The present study uses the differential transformation method to solve the governing
equations of the coronary artery system and then analyzes the dynamic behavior of
the system by means of phase portraits, power spectra, bifurcation diagrams, and
Poincaré maps. Also, a masterslave control system is proposed to suppress the
nonlinear chaotic behavior of the coronary artery system. The results show that the
dynamic behavior of the coronary artery system is significantly dependent on the
magnitude of the vibrational amplitude. Specifically, the motion changes from
Arrhythmias are heartrhythm problems in which the heart beats too fast, too slow, or irregularly. While many arrhythmias are harmless and cause only relatively minor symptoms such as shortness of breath, dizziness, and sudden weakness, in severe cases arrhythmias can lead to cardiac arrest and even death. There are many reasons for the rhythm change of the coronary artery (CA) system, including thyroid disease, high blood pressure, heartvalve problems, abnormalities of the heart ectopic pacemaker, and irregular dynamic phenomena such as the frequency of the conduction system. The prediagnosis of arrhythmia enables the selection of appropriate antiarrhythmic drugs, thus improving the arrhythmia and reducing the probability of sudden heart problems. Arrhythmia is generally detected by means of electrocardiography (ECG), in which the electrical activity of the heart is measured by means of electrodes attached to the arms, legs, and chest and is then printed out on paper [
A typical electrocardiogram includes three basic waveforms, namely, a P wave, a QRS wave, and a T wave, where a P wave represents the wave of depolarization, a QRS complex wave represents ventricular depolarization, and a T wave represents ventricular repolarization [
This paper applies the differential transformation method to solve the governing equations for the coronary artery (CA) system. The nonlinear dynamic behavior of the system is then investigated by means of phase portraits, power spectra, bifurcation diagrams, and Poincaré maps. In addition, a masterslave control system is proposed to suppress the nonlinear chaotic behavior of the CA system. The simulation results show that the proposed controller enables the abnormal slave CA system to be synchronized with the normal master CA system despite the presence of system uncertainties.
The governing equations of the CA system were originally derived by Guan [
A nonlinear chaotic behavior of the CA system may result in various cardiovascular problems, including myocardial infarction, angina, and even death. Therefore, the present study proposes a masterslave CA synchronization control system based on the following state equations:
In designing the proposed controller, the CA system parameters are assigned the values shown in Table
System parameters [
Parameter 





Value  0.15  −1.7  −0.65  1 
Applying the differential transformation method [
In the present study, the dynamic behavior of the CA system is characterized by means of the phase portraits, power spectra, bifurcation diagrams, and Poincaré maps produced using the timeseries data for parameters
Figure
Phase portraits of coronary artery system at
Figures
Power spectra of coronary artery system at
Figures
Bifurcation diagrams for variation of blood vessel diameter
Poincaré maps of coronary artery system at different values of vibrational amplitude
From the above discussions, it is clear that the dynamic response of the CA system depends heavily on the magnitude of the vibrational amplitude. The various motions performed by the system as the vibrational amplitude is increased from
Variation of coronary artery system response with vibrational amplitude over interval

Dynamic behavior 


T 

Chaos 

T 

2T 

4T 

8T 

Chaos 

8T 

6T 

3T 

4T 
From the above analysis, it can be seen that the coronary artery system have very complex behaviour. In this section, it will be studied to design a controller to synchronize the abnormal coronary artery system to a normal coronary artery system.
Consider the following masterslave coronary artery systems.
Master system is
From [
In this study, a fuzzy logic controller (FLC) by Yau and Shieh [
In consequence, to achieve this control goal for chaotic coronary artery systems with uncertainties, there exist two major phases. First, we let the control input
According to the state transformation by Yau and Shieh [
Rule table of FLC.
Rule  Antecedent  Consequent  



 
1  P  P 

2  P  Z 

3  P  N 

4  Z  P 

5  Z  Z 

6  Z  N 

7  N  P 

8  N  Z 

9  N  N 

For the overall control systems (
Time responses of chaos synchronization of coronary artery systems: master and slave system outputs are
The time response of error states with control
The control action versus time with control
This study has applied the differential transformation method to investigate the dynamic behavior of a CA system. Phase trajectories, Poincaré maps, and bifurcation diagrams have been used to characterize the dynamic response of the system as a function of the vibrational amplitude and to detect the onset of chaotic motion. In general, the results have shown that, as the vibrational amplitude is increased from 0.1 to 20.0, the system motion changes initially from
In this paper, nonlinear FLC theory has been exploited to design a controller for chaos synchronization with system uncertainties. It can synchronize the abnormal CA system to a normal CA system. It shows that the FLC in this paper is realizable for implementation and it can reduce the actuator saturation phenomenon in real physics system. The other types of CA synchronization control could also be synchronized by using the same control scheme proposed in this study.
Let
At
If
If
Using the differential transformation method, a differential equation in the domain of interest can be transformed to an algebraic equation in the
In order to accelerate the rate of convergence and improve the accuracy of the calculations, the overall
The approach described above is used to split the time domain into a total of
Time step diagram.
Note that
In general, the function
Using the
Table
Operation in the
Operator  

Spectrum 

Function 

Convolution 

Derivative 

If
Therefore, the differential transform of
The authors declare no conflict of interests.
The financial support provided to this study by the National Science Council of Taiwan under Grant no. NSC1002628E269016MY2 is greatly appreciated.