Limit Cycles of a Class of Hilbert's Sixteenth Problem Presented by Fractional Differential Equations

Abstract

The second part of Hilbert's sixteenth problem concerned with the existence and number of the limit cycles for planer polynomial differential equations of degree n. In this article after a brief review on previous studies of a particular class of Hilbert's sixteenth problem, we will discuss the existence and the stability of limit cycles of this class in the form of fractional differential equations.

1. Introduction

The second part of the well-known Hilbert's 16th problem is still unsolved since Hilbert proposed it in 1900. This problem is concerned with the maximum number of limit cycles and their relative distributions of the real planar polynomial systems of degree n in the form of

(1.1)

where and are polynomial of degree with real coefficients. The general form of this problem, even for , is yet an open problem that has attracted more researches but it is remarkably inflexible. With the development of computer's and graphical software, many recent new improvement results have been obtained. Some survey articles can be found in [1–5] and references therein. One of the classical methods to produce and study limit cycles in such system (1.1) is by perturbing a system which has a centre (e.g., see [6, 7]). In such methods the limit cycles are produced in the perturbed system from the periodic orbits of the periodic annulus of the unperturbed system. As we can see in [8] by perturbing the linear centre   , using arbitrary polynomials and of degree n, limit cycles bifurcated with the bifurcation parameter of order one. Almost the same argument can be seen in [9] by perturbing the system    with maximum n limit cycles. By perturbing the Hamiltonian centre given by in the polynomial differential systems of odd degree n, we can obtain limit cycles [10]. Several other similar investigations have been done using the perturbed polynomial differential systems of second, third, or even more degree. For example, see [11–13] and references therein.

Based on the above studies, some of the authors of this article investigated the number of limit cycles of perturbed quintic Hamiltonian systems with different degree polynomials [14, 15]. In these former articles a weakened Hilbert's 16th problem in the following form is considered:

(1.2)

In system (1.2) is a real polynomial of degree n, and and are two real polynomial of degree m. Moreover, system (1.2) contains at least a family of closed orbits for any level curve with and . A full investigation of this planar system for the number of limit cycles and their stabilities can be found in [15]. In this article we study the existence of limit cycles and their stabilities for such system in the form of Fractional Differential Equations (FDEs). Recently great considerations have been made to the systems of FDE. The most essential property of these systems is their nonlocal property which does not exist in the integer-order differential operators. We mean by this property that the next state of a system depends not only upon its current state but also upon all of its historical states. This is a more realistic and is one reason why fractional calculus has become more and more popular. On the other hand, the integer-order differential operator is indifferent to its history. Furthermore, there have been several recent mathematical discoveries that have helped to unlock the power of the fractional derivative [16]. One such discovery is that of fractal functions. Indeed, most of the functions that we are familiar with are smooth. This means that locally they can be approximated by a straight line segment. For example, the function is well approximated by at the point . The derivative of the function at a particular point provides the slope of the straight line approximation or tangent to the curve. Fractal functions are not smooth. They have details on all scales and they cannot be approximated locally by straight line segments. An example is the Weierstrass function which can be written as the infinite sum of cosine functions, . For this function at the point , the tangent changes orientation under increasing magnification. Functions such as the Weierstrass function cannot be differentiated (a whole number of times). But it turns out that these fractal functions can be differentiated a fractional number of times, and the fractional calculus is important for studying these differentiability properties. Fractals are characterized by scaling laws and the fractional derivative at a point can reveal this law. In recent research, scientists at the Mount Sinai School of Medicine have shown that the surfaces of breast cells are fractals and they have found clear differences in the scaling laws for benign cells and malignant cells. The different scaling laws have enabled accurate diagnosis of breast cancers. Another important new discovery that has brought fractional calculus into prominence is that many physical processes are modeled by fractional differential equations. Obviously, the importance of a mathematical model is that it can be used to make predictions and to give insight into the physical process that underlies the behavior. One area where mathematical models have been employed extensively is that of diffusion and transport processes. For example, the dispersion of pollutants in the ocean and the motion of electronic charges in conductors are diffusion processes. Here, a probabilistic description leads to a (whole number) differential equation which can be solved to predict average properties of the system. Similar types of equations are used by financial analysts to model stock prices. It has recently been discovered that processes governed by diffusion which is enhanced or hindered in some fashion are better modeled by FDEs than by integer-order differential equations. These FDEs are finding numerous applications in areas ranging from financial mathematics to ocean-atmosphere dynamics to mathematical biology [16].

These and the other applications of FDEs provide a good motivation for study such Hilbert's 16th problem of system (1.2) in the form of FDE. So, in the next section we will consider system (1.2) in the form of FDEs and to be more specific we will take , as polynomials of degree 1 and , as polynomials of degrees 3 and 5, respectively. Due to the existence of Riemann-Liouville integral operator in the definition of FDE in the Caputo sense [17], direct analytical solution for FDE is too rare, and so using the numerical methods is inevitable. In order to use a reliable numerical method we should first discretized the given FDE. However, discretization schemes that produce difference equations whose dynamics resemble that of their continuous counterparts are a major challenge in numerical analysis. To this end we will apply the Mickens nonstandard discretization scheme [18] to the Grunwald-Letnikov discretization process for our system of FDE. As we will see in Section 3 this discretization scheme leads to the fast convergence with more accurate results in solving the original system (1.2) with integer-order derivative one. Therefore, we are expecting the same accurate results for system (1.2) in the form of FDE with different noninteger-order derivative. Then in Section 4 we will discuss the stability of limit cycle which exists in our system and illustrate the numerical results. We will summarize the results with some final comments in Section 5.

2. Specific Case of the Weakened Hilbert's 16th Problem

We consider the specific case of system (1.2) as

(2.1)

where , and are real constants. As discussed in [15] there is a closed relation between the number of the limit cycles in (2.1) and the number of zeros of its related Abelian integral [19]. The related Abelian integral of (2.1) stated as

(2.2)

where is the area surrounded by the first integral curves of (2.1), that is,

(2.3)

as and . Now to evaluate the Abelian integral in (2.2), first we note that the limits of the double integrals can be found by solving the first integral (2.3) for y, that is, and then for by solving when , which yields with . Hence, with the symmetry of which exists with respect to and noting that , integral (2.2) yields . After evaluating and simplifying this equation as a polynomial of , we get

(2.4)

Note that here we replace where . Finally, with this brief discussion the existence and stability of limit cycle for perturbed system (2.1) can be finalized in the following theorem.

Theorem 2.1.

The perturbed system (2.1) has no limit cycle for and one limit cycle for . In the former case the unique limit cycle is stable for and unstable for .

For the proof of this theorem, as discussed above, we need to find the zero of the Abelian integral (2.4) which leads to a polynomial of degree 3 with respect to . Then it is straight forward to see that this polynomial has no positive root for and at least one positive real root for . That is, in the first case system (2.1) has no limit cycle and in the former case there is one limit cycle. For the detail proof of this theorem refer to [15].

3. System (2.1) in the Form of Fractional Differential Equations and Its Discretization

In general, and is a single initial value FDE, where denotes the fractional derivative in the Caputo sense [17] and defined by . Here , and is the -order Riemann-Liouville integral operator defined as

(3.1)

with A limited number of methods have been utilized to solve this initial value problem. In order to apply Mickens' nonstandard discretization scheme [18] in our numerical scheme we choose the Grunwald-Letnikov method to approximate the one-dimensional fractional derivative as follows [20]:

(3.2)

where denotes the integer part of and is the step size. In this case the above initial value problem is discretized as

(3.3)

where and are the Grunwald-Letnikov coefficients defined as , or recursively and .

Now using this definition for FDE with Grunwald-Letnikov discretization method, system (2.1) in the form of FDE is discretized as follows:

(3.4)

We assert that nonstandard discretization method is a numerical attempt which can be used in discretization process of FDE to get the better results and preserves their crucial property, that is, nonlocal property. In order to do this, we apply the Mickens nonstandard discretization scheme [18] to the Grunwald-Letnikov discretization process for FDE system (3.4). Indeed, the derivative term, , in the Mickens schemes is replaced by , where is a continuous function of step size . In addition the nonlinear terms such as are either replaced by , or left untouched depending upon the context of the differential equation. There is no appropriate general method for choosing the function , but some special techniques may be found in [18, 21].

Now we first write system (3.4) as follows:

(3.5)

Here, we replaced and by and , respectively. Later on, following Mickens' method in the next section, for finding the better results we replace the nonlinear terms in system (3.5) by appropriate combination of the variables in different levels of times.

4. Stability of the Limit Cycles in System (3.5) and Numerical Results

First we note that the linearized system (3.5), around a stationary point or simply , will be where matrix is evaluated as

(4.1)

Without losing our generality, we suppose here to be positive. Now, from theory of dynamical systems, the limit cycle exists in system (3.5) whenever the characteristic equation of matrix,, has two solutions with module one.

In addition, for the stability of this limit cycle we can use the stability analysis which is thoroughly investigated by Matignon in [22]. To utilize this theorem for our problem, first we consider the linearization of system (2.1) in the form of FDE with the derivative order in both equations around a given stationary point . This linearized system can be written as where matrix is similar to matrix in (4.1) with and . Now the Matignon stability theorem for our problem can be stated as the following theorem.

Theorem 4.1.

The linearized system of fractional differential equations is asymptotically stable if and only if .

Note that the stability exists if and only if either asymptotically stability exists or those eigenvalues which satisfy have geometric multiplicity one.

Now we will implement our numerical method described above for the existence of limit cycles in system (2.1) in the form of FDE for different values of fractional order . In order to be consistence with the results in [15], in system (3.5) we let the constants , be one and choose , according to the following discussion with . By these assumptions characteristic equation of matrix in (4.1) at the point (1, 1) will be

(4.2)

which yields

(4.3)

Equation (4.3) has two solutions with modular one if

(4.4)

First, we note that for the inequality and equality occur in (4.4) if . That is, for some choice of and with opposite signs there is a limit cycle for system (3.5). This result agrees with the consequence of Theorem 2.1. Suppose that we choose and ; then the numerical solutions results of system (3.5) are illustrated in Figure 1. As we stated before, in order to get the better results in solving this system, following Mickens' method, we replace nonlinear terms , and with , , and , respectively. Note that, given , , and from conditions (4.4) we can evaluate the best choice for (or ). Here, we get .

Figure 1

Stable limit cycle of system (3. 5) for , , , and with starting point (1, 1).

For the stability of this limit cycle, since , by the well-known theories of dynamical systems we should find the eigenvalues of the linearized system (2.1) at the given point (1,1). In this case, with the choice of the above parameters, these eigenvalues are . Obviously, since the signs of real parts of and are negative, the limit cycle in Figure 1 is stable. With similar discussion, if we choose to be a positive constant, say 2, then the related eigenvalues will be with the positive real parts. In this case, as we can see in Figure 2, the limit cycle is unstable. These results agree with consequence of Theorem 2.1.

Figure 2

Unstable limit cycle of system (3. 5) for , and with starting point (1, 1).

Now for the fractional order , say , with the same values as above for , and , conditions (4.4) are satisfied. So by these values of the parameters, there is a limit cycle for the system (3.5). As illustrated in Figures 2, 3, 4, and 5 these limit cycles exist for different values of . For the stability of these limit cycles we may apply Theorem 4.1. For example, for corresponding eigenvalues of matrix at the point can be found as which yields with argument . Obviously, this value of is beiger than for , which proves the stability of the limit cycles whenever exist.

Figure 3

Stable limit cycle of system (3. 5) for , , and with starting point (1, 1).

Figure 4

Stable limit cycle of system (3. 5) for , , , and with starting point (1, 1).

Figure 5

Stable but sensitive limit cycle of system (3. 5) for , , , and with starting point (1, 1).

5. Final Comments

In this article we discussed the existence and stability of the limit cycle for special case of perturbed Hilbert's 16th problem. We found these limit cycles for different values of fractional order using discretized system (3.5), provided by Grunwald-Letnikov numerical method for solving FDEs and applying Mickens' nonstandard method for more accurate results. The difficulties that we are facing here are in solving system (3.5) for values of . Though this is the case for different nonlinear systems of FDEs, existing numerical methods are not capable for solving such these systems for small fractional derivative order . In other words, for the small efficient dimension, which is the sum of fractional derivatives of the equations in the systems such as system (2.1), the numerical results are not accurate enough. Here, for the values of with the same values for other parameters as above, the results are found by solving system (3.5) all converging to zero (see Figure 6).

Figure 6

Numerical results of system (3. 5) which is converging to zero for , , , and with starting point (1, 1).

Another difficulty exists in choosing and for conditions in (4.4) to be satisfied. That is, whenever by choosing small positive values for and conditions (4.4) are satisfied, but the numerical limit cycle cannot be found in system (3.5) even in unstable form. Nevertheless, as we saw the limit cycles exist for the values and with different signs. In particular these limit cycles are stable, easy to find for values , and agreed with the stability condition in Theorem 4.1.