NUMERICAL INVESTIGATION OF THE INDEX OF THE VECTOR FIELD OF HOLLING-TANNER MODEL BY THE FAST FOURIER TRANSFORM

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The following Van der Pol and Holling-Tanner equations is analyzed from the qualitative viewpoint by investigating their vector fields and analyzing the nature of the stationary points of these equations. The winding numbers (indices) of the stationary points are investigated by calculating the Poincare integrals. This calculation is performed by a novel method which is based on application of the fast Fourier transform (FFT) formation to the Poincare integrand.


INTRODUCTION
The vector field represents vectors with components X = X(x, y) and Y = Y (x, y) in Oxy rectangular coordinated system, where X and Y are right hand sides/parts of the system of equations: Stationary points of this system are calculated as solution of the system of equation Y (x, y) = 0.
Index (winding number) of the stationary points is calculated by the Poincare integral: where d p is a differential along the curve AB in Oxy-coordinate system, surrounding the stationary point, x = x(p), and y = y(p). According to [1], a Lipschitz continuous function of a plane vector field F = [X(x, y),Y (x, y)] over a smooth Jordan curve is considered over the integral (Poincare's integral).
Due to 2π-periodicity of f (p), we can expand it in the Fourier series In these terms, the Poincare integral and can be found numerically by any discrete Fourier transformation (DFT). For example, the FFT, which is calculated from N = 2 n (n is positive integer) samples of f (p) which are equidis- One of the advantages of FFT is its speed which it gets by reducing the number of calculations needed to analyze waveform [2]. This is an alternative to DFT, because of the speed the algorithm reduces an n-points of Fourier transform to about n 2 log2(n). FFT is widely used in many applications in engineering, science, and mathematics such as speech processing and frequency estimation [3]. The fundamental idea of the algorithm was firstly derived in 1805 according to [4]. Recently the FFT algorithm was used to analyzed and detected the frequency with respect to the noise level using the white Gaussian noise [5]. The FFT algorithm (also known as Cooley-Turkey algorithm) was formulated by James W. Cooley and John W. Tukey in 1965 [6] and it is the most important numerical algorithm in applications.
Examples of FFT algorithm are 1 Rader's FFT algorithm, 2 Prime-factor FFT algorithm, 3 Chirp Z-transform, 4 Bruun's FFT algorithm, 5 Bluestein's FFT algorithm, 6 Hexagonal FFT, etc. It is of great importance to numerically investigate the dynamical systems (Van der pol and Holling-Tanner equation) considered in this study, through software MathCad ® and Mathematica ® for the vector plot, equilibrium points, and indices of the stationary points of the vector fields.

PROBLEM STATEMENT
This paper is devoted to the numerical analysis of the vector fields of the Van der Pol and Holling-Tanner systems by the built-in methods of the Mathematica ® by means of numerical calculations of the indices of these fields by the FFT. The aims of the study were to: • Obtain the equilibrium points of the Van der Pol and Holling-Tanner systems.
• Investigate the vector fields of the Van der Pol and Holling-Tanner systems by the builtin functions of Mathematica ® .
• Calculate the indices of stationary points of the vector fields of the Van der Pol and Holling-Tanner models.

MODEL EXAMPLE: VECTOR FIELD AND INDEX OF THE STATIONARY POINT OF VAN DER POL EQUATION
In this section, we consider the well known Van der Pol equation and illustrate our approach to calculate the index of the stationary point of the vector field of this equation.

Let us represent Van der Pol equationẍ
in the Cauchy form as and hence, the characterization equation is from which it follows that the eigenvalues are where i is complex unit (i 2 = −1).
Hence, the stationary point x = y = 0 is unstable focus (source). In the case of sources and sinks, the index of the stationary points is equal to +1 and in the core of the sinks, its equal to −1. To calculate the index numerically, we assume and Let ψ(θ ) and κ(θ ) be represented in Equation (17) Therefore, is calculated by Maple ® exactly.
For its calculation by the FFT-method,we:  Values of a 0 -factor for ρ = 0.05 differentiation values of n is shown in Table (1). As we see, it is necessary to have N = 2 4 points of discretization in this case to achieve the value of the index of the stationary point. When selecting number of points for the integrand discretization, it is necessary to keep in mind that the spectrum of the functions contains main cosine and sine harmonics so that higher harmonics in the spectrum have relatively small amplitude. Otherwise value of the index can be substantially deteriorated by the aliasing effect.
This effect is illustrated in the next section.

VECTOR FIELD AND INDICES OF STATIONARY POINTS OF HOLLING-TANNER SYS-TEM
In the Holling-Tanner system of equation, we assume that Assuming that only non-negative y and positive x are considered, we can find three stationary points from equations X = Y = 0; In these cases, Eigenvalues of the linearized system of Equation (31) are and hence, the stationary point is saddle.
At Equation (28), x 2 = 0.32134, y 2 = 0.21422, the linearized Holling-Tanner system in the vicinity of x 2 , y 2 is with corresponding eigenvalues hence, the stationary point is saddle.
In the third case Equation (29) at x 3 ≈ 3.11200, y 3 ∼ = 2.07466, the linearized Holling-Tanner system in the vicinity of x 3 , y 3 is  To analyze the behaviour of the first stationary point x 1 = 10, y 1 = 0, we make change of variable.
where x 1 = ρ cos θ , y 1 = ρ sin θ and compose the Poincare integral with The results of discretization of the integrand in N = 2 5 points with ρ = 1 100 are shown in Figure (4). Results of the FFT of the discretized function v k = f 1 (θ k ), where θ k = 2π k · k, k = 1, 2, . . . , N − 1 and N = 2 n are represented in Table (2).  As we see the index of the stationary point x 1 = 10, y 1 = 0 equals to −1, which corresponds to the saddle, can be obtained with minimum possible points of discretization.
Discretizing the Poincare integrand in N = 2 6 points , we obtain graph shown in Figure (6). Results of the corresponding FFTs for the different and N = 2 n are given in Table (4).  The results of incorrect calculation of the index of stationary points (x 3 , y 3 ) is explained by the sharp positive spikes in the graph of the integrand and hence, the aliasing effect due to insufficient number of harmonics taken into consideration in the spectrum of the integrand. As we can see, the index of the stationary point (x 3 , y 3 ) is equal to 1.