Chaos Analysis and Synchronization Control of Coronary Artery Systems

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Introduction
Arrhythmias are heart-rhythm 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, heart-valve 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 [1][2][3].However, while electrocardiograms are essential tools in diagnosing heart disease, the ECG signal is highly nonlinear and is therefore not easily analyzed using traditional methods.Thus, various machine-assisted solutions for diagnosing and monitoring arrhythmia have been proposed in recent decades, including Bayesian methods, heuristic methods, expert systems, Markov models, and artificial neural networks (ANNs) [4].
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 [5].These waves are the main factors affecting the movement of the heart, and thus their dynamic behaviors must be properly understood if the mechanisms of the HBV system are to become clear [6].Furthermore, if the cardiac motion can be precisely controlled, the chances of both diagnosing and curing heart disease and abnormalities can be significantly enhanced [7].
This paper applies the differential transformation method to solve the governing equations for the coronary artery (CA) system.The non-linear dynamic behavior of the system is then investigated by means of phase portraits, power spectra, bifurcation diagrams, and Poincaré maps.In addition, a master-slave control system is proposed to suppress the non-linear 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.

Nonlinear Dynamic Analysis
2.1.Mathematical Modeling.The governing equations of the CA system were originally derived by Guan [8] in 2002 and were subsequently converted to the following form by Gong et al. [9] a few years later: where  is the variation of the blood vessel diameter,  is the change in the blood pressure,  cos  is the external disturbance factor acting on the blood vessels, and , , and  are the system parameters.
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 master-slave 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 1.

Differential Transformation Method.
Applying the differential transformation method [10][11][12] to (2) with respect to the time domain , the two state equations of the coronary artery system become respectively (note that the fundamental properties of the Taylor transformation are provided in the appendix).
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 time-series data for parameters  and  of the CA system.Note that, sin producing the various plots, the time-series data corresponding to the first 1000 revolutions are deliberately excluded in order to ensure that the results relate to steady-state conditions.

Numerical Results for Nonlinear Dynamic Behavior
2.3.1.Phase Portraits and Power Spectra. Figure 1 shows the phase portraits of  and  for various values of the vibrational amplitude, .Figures 1(a) and 1(c) show that the orbit is regular at  = 0.1 and 4.3 but is irregular and nonperiodic at  = 0.3 and 4.5 as shown in Figures 1(b) and 1(d).Furthermore, at higher values of the vibrational amplitude, that is,  = 5.9 and 16.3, respectively, the orbit exhibits regular, periodic motion (see Figures 1(e) and 1(f)).
Figures 2(a)-2(f) present the power spectra for the variation of the blood vessel diameter and the change in blood pressure of the CA system.It is seen that, for vibrational amplitudes of  = 0.1 and 4.3, the CA system performs periodic and 8-periodic motion.However, for vibrational amplitudes of  = 0.3 and 4.5, the system exhibits chaotic motion.Finally, for vibrational amplitudes of  = 5.9 and 16.3, the system exhibits multiperiodic behavior.show that, at lower values of the vibrational amplitude, that is,  < 0.3, the variation of the bloodvessel diameter () and the change in blood pressure () both exhibit a dynamic periodic response.Figure 4(a) presents the Poincaré map corresponding to  = 0.1.It is seen that the orbit behaves periodic motion as shown and proven in Figure 1(a) and corresponding to a single point on Poincaré map.As the value of the vibrational amplitude is increased from  = 0.3 to  = 0.59, the system performs chaotic motion, as shown in Figure 4(b).At  = 0.6, the chaotic motion is replaced by -periodic motion.Figure 3 shows that this -periodic motion is maintained for all values of the vibrational amplitude in the range of 0.6 ≤  < 2.1.However, as the vibrational amplitude is increased to  = 3.8, the -periodic motion loses its stability and is replaced by 2-periodic (subharmonic) motion.As  is further increased over the interval 3.8 ≤  < 4.5, the system exhibits multiperiodic motion comprising both 4and 8-periodic motions, as shown in Figures 4(c) and 4(d).However, at  = 4.5, the multiperiodic motion is replaced by chaotic motion [13,14].For values of the vibrational amplitude in the range of 4.5 ≤  < 5.9, the system performs chaotic motion (see Figure 4(e)).However, as  is increased over the interval 5.9 ≤  ≤ 20.0, the system exhibits multiperiodic motion once again, including 8-periodic, 6-periodic, 3-periodic, and 4-periodic, as shown in Figure 4(f), corresponding to  = 16.3.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  = 0.1 to  = 20.0 are summarized in Table 2.In general, the results show that, depending on the value of the vibrational amplitude, the CA system may exhibit periodic behavior, that is, -, 2-, 4-, or 8-periodic motion, or a chaotic response.In addition, it is noted that an explosive bifurcation occurs at a vibrational amplitude of  = 0.3; the system has a chaotic state with windows of periodic motion.

Chaos Synchronization Control
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   4T abnormal coronary artery system to a normal coronary artery system.

Control System Description.
Consider the following master-slave coronary artery systems.
Master system is and Slave system: where   , and   ( = 1, 2) are state variables.It is assumed that the master system ( 4) is a normal coronary artery system and demonstrates chaotic motion in the region of  1 ∈ [0.3, 0.6).
The slave system ( 5) is an abnormal coronary artery system and also demonstrates chaotic motion in the region of  2 ∈ [4.5, 5.9).() in ( 5) is a control input and Δ is a bounded unmodelled system structure; that is, |Δ| ≤  ( is positive).Now define the error states as The dynamics of the error system is determined directly from ( 4)-( 5) as follows: From [15], it can be seen that, if (7a) and (7b) are asymptotically stable, then the error states  1 and  2 will approach zero and the systems (4) and (5) will be synchronized.Therefore, the abnormal chaotic coronary artery system will be controlled to the normal chaotic coronary artery system.The considered goal is that, for any given chaotic coronary artery systems as (4) and ( 5), a controller is designed such that the resulting tracking error can be driven to zero; that is, lim In this study, a fuzzy logic controller (FLC) by Yau and Shieh [15] is used to achieve the control goal.It means that the fuzzy logic control input  (e.g., nitroglycerin) is quickly absorbed; blood vessels dilate and increase blood supply to the heart muscle, which effectively relieves or eliminates angina symptoms.
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 () =  eq +   and  eq = − 3  1 +  3 1 + ( 1 −  2 ) cos(); then the error dynamics becomes  According to the state transformation by Yau and Shieh [15], the state transformation is defined as Substituting ( 10) into (9a) and (9b) yields where  1 = ( − ) and  =  + ( + 1).Second, it needs to determine a FLC such that the error dynamic system (11) is asymptotically stable and the error states  1 and  2 will approach zero.Therefore, the FLC design process is the same as Duffing-Holmes system studied in [15].The consequent part in Table 1 is shown in the following: Similarly, we also can show that all the rules in Table 3 also satisfy V < 0 and the proof is omitted.Hence, all of the rules in the FLC can lead to Lyapunov stable subsystems under the same Lyapunov function.Furthermore, the closed-loop rulebased systems (11) are asymptotically stable for each derivate of the Lyapunov function that satisfies V < 0. That is, the error states  1 and  2 guarantee convergence to zero, and the chaotic coronary artery systems (4) and ( 5) are synchronized.

Numerical Results of Synchronization Control.
For the overall control systems ( 4) and ( 5), the parameters are  = 2, and  2 (0) = 0.2 are shown in Figures 5-7. Figure 5 shows that the slave system and the master system can reach synchronization with control operation.In addition, the time responses of error states and control input are shown in Figures 6 and 7.It can be seen that the system error states are regulated to zero asymptotically even when the overall system is undergoing system uncertainty.

Conclusion
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 -periodic to chaotic, -periodic, 2-periodic, and multiperiodic and then from 8-periodic to chaos and is finally transferred to multiperiodic motion with windows of periodic motion.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.

Figure 6 :Figure 7 :
Figure 6: The time response of error states with control () activated at  = 50 sec.

Table 2 :
Variation of coronary artery system response with vibrational amplitude over interval 0.1 ≦  ≦ 20.0.

Table 3 :
Rule table of FLC.