# Chaos Analysis and Synchronization Control of Coronary Artery Systems

Authors: Cheng-Chi Wang; Her-Terng Yau
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

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 master-slave 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 fromT-periodic to 2T-periodic, then from 4T-periodic to 8T-periodic, and finally to chaotic
motion with windows of periodic motion as the vibrational amplitude is increased
from 0.3 to 0.6 and from 4.5 to 5.9. In addition, it is shown that the proposed control
scheme enables the coronary artery system to be synchronized to any state
asymptotically such that the risk of cardiopathy is reduced.


---

## Body

## 1. 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 [1–3]. 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.


## 2. 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:
(1)x˙+bx+cy=0,y˙+(λ+bλ)x+(λ+cλ)y-λx3-Ecosωt=0,
where x is the variation of the blood vessel diameter, y is the change in the blood pressure, Ecosωt is the external disturbance factor acting on the blood vessels, and b, c, and λ 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:(2)x˙=-bx-cy,y˙=-(λ+bλ)x-(λ+cλ)y+λx3+Ecosωt,In designing the proposed controller, the CA system parameters are assigned the values shown in Table1.Table 1
System parameters [9].
Parameter
b
C
λ
ω
Value
0.15
−1.7
−0.65
1


### 2.2. Differential Transformation Method

Applying the differential transformation method [10–12] to (2) with respect to the time domain t, the two state equations of the coronary artery system become
(3)k+1HX-(k+1)=-bX-(k)-cY-(k),k+1HY-(k+1)=-(λ+bλ)X-(k)-(λ+cλ)Y-(k)+λX-⊗X-⊗X-+E(ωH)k!cos(πk2),
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 parametersx and y 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.


### 2.3. Numerical Results for Nonlinear Dynamic Behavior

#### 2.3.1. Phase Portraits and Power Spectra

Figure1 shows the phase portraits of X and Y for various values of the vibrational amplitude, E. Figures 1(a) and 1(c) show that the orbit is regular at E=0.1 and 4.3 but is irregular and nonperiodic at E=0.3 and 4.5 as shown in Figures 1(b) and 1(d). Furthermore, at higher values of the vibrational amplitude, that is, E=5.9 and 16.3, respectively, the orbit exhibits regular, periodic motion (see Figures 1(e) and 1(f)).Phase portraits of coronary artery system atE= (a) 0.1, (b) 0.3, (c) 4.3, (d) 4.5, (e) 5.9, and (f) 16.3.
(a)
(b)
(c)
(d)
(e)
(f)Figures2(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 E=0.1 and 4.3, the CA system performs T-periodic and 8T-periodic motion. However, for vibrational amplitudes of E=0.3 and 4.5, the system exhibits chaotic motion. Finally, for vibrational amplitudes of E=5.9 and 16.3, the system exhibits multiperiodic behavior.Power spectra of coronary artery system atE= (a) 0.1, (b) 0.3, (c) 4.3, (d) 4.5, (e) 5.9, and (f) 16.3.
(a)
(b)
(c)
(d)
(e)
(f)


#### 2.3.2. Bifurcation Analysis

Figures3(a) and 3(b) show the bifurcation diagrams for the variation of the blood vessel diameter and change in blood pressure, respectively. In addition, Figures 4(a)–4(f) present the Poincaré maps of the system trajectory given vibrational amplitudes of E = 0.1, 0.3, 0.6, 2.1, 3.8, 4.3, 4.4, 4.5, 5.9, 6.0, 6.2, and 16.3, respectively. Figures 3(a) and 3(b) show that, at lower values of the vibrational amplitude, that is, E<0.3, the variation of the bloodvessel diameter (X) and the change in blood pressure (Y) both exhibit a dynamic periodic response. Figure 4(a) presents the Poincaré map corresponding to E=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 E=0.3 to E=0.59, the system performs chaotic motion, as shown in Figure 4(b). At E=0.6, the chaotic motion is replaced by T-periodic motion. Figure 3 shows that this T-periodic motion is maintained for all values of the vibrational amplitude in the range of 0.6≤E<2.1. However, as the vibrational amplitude is increased to E=3.8, the T-periodic motion loses its stability and is replaced by 2T-periodic (subharmonic) motion. As E is further increased over the interval 3.8≤E<4.5, the system exhibits multiperiodic motion comprising both 4T- and 8T-periodic motions, as shown in Figures 4(c) and 4(d). However, at E=4.5, the multiperiodic motion is replaced by chaotic motion [13, 14]. For values of the vibrational amplitude in the range of 4.5≤E<5.9, the system performs chaotic motion (see Figure 4(e)). However, as E is increased over the interval 5.9≤E≤20.0, the system exhibits multiperiodic motion once again, including 8T-periodic, 6T-periodic, 3T-periodic, and 4T-periodic, as shown in Figure 4(f), corresponding to E=16.3.Bifurcation diagrams for variation of blood vessel diameter(X) and change in blood pressure (Y) using vibrational amplitude (E) as bifurcation parameter: (a) X(nT) and (b) Y(nT).
(a)
(b)Poincaré maps of coronary artery system at different values of vibrational amplitude(E): (a) 0.1, (b) 0.3, (c) 3.8, (d) 4.3, (e) 4.5, and (f) 16.3.
(a)
(b)
(c)
(d)
(e)
(f)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 fromE=0.1 to E=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, T-, 2T-, 4T-, or 8T-periodic motion, or a chaotic response. In addition, it is noted that an explosive bifurcation occurs at a vibrational amplitude of E=0.3; the system has a chaotic state with windows of periodic motion.Table 2
Variation of coronary artery system response with vibrational amplitude over interval0.1≦E≦20.0.
E
Dynamic behavior
[
0.1,0.3
)
T
[
0.3,0.6
)
Chaos
[
0.6,2.1
)
T
[
2.1,3.8
)
2T
[
3.8,4.3
)
4T
[
4.3,4.5
)
8T
[
4.5,5.9
)
Chaos
[
5.9,6.0
)
8T
[
6.0,6.2
)
6T
[
6.2,16.3
)
3T
[
16.3,20.0
]
4T


## 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:
(1)x˙+bx+cy=0,y˙+(λ+bλ)x+(λ+cλ)y-λx3-Ecosωt=0,
where x is the variation of the blood vessel diameter, y is the change in the blood pressure, Ecosωt is the external disturbance factor acting on the blood vessels, and b, c, and λ 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:(2)x˙=-bx-cy,y˙=-(λ+bλ)x-(λ+cλ)y+λx3+Ecosωt,In designing the proposed controller, the CA system parameters are assigned the values shown in Table1.Table 1
System parameters [9].
Parameter
b
C
λ
ω
Value
0.15
−1.7
−0.65
1


## 2.2. Differential Transformation Method

Applying the differential transformation method [10–12] to (2) with respect to the time domain t, the two state equations of the coronary artery system become
(3)k+1HX-(k+1)=-bX-(k)-cY-(k),k+1HY-(k+1)=-(λ+bλ)X-(k)-(λ+cλ)Y-(k)+λX-⊗X-⊗X-+E(ωH)k!cos(πk2),
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 parametersx and y 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.


## 2.3. Numerical Results for Nonlinear Dynamic Behavior

### 2.3.1. Phase Portraits and Power Spectra

Figure1 shows the phase portraits of X and Y for various values of the vibrational amplitude, E. Figures 1(a) and 1(c) show that the orbit is regular at E=0.1 and 4.3 but is irregular and nonperiodic at E=0.3 and 4.5 as shown in Figures 1(b) and 1(d). Furthermore, at higher values of the vibrational amplitude, that is, E=5.9 and 16.3, respectively, the orbit exhibits regular, periodic motion (see Figures 1(e) and 1(f)).Phase portraits of coronary artery system atE= (a) 0.1, (b) 0.3, (c) 4.3, (d) 4.5, (e) 5.9, and (f) 16.3.
(a)
(b)
(c)
(d)
(e)
(f)Figures2(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 E=0.1 and 4.3, the CA system performs T-periodic and 8T-periodic motion. However, for vibrational amplitudes of E=0.3 and 4.5, the system exhibits chaotic motion. Finally, for vibrational amplitudes of E=5.9 and 16.3, the system exhibits multiperiodic behavior.Power spectra of coronary artery system atE= (a) 0.1, (b) 0.3, (c) 4.3, (d) 4.5, (e) 5.9, and (f) 16.3.
(a)
(b)
(c)
(d)
(e)
(f)


### 2.3.2. Bifurcation Analysis

Figures3(a) and 3(b) show the bifurcation diagrams for the variation of the blood vessel diameter and change in blood pressure, respectively. In addition, Figures 4(a)–4(f) present the Poincaré maps of the system trajectory given vibrational amplitudes of E = 0.1, 0.3, 0.6, 2.1, 3.8, 4.3, 4.4, 4.5, 5.9, 6.0, 6.2, and 16.3, respectively. Figures 3(a) and 3(b) show that, at lower values of the vibrational amplitude, that is, E<0.3, the variation of the bloodvessel diameter (X) and the change in blood pressure (Y) both exhibit a dynamic periodic response. Figure 4(a) presents the Poincaré map corresponding to E=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 E=0.3 to E=0.59, the system performs chaotic motion, as shown in Figure 4(b). At E=0.6, the chaotic motion is replaced by T-periodic motion. Figure 3 shows that this T-periodic motion is maintained for all values of the vibrational amplitude in the range of 0.6≤E<2.1. However, as the vibrational amplitude is increased to E=3.8, the T-periodic motion loses its stability and is replaced by 2T-periodic (subharmonic) motion. As E is further increased over the interval 3.8≤E<4.5, the system exhibits multiperiodic motion comprising both 4T- and 8T-periodic motions, as shown in Figures 4(c) and 4(d). However, at E=4.5, the multiperiodic motion is replaced by chaotic motion [13, 14]. For values of the vibrational amplitude in the range of 4.5≤E<5.9, the system performs chaotic motion (see Figure 4(e)). However, as E is increased over the interval 5.9≤E≤20.0, the system exhibits multiperiodic motion once again, including 8T-periodic, 6T-periodic, 3T-periodic, and 4T-periodic, as shown in Figure 4(f), corresponding to E=16.3.Bifurcation diagrams for variation of blood vessel diameter(X) and change in blood pressure (Y) using vibrational amplitude (E) as bifurcation parameter: (a) X(nT) and (b) Y(nT).
(a)
(b)Poincaré maps of coronary artery system at different values of vibrational amplitude(E): (a) 0.1, (b) 0.3, (c) 3.8, (d) 4.3, (e) 4.5, and (f) 16.3.
(a)
(b)
(c)
(d)
(e)
(f)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 fromE=0.1 to E=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, T-, 2T-, 4T-, or 8T-periodic motion, or a chaotic response. In addition, it is noted that an explosive bifurcation occurs at a vibrational amplitude of E=0.3; the system has a chaotic state with windows of periodic motion.Table 2
Variation of coronary artery system response with vibrational amplitude over interval0.1≦E≦20.0.
E
Dynamic behavior
[
0.1,0.3
)
T
[
0.3,0.6
)
Chaos
[
0.6,2.1
)
T
[
2.1,3.8
)
2T
[
3.8,4.3
)
4T
[
4.3,4.5
)
8T
[
4.5,5.9
)
Chaos
[
5.9,6.0
)
8T
[
6.0,6.2
)
6T
[
6.2,16.3
)
3T
[
16.3,20.0
]
4T


## 2.3.1. Phase Portraits and Power Spectra

Figure1 shows the phase portraits of X and Y for various values of the vibrational amplitude, E. Figures 1(a) and 1(c) show that the orbit is regular at E=0.1 and 4.3 but is irregular and nonperiodic at E=0.3 and 4.5 as shown in Figures 1(b) and 1(d). Furthermore, at higher values of the vibrational amplitude, that is, E=5.9 and 16.3, respectively, the orbit exhibits regular, periodic motion (see Figures 1(e) and 1(f)).Phase portraits of coronary artery system atE= (a) 0.1, (b) 0.3, (c) 4.3, (d) 4.5, (e) 5.9, and (f) 16.3.
(a)
(b)
(c)
(d)
(e)
(f)Figures2(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 E=0.1 and 4.3, the CA system performs T-periodic and 8T-periodic motion. However, for vibrational amplitudes of E=0.3 and 4.5, the system exhibits chaotic motion. Finally, for vibrational amplitudes of E=5.9 and 16.3, the system exhibits multiperiodic behavior.Power spectra of coronary artery system atE= (a) 0.1, (b) 0.3, (c) 4.3, (d) 4.5, (e) 5.9, and (f) 16.3.
(a)
(b)
(c)
(d)
(e)
(f)


## 2.3.2. Bifurcation Analysis

Figures3(a) and 3(b) show the bifurcation diagrams for the variation of the blood vessel diameter and change in blood pressure, respectively. In addition, Figures 4(a)–4(f) present the Poincaré maps of the system trajectory given vibrational amplitudes of E = 0.1, 0.3, 0.6, 2.1, 3.8, 4.3, 4.4, 4.5, 5.9, 6.0, 6.2, and 16.3, respectively. Figures 3(a) and 3(b) show that, at lower values of the vibrational amplitude, that is, E<0.3, the variation of the bloodvessel diameter (X) and the change in blood pressure (Y) both exhibit a dynamic periodic response. Figure 4(a) presents the Poincaré map corresponding to E=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 E=0.3 to E=0.59, the system performs chaotic motion, as shown in Figure 4(b). At E=0.6, the chaotic motion is replaced by T-periodic motion. Figure 3 shows that this T-periodic motion is maintained for all values of the vibrational amplitude in the range of 0.6≤E<2.1. However, as the vibrational amplitude is increased to E=3.8, the T-periodic motion loses its stability and is replaced by 2T-periodic (subharmonic) motion. As E is further increased over the interval 3.8≤E<4.5, the system exhibits multiperiodic motion comprising both 4T- and 8T-periodic motions, as shown in Figures 4(c) and 4(d). However, at E=4.5, the multiperiodic motion is replaced by chaotic motion [13, 14]. For values of the vibrational amplitude in the range of 4.5≤E<5.9, the system performs chaotic motion (see Figure 4(e)). However, as E is increased over the interval 5.9≤E≤20.0, the system exhibits multiperiodic motion once again, including 8T-periodic, 6T-periodic, 3T-periodic, and 4T-periodic, as shown in Figure 4(f), corresponding to E=16.3.Bifurcation diagrams for variation of blood vessel diameter(X) and change in blood pressure (Y) using vibrational amplitude (E) as bifurcation parameter: (a) X(nT) and (b) Y(nT).
(a)
(b)Poincaré maps of coronary artery system at different values of vibrational amplitude(E): (a) 0.1, (b) 0.3, (c) 3.8, (d) 4.3, (e) 4.5, and (f) 16.3.
(a)
(b)
(c)
(d)
(e)
(f)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 fromE=0.1 to E=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, T-, 2T-, 4T-, or 8T-periodic motion, or a chaotic response. In addition, it is noted that an explosive bifurcation occurs at a vibrational amplitude of E=0.3; the system has a chaotic state with windows of periodic motion.Table 2
Variation of coronary artery system response with vibrational amplitude over interval0.1≦E≦20.0.
E
Dynamic behavior
[
0.1,0.3
)
T
[
0.3,0.6
)
Chaos
[
0.6,2.1
)
T
[
2.1,3.8
)
2T
[
3.8,4.3
)
4T
[
4.3,4.5
)
8T
[
4.5,5.9
)
Chaos
[
5.9,6.0
)
8T
[
6.0,6.2
)
6T
[
6.2,16.3
)
3T
[
16.3,20.0
]
4T


## 3. 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 abnormal coronary artery system to a normal coronary artery system.


### 3.1. Control System Description

Consider the following master-slave coronary artery systems.Master system is(4)x˙1=-bx1-cx2,x˙2=-(b+1)λx1-(c+1)λx2+λx13+E1cosωt
and Slave system:
(5)y˙1=-by1-cy2,y˙2=-(b+1)λy1-(c+1)λy2+λy13+E2cosωt+Δf+u,
where xi, and yi(i=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 E1∈[0.3,0.6). The slave system (5) is an abnormal coronary artery system and also demonstrates chaotic motion in the region of E2∈[4.5,5.9). u(t) in (5) is a control input and Δf is a bounded unmodelled system structure; that is, |Δf|≤α (α is positive). Now define the error states as
(6)e1=y1-x1;e2=y2-x2.
The dynamics of the error system is determined directly from (4)-(5) as follows:(7a)e˙1=-be1-ce2,(7b)e˙2=-(b+1)λe1-(c+1)λe2+λy13-λx13+(E2-E1)cos(ωt)+Δf+u.From [15], it can be seen that, if (7a) and (7b) are asymptotically stable, then the error states e1 and e2 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,
(8)limt→∞|ei|⟶0,i=1,2.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 u (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 inputu(t)=ueq+uL and ueq=-λy13+λx13+(E1-E2)cos(ωt); then the error dynamics becomes(9a)e˙1=-be1-ce2,(9b)e˙2=-(b+1)λe1-(c+1)λe2+Δf+uL.According to the state transformation by Yau and Shieh [15], the state transformation is defined as
(10)[e-1e-2]=[-1c0bc1][e1e2].
Substituting (10) into (9a) and (9b) yields
(11)e-1˙=e-2,e-˙2=p1e-1-pe-2+Δf+uL,
where p1=(b-c)λ and p=b+(c+1)λ. Second, it needs to determine a FLC such that the error dynamic system (11) is asymptotically stable and the error states e1 and e2 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:
(12)uL1=uL4=uL7=(p1-1)e-1+pe-2-α,uL3=uL6=uL9=(p1-1)e-1+pe-2+α,uL2=-sgn(e2)+p1e-1+pe-2-α,uL8=-sgn(e2)+p1e-1+pe-2+α,uL5=0.
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 rule-based systems (11) are asymptotically stable for each derivate of the Lyapunov function that satisfies V˙<0. That is, the error states e1 and e2 guarantee convergence to zero, and the chaotic coronary artery systems (4) and (5) are synchronized.Table 3
Rule table of FLC.
Rule
Antecedent
Consequent
e
1
e
2
u
L
i
1
P
P
u
L
1
2
P
Z
u
L
2
3
P
N
u
L
3
4
Z
P
u
L
4
5
Z
Z
u
L
5
6
Z
N
u
L
6
7
N
P
u
L
7
8
N
Z
u
L
8
9
N
N
u
L
9


### 3.2. Numerical Results of Synchronization Control

For the overall control systems (4) and (5), the parameters are b=0.15,  c=-1.7,  λ=-0.65,  E1=0.3,  E2=4.5, and ω=1; the master system (4) displaces chaotic behavior. It is supposed that the uncertainty Δf=0.2·cos(πy1); that is |Δf|≤α=0.2. The simulation results with initial conditions x1(0)=0.4,x2(0)=-0.1,  y1(0)=0.2, and y2(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.Time responses of chaos synchronization of coronary artery systems: master and slave system outputs arex1, x2 (solid) and y1, y2 (dashed), respectively. The control u(t) is activated at t=50 sec.
(a)
(b)Figure 6
The time response of error states with controlu(t) activated at t=50 sec.Figure 7
The control action versus time with controlu(t) activated at t=50 sec.


## 3.1. Control System Description

Consider the following master-slave coronary artery systems.Master system is(4)x˙1=-bx1-cx2,x˙2=-(b+1)λx1-(c+1)λx2+λx13+E1cosωt
and Slave system:
(5)y˙1=-by1-cy2,y˙2=-(b+1)λy1-(c+1)λy2+λy13+E2cosωt+Δf+u,
where xi, and yi(i=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 E1∈[0.3,0.6). The slave system (5) is an abnormal coronary artery system and also demonstrates chaotic motion in the region of E2∈[4.5,5.9). u(t) in (5) is a control input and Δf is a bounded unmodelled system structure; that is, |Δf|≤α (α is positive). Now define the error states as
(6)e1=y1-x1;e2=y2-x2.
The dynamics of the error system is determined directly from (4)-(5) as follows:(7a)e˙1=-be1-ce2,(7b)e˙2=-(b+1)λe1-(c+1)λe2+λy13-λx13+(E2-E1)cos(ωt)+Δf+u.From [15], it can be seen that, if (7a) and (7b) are asymptotically stable, then the error states e1 and e2 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,
(8)limt→∞|ei|⟶0,i=1,2.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 u (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 inputu(t)=ueq+uL and ueq=-λy13+λx13+(E1-E2)cos(ωt); then the error dynamics becomes(9a)e˙1=-be1-ce2,(9b)e˙2=-(b+1)λe1-(c+1)λe2+Δf+uL.According to the state transformation by Yau and Shieh [15], the state transformation is defined as
(10)[e-1e-2]=[-1c0bc1][e1e2].
Substituting (10) into (9a) and (9b) yields
(11)e-1˙=e-2,e-˙2=p1e-1-pe-2+Δf+uL,
where p1=(b-c)λ and p=b+(c+1)λ. Second, it needs to determine a FLC such that the error dynamic system (11) is asymptotically stable and the error states e1 and e2 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:
(12)uL1=uL4=uL7=(p1-1)e-1+pe-2-α,uL3=uL6=uL9=(p1-1)e-1+pe-2+α,uL2=-sgn(e2)+p1e-1+pe-2-α,uL8=-sgn(e2)+p1e-1+pe-2+α,uL5=0.
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 rule-based systems (11) are asymptotically stable for each derivate of the Lyapunov function that satisfies V˙<0. That is, the error states e1 and e2 guarantee convergence to zero, and the chaotic coronary artery systems (4) and (5) are synchronized.Table 3
Rule table of FLC.
Rule
Antecedent
Consequent
e
1
e
2
u
L
i
1
P
P
u
L
1
2
P
Z
u
L
2
3
P
N
u
L
3
4
Z
P
u
L
4
5
Z
Z
u
L
5
6
Z
N
u
L
6
7
N
P
u
L
7
8
N
Z
u
L
8
9
N
N
u
L
9


## 3.2. Numerical Results of Synchronization Control

For the overall control systems (4) and (5), the parameters are b=0.15,  c=-1.7,  λ=-0.65,  E1=0.3,  E2=4.5, and ω=1; the master system (4) displaces chaotic behavior. It is supposed that the uncertainty Δf=0.2·cos(πy1); that is |Δf|≤α=0.2. The simulation results with initial conditions x1(0)=0.4,x2(0)=-0.1,  y1(0)=0.2, and y2(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.Time responses of chaos synchronization of coronary artery systems: master and slave system outputs arex1, x2 (solid) and y1, y2 (dashed), respectively. The control u(t) is activated at t=50 sec.
(a)
(b)Figure 6
The time response of error states with controlu(t) activated at t=50 sec.Figure 7
The control action versus time with controlu(t) activated at t=50 sec.


## 4. 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 fromT-periodic to chaotic, T-periodic, 2T-periodic, and multiperiodic and then from 8T-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.



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*Source: 209718-2013-11-13.xml*