Chaos and Complexity Analysis of a Discrete Permanent-Magnet Synchronous Motor System

. In this paper, a spectral entropy algorithm has been used successfully for complexity measure of a discrete permanent-magnet synchronous motor (PMSM) system. Firstly, the discrete PMSM system is achieved using the forward Euler scheme. Secondly, by adopting the bifurcation diagram, phase portraits, 0-1 test, and largest Lyapunov exponent, the chaotic dynamics of the discrete PMSM system are analyzed. The complexity of the discrete PMSM system is discussed by employing the spectral entropy algorithm. It shows that the spectral entropy complexity analysis is an e ﬃ cient tool to study chaotic dynamics. Finally, we illustrate this result through numerical experiments.


Introduction
Due to its potential applications in the fields of secure communication and information encryption, chaos as a hot topic has been widely studied for nearly three decades, especially the chaotic dynamics and complexity analysis of chaotic sequences.
In recent years, a discrete chaotic system, which is obtained by the forward Euler scheme for its continuous system, is widely studied.A natural question to follow is whether these discrete chaotic systems are more complicated than its continuous systems or not?How to select the system parameters and step size to get more complex dynamics?Therefore, it is worthwhile to take the fluctuation of step size into full account for the dynamic problem of a chaotic system.To our knowledge, the relevant researches of these questions keep open, and these topics deserve further investigation.
Based on the above questions, the major task of this paper is to deal with complexity of chaotic sequences by means of the spectral entropy algorithm.As a result, a discrete permanent-magnet synchronous motor (PMSM) system, which is obtained by a continuous PMSM system, is considered.The PMSM system is a kind of highefficient and high-powered motor, which has been widely used in the industry [23].The chaotic dynamics of the continuous PMSM system becomes one of most active research areas.Many important research works on the PMSM system are found in [24][25][26][27] and the references cited therein.This paper is organized as follows.In Section 2, the chaotic dynamics of the discrete PMSM system, which is obtained by using the forward Euler scheme, is investigated by means of the bifurcation diagram, phase portraits, 0-1 test, and largest Lyapunov exponent.In Section 3, the spectral entropy algorithm is described, and the complexity of chaotic sequences in the discrete PMSM system is calculated.Finally, conclusions and future direction are presented in Section 4.

Dynamics of the Three-Dimensional Discrete PMSM System
2.1.The Three-Dimensional Discrete PMSM System.This section illustrates the application of the forward Euler method to a continuous PMSM system.
where a and b are positive parameters.When a = 28 and b = 3, the PMSM system is chaotic.Its typical butterfly chaotic attractor is shown in Figure 1.When the forward Euler is employed, we obtain the following three-dimensional discrete PMSM system: 2 Complexity where a and b are positive parameters and δ > 0 is the step size.

Fixed
Points and Their Stability.The fixed points 2) are obtained by solving the equations: In order to study the stability of the fixed point O 1 , we calculate the Jacobian matrix of system (2) at O 1 as and its characteristic polynomial equation is According to the Schur-Cohn criterion [28], we derive the stability conditions of O 1 as follows: (a) α + γ < 1 + β and (b) As a result, by taking b = 3 and δ = 0 01, the stability condition of O 1 is 0 < a < 1.Here, we only consider stability of the fixed point O 1 , and the stability of the fixed points O 2 and O 3 can be studied in a similar manner as for O 1 .

Chaos in the Three-Dimensional Discrete PMSM System.
In this section, by means of the phase portraits, bifurcation diagram, 0-1 test and largest Lyapunov exponent, the dynamics of system (2) is researched.
In particular, variations of the parameter a are considered by keeping b = 3 and δ = 0 02.The largest Lyapunov exponent diagram of system (2) is obtained using the Jacobi matrix method as shown in Figure 2. Letting δ = 0 02 and fixing b = 3, the bifurcation diagram of system (2) in Figure 3 is obtained for a in the range 0, 30 .From Figures 2 and 3, of particular interest is that it has a chaotic attractor with one positive Lyapunov exponent in a large range of parameters.As a result, starting with a = 8, a stable solution is obtained, and the corresponding phase diagram is given in Figure 4(a).By increasing the value of a, a two-scroll chaotic attractor is obtained for a = 28 as shown in Figure 4(b).By adopting 0-1 test [29], the corresponding trajectories in the p − s plane are shown in Figures 5(a) and 5(b).Here, p and s are two real valued functions, respectively, where θ j = jc + ∑ j i=1 ϕ i , j = 1, 2, 3 ⋯ , n, and c is a random number in the range π/5, 4π/5 .Also note that bounded trajectories in the p − s plane correspond to regular dynamics, whereas Brownianlike (unbounded) trajectories correspond to chaotic dynamics [29].See Gottwald and Melbourne [29] for details. 3 Complexity From Figures 2-4, it is clear that the stable, periodic solutions and ensuing transition to chaos are observed.

Complexity Analysis
In this section, the complexity of system (2) is discussed by employing the spectral entropy algorithm.
The spectral entropy algorithm is a method that deals with the complexity of a time series.The spectral entropy reflects the disorder in the Fourier transformation domain, if a flatter spectrum has a larger value of spectral entropy, which shows a higher complexity of the time series [16].Consider a set of discrete data ϕ n , n = 0, 1, 2, ⋯, N − 1 with a length of N, representing a one-dimensional observable data set.According to discrete Fourier transformation approach, the spectral entropy algorithm [13][14][15][16] is briefly described as follows.
Step 1. Data processing.In order to better reflect the energy information of the data ϕ n , transform the data ϕ n as follows: where ϕ = 1/N ∑ N−1 n=0 ϕ n .
Step 2. Discrete Fourier transformation.Taking the discrete Fourier transform for the data ϕ′ n and its corresponding discrete Fourier transform is defined as where k = 0, 1, … N − 1 and j = −1 is the unit imaginary.

Complexity
Step 3. Calculating relative power spectrum.If the power of a discrete power spectrum with the kth frequency is Φ k 2 , then the "probability" of this frequency is defined as Step 4. Spectral entropy is calculated.Applying the P k , the spectral entropy is defined by Based on the above four steps, the spectral entropy of a given time series with length N is calculated, without choosing any other parameters [15].
Here, we choose parameter planes a, b , a, δ , and b, δ to calculate its complexity, for example, to see how the complexity changes as the parameters a, b, and δ increase (or decrease).As a result, the length of data ϕ n , n = 1, 2 ⋯ for spectral entropy is 4 × 10 4 after removing the first 1 × 10 4 points of data [15].By taking δ = 0 02, the spectral entropy complexity of x series of system ( 2) is calculated and illustrated in Figure 6.
From Figure 6, it can be seen that the parameter b has a larger range of choices when parameter a is smaller.For example, the parameter a is selected in the range 0, 1 ; the system (2) maintains a high complexity for parameter 5 Complexity b ∈ 0, 30 .When the parameters a and b are selected in the two light-colored areas in the upper right, the system (2) maintains the largest complexity.Hence, for this chaotic system (2), the complexity depends on parameters a and b sensitively.
Next, we calculate the complexity of system (2), when the step size δ is selected in the range 0, 0 035 .By taking b = 3, the spectral entropy complexity in Figure 7 is obtained for parameters a and δ in the range 0, 30 and 0, 0 035 , respectively.
From Figure 7, it can be seen that the largest value of spectral entropy complexity of system (2) fluctuates within a small range for parameters a and δ.Similarly, by taking a = 28, the spectral entropy complexity in Figure 8 is obtained for parameters b and δ in the range 0, 30 and 0, 0 035 , respectively.According to Figure 8, it can be seen that the largest value of spectral entropy complexity of system (2) is 0.7.For parameters b and δ, they are roughly in the range of 3, 10 and 0 015, 0 0 025 , respectively.

Complexity
Based on the discussion above, we found that spectral entropy complexity analysis is a convenient method to choose parameters for a discrete system.
In the following discussion, two of the parameters are fixed and the spectral entropy complexity is calculated.By taking b = 3 and δ = 0 02, the spectral entropy complexity in Figure 9(a) is obtained for parameter a in the range 0, 30 .Similarly, fixing a = 28 and δ = 0 02, the spectral entropy complexity is calculated for parameter b in the range 0, 30 as shown in Figure 9(b).Combined with Figure 9, we find that the spectral entropy complexity calculation is similar to that of the Lyapunov exponent, which reflects the complexity of chaos.In Figure 9(a), letting a = 15, b = 3, and δ = 0 02, the 0-1 test is obtained as shown in Figure 10.In Figure 9(b), letting a = 28, b = 7, and δ = 0 02, the 0-1 test is obtained as shown in Figure 11.
In addition, it is worth noting that the spectral entropy complexity of system (2) increases with the increasing step size δ.In particular, variations of the step size δ is considered by keeping a = 28 and b = 3.The spectral entropy complexity is shown in Figure 12.

Conclusions
In this paper, by means of the largest Lyapunov exponent, phase diagram, bifurcation diagram, 0-1 test, and spectral entropy algorithm, we investigated the chaotic dynamics and complexity of the discrete PMSM system obtained by the forward Euler scheme.The conclusions are drawn as follows.
The discrete PMSM system contains rich dynamical behaviors.The two-scroll stronger chaotic attractor is observed with the increasing parameter a.Moreover, the discrete PMSM system is chaotic in large range for parameter a.Furthermore, by calculating the chaos complexity of discrete PMSM system, we found that it has a high complexity.Therefore, the discrete PMSM system has higher application value in the real-word applications.For example, this system can be considered for chaotic encryption.
We also found an interesting phenomenon that the spectral entropy complexity of discrete PMSM system increases with increasing step size δ.Thus, the spectral entropy complexity analysis is a convenient method to choose step size.In fact, it also provides a reference for Euler method to study the chaotic dynamics of continuous dynamical system.
It is expected that the results enrich our knowledge of the dynamics of discrete systems, which is obtained by the forward Euler scheme.Furthermore, it suggests that the results are beneficial to researchers who explore other dynamical behaviors.