𝑝,𝑠) space with 𝑞1=0.98, 𝑞3=0.97, 𝑞4=0.78, 𝑎=3, 𝑏=0.4, 𝑐=0.4, and 𝑑=0.1.

(a) 𝑥-𝑦-𝑧 space

(b) 𝑥-𝑦-𝑢 space

Figure 7 

Phase portraits of system (3) for 𝑞1=0.98, 𝑞2=0.4, 𝑞3=0.97, 𝑞4=0.78, 𝑎=3, 𝑏=0.3, 𝑐=0.8, and 𝑑=0.1.

(a) 𝑥-𝑦-𝑧 space

(b) 𝑥-𝑦-𝑢 space

Figure 8 

Phase portraits of system (3) for 𝑞1=0.98, 𝑞2=0.89, 𝑞3=0.97, 𝑞4=0.78, 𝑎=3, 𝑏=0.3, 𝑐=0.8, and 𝑑=0.1.

5. Conclusion

A government is used to employing investment incentive to regulate financial systems. In this paper, a simple investment incentive is introduced into the classical chaotic financial system, and a 4-D chaotic financial system is obtained. What is more, there are some memory effects in the financial systems, so the 4-D fractional-order chaotic financial system is proposed by introducing the fractional calculus into the mentioned 4-D integer-order financial system. With the improved Adams-Bashforth-Moulton predictor-corrector scheme, phase portraits and bifurcation diagrams are illustrated, and then 𝐾 value diagram and Plots in new coordinates (𝑝,𝑠) space are given by using the 0-1 test algorithm. Thus chaotic dynamics is distinguished from regular dynamics in the 4-D fractional-order chaotic financial system by the abovementioned methods.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported partly by Excellent Young Scientist Foundation of Shandong Province (Grant no. BS2011SF018), National Social Science Foundation of China (Grant no. 12BJY103), Humanities and Social Sciences Foundation of the Ministry of Education of China (Grant no. 11YJCZH200), and National Natural Science Foundation of China (Grant no. 71272148).