Chaotic analysis of Atangana–Baleanu derivative fractional order Willis aneurysm system
School of Science, Wuhan University of Technology, Wuhan 430070, China
† Corresponding author. E-mail:
hgaofei@gmail.com
1. IntroductionThe dynamic behaviors of chaotic systems have received increasing attention in recent decades. Some research revolved around the dynamic behavior of the models. In Ref. [1], a new model that can reduce the chaotic region was obtained by introducing a new variable. Wang et al.[2] proposed a pseudo-random coupling method which can accelerate the chaotic behavior of the full space. In Ref. [3], the complex dynamics of a geomagnetic field system were explored by introducing a non-smooth factor. The Willis aneurysm system is a biomedical chaotic system that has attracted researchers’ interest in recent years. Cerebral aneurysms are prone to rupture, which leads to high disability and mortality,[4] and the circle of Willis is a high incidence area of cerebral aneurysms. Therefore, it is necessary to predict and control the formation and growth of cerebral aneurysms by analyzing the chaotic dynamics of the Willis aneurysms system. The research on the Wills aneurysm system can be divided into the following three aspects: periodic solution, control method, and fractional order system.
In Ref. [5], Austin explained that a physical circuit can be used to simulate the flow of blood in a brain aneurysm, and a biomechanical model of Willis cerebral aneurysm was presented. This model reveals the non-linear relationship between the volumes and the pressure in the cerebral aneurysm. Some information to prevent rupture and swelling of the cerebral aneurysm was presented. However, the Austin model has limitations due to the omission of damping and certain constraints. Therefore, in Refs. [6] an [7], a Duffing equation with damping terms and forced external excitation was obtained by reanalyzing the analog circuit of Austin. Some studies were based on this new equation. Cao et al.[8] proved that the equation has at least one periodic solution under certain conditions. The condition of Devaney chaos in the Willis brain aneurysm model was obtained by the method of Melnikov function in Ref. [9]. The condition for the existence of the periodic solution of the system was obtained by Feng.[10] Nieto et al.[11] considered the existence of the solution of the periodic boundary value problem for the Willis aneurysm system.
The chaotic state of the Willis aneurysm system is harmful and chaos should be avoided as much as possible. In view of this problem, some scholars have discussed the control of the Willis aneurysm system. For the variable parameter Willis aneurysm system, based on the adaptive control idea, synchronization of two parameters of the Willis aneurysm system was obtained through the construction parameter adaptive law in Ref. [12]. Aiming at the control problem of the Willis aneurysm system with uncertainty, Peng et al.[13] proposed an adaptive fuzzy sliding mode control method. Sun et al.[14] studied a new Willis ring brain aneurysm mathematical model by introducing the antihypertensive drug excitation term function. The control method of antihypertensive drugs was presented.
All the above literatures only focussed on the integer-order Willis aneurysm system. Considering that the integer-order Willis aneurysm system has certain defects in describing the complex dynamics of blood flow, Cao et al.[15] studied a fractional order Willis aneurysm system based on Caputo derivative. This system was effectively controlled by the method of designing a controller and the drug excitation term function.
Many researches have analyzed the merits and drawbacks of fractional derivatives and put forward different definitions. Riemann–Liouville fractional derivative and Caputo fractional derivative are based on the power law functions. These two fractional derivatives have been in practice for a long time and applied to various practical problems successfully.[16,17] However, their kernel has a singularity, the kernel of Caputo derivative is a singular and the derivative of a constant is not zero in case of Riemann–Liouville fractional derivative.[18,19] Therefore, Caputo and Fabrizio[18] presented a new definition having non-singularity in the kernel to overcome the problem in 2015. Some research[20,21] revolved around the Caputo–Fabrizio fractional derivative. Besides the definition of Caputo–Fabrizio fractional derivative, Atangana and Baleanu[22] presented a new fractional definition in 2016. The Atangana–Baleanu (AB) fractional-order derivative is known to possess nonsingularity as well as nonlocality of the kernel, which adopts the generalized Mittag–Leffler function. The AB fractional derivative has all the benefits of the above fractional derivatives.[23] Recently, the Atangana–Baleanu fractional derivative has been applied in many fields, for instance, in biology and ecology,[24,25] electrical circuit,[26–28] fluid model,[29–32] groundwater,[33,34] and chaos theory[35,36] among many others.
Firstly, blood as a viscoelastic body[37] in soft matter[38] has the fluid viscosity and the elasticity of solids. Thus it has a complex flow pattern. Besides, the vascular system is a relatively complicated elastic piping system. Secondly, the brain aneurysm is a swelling area in the cerebral artery wall, and the system of Willis aneurysm involves the state and nature of the blood and vascular system. Therefore, the Willis aneurysm system has a complex dynamic mechanism. Finally, the asymptotic behavior of AB derivative makes it more advantageous than simple Caputo derivative in simulating complex phenomena.[39] The fractional order Willis aneurysm system with AB derivative has not considered before. Therefore, on the basis of the above three points, we consider a fractional order Willis aneurysm system using the new AB fractional derivative.
The main structure of this paper is as follows. The basic definitions of AB derivative are presented in Section 2. In Section 3, the Atangana–Baleanu fractional Willis aneurysm system (ABWAS) is created and the simulation results show that the ABWAS system has the chaotic phenomenon. In addition, we compare the effects of system order and pulse pressure on ABWAS and Caputo fractional WAS (FWAS). The effects of blood flow damping coefficient and reciprocal heart rate on the ABWAS are also given. In Section 4, the control of ABWAS is studied by the method of designing a reasonable controller through stability conditions. Furthermore, we compare the control effect for ABWAS and FWAS. Finally, this article is concluded in Section 5.
2. PreliminariesThe definitions of the Atangana–Baleanu fractional derivative and integral are shown in this section. Besides, we have the conclusion that the AB fractional Willis aneurysm system with initial conditions is stable.
Definition 1[22] For
,
,
, with the function f differentiable, the definition of the new fractional derivative (Atangana–Baleanu derivative in Caputo sense) is given as
where
B satisfies the following
, and
.
Definition 2[22] The fractional integral associate to the new fractional derivative with nonlocal kernel (Atangana–Baleanu fractional integral) is defined as
Theorem 1 The AB fractional system with initial conditions is as follow:
where
satisfies the Lipschitz condition and
is always zero. When the system order is in the interval (0, 1), for any state variable
,
, the system (
3) is stable under the conditions that we can find positive definite matrix
such that
Proof Let
, according to the AB fractional integral, the solution of system (3) is shown as follow:
According to the definition of the derivative, we obtain
The above equation is simplified as follow:
Thus,
Constructing a positive definite function
, we have
By using the assumptions of the theorem, the following formula can be derived:
satisfies the Lipschitz condition, therefore
According to the Lyapunov stability theory,
. Therefore,
and the system (
3) is stable. This completes the proof.
3. Willis aneurysm system with Atangana–Baleanu derivative3.1. Establishment of ABWASFrom Ref. [15], the Caputo fractional Willis aneurysm system is presented as follow:
where
x denotes the blood flow velocity,
y represents the blood flow change rate,
F is the pulse pressure,
μ stands for the damping coefficient of the blood flow,
ω denotes the reciprocal of heart rate. The parameters
α,
β, and
γ are related to the blood flow resistance and the elasticity of the vessel wall. These parameters are related to the human physiological structure and aneurysm status.
By changing the Caputo fractional differential of system (13) into the Atangana-Beleanu fractional differential, we obtain the ABWAS
3.2. Numerical approximations of ABWASFor the sake of simplicity, equation (14) is rewritten as follow:
Applying the Atangana–Baleanu fractional integral to system (
15), we obtain
For the ABWAS system (
15), by employing the prediction correction method,
[40] the iterative formulas are derived as follows:
The iteration formulas can further be simplified as follows:
3.3. Comparison of results between FWAS and ABWASFor system (14) and system (13), the parameters are taken as follows: α =0.9, β =3, γ =2, F=0.11, μ =0.1, ω =1. If there are dense points and these points have a hierarchical structure on the Poincaré section, then it can be judged that the system is in a chaotic state. Figure 1 shows the curves of x–t and y–t, time course, phase diagram, and Poincaré section of system (14) for q1 =1.02, q2 =0.97. As can be seen in the Poincaré section (Fig. 1(d)), system (14) exhibits complex chaos.
Since systems (13) and (14) are affected by many parameters, it is difficult to find the parameters that make the results of the two systems completely equal. However, when the parameters of system (14) are taken as follows: q1 =1.02, q2 =0.97, α =0.9, β =3, γ =2, F=0.11, μ =0.1, ω =1; and the parameters of system (13) are taken as follows: q1 =1.03, q2 =0.96, α =0.9, β =3, γ =2, F=0.11, μ =0.1, ω =1, we obtain approximately the same results of the two systems. Figure 2 shows the phase diagram and Poincaré section of the two systems when the parameters are set as above. From Fig. 2, it can be seen that the phase diagram and Poincaré section of systems (14) and (13) are basically coincident in shape and structure.
In order to compare the results of FWAS and ABWAS when
,
, we consider the cases with F=0.095 (Figs. 3(a) and 3(d)), F=0.097 (Figs. 3(b) and 3(e)), and F=0.1 (Figs. 3(c) and 3(f)). In addition, other parameters are unchanged. The first three figures in Fig. 3 show the the phase diagram of FWAS when
,
. In the last three figures of Fig. 3, we present the phase diagram of ABWAS when
,
. The results of Fig. 3 show that the phase diagram structure is basically the same when the system order of ABWAS and FWAS is equal to 1. However, when the value of F is determined, there are numerical differences between the phase diagrams of the two systems.
3.4. Effects of pulse pressure on ABWAS and FWASThe following system is obtained with the parameters of q1 =1.02, q2 =0.97, α =0.9, β =3, γ =2, μ =0.1, ω =1:
The influence of pulse pressure here mainly refers to the influence of blood pressure on ABWAS. Figure
4 shows the
x–
F and
y–
F bifurcation diagrams (system (
25)), where
F is in the range of [0.05, 0.2]. From Fig.
4, we can see that small changes in the pulse pressure
F can significantly affect the chaotic state. The system (
25) is chaotic in the interval
(Figs.
4(a), and
4(b)). The results in Fig.
4 show that when
F is greater than 0.121 the system (
25) is characterized as stable behavior.
For system (13), we set the same parameters as those of system (25). The results of the numerical simulation are shown in Fig. 5. The system (13) is chaotic in the interval
(Fig. 5(a)). In addition, the system (13) is chaotic in the interval
(Fig. 5(b)). It is clear that if F is greater than 0.124 (Fig. 5(a)) or 0.123 (Fig. 5(b)) the system (13) is characterized as stable behavior.
It can be seen that when the blood pressure is stable, the blood flow in the cerebral aneurysm is relatively stable. Conversely, the blood flow in the cerebral aneurysm is chaotic, and the cerebral aneurysm is prone to rupture.
3.5. Influence of system order q1 and q2 on ABWAS and FWASThe effects of fractional order q1 and q2 on system (14) are further discussed for F=0.11 and other parameters remain unchanged. Let the fractional order q1 varied from 0.9 to 1.044. Figure 6(a) shows the bifurcation diagram (system (14)) of x as a function of q1. The system (14) is chaotic in the interval
(Fig. 6(a)). In Fig. 6(b), we present x–q2 bifurcation diagram (system (14)), where q2 is in the range of [0.9, 0.99]. Furthermore, the system (14) is chaotic in the interval
(Fig. 6(b)).
We can learn from Fig. 6 that the small changes of the order q1 and q2 can significantly affect the chaotic state of ABWAS, and the system (14) reaches the chaotic state through the double-cycle bifurcation road.
For system (13), we set the same parameters as those of system (14). The results of the numerical simulation are shown in Fig. 7. The system (13) is chaotic in the intervals
(Fig. 7(a)) and
(Fig. 7(b)). It is clear that when q1 is greater than 1.027 (Fig. 7(a)) or q2 is greater than 0.977 (Fig. 7(b)) the system (13) is characterized as stable behavior.
Compared with that of system (13), the bifurcation diagram of system (14) appears fewer chaotic phenomena.
The system order we select should help us to analyze the dynamic behavior of ABWAS. Further, the chaotic state of ABWAS shows unstable blood flow and rupture of cerebral aneurysms. Therefore, the system order that makes ABWAS in a chaotic state is our focus. It can be seen from Fig. 66 that ABWAS is in chaos when q1 =1.02 and q2 =0.97, thus we choose q1 =1.02 and q2 =0.97 for research.
3.6. Influence of blood flow damping coefficient μ and reciprocal of heart rate ω on ABWASIn this section, the parameters of system (14) are taken as follows: α =0.9, β =3, γ =2, F=0.11, q1 =1.02, q2 =0.97. Let the blood flow damping coefficient μ varied from 0 to 1.2. Figure 8(a) shows the μ–x bifurcation diagrams. In Fig. 8(b), we present the ω–x bifurcation diagram (system (14)), where ω is in the range of [0.01, 2].
In Fig. 8(a), the velocity of blood flow tends to be stable with the increase of the blood flow damping coefficient. In Ref. [41], the authors proposed that coil embolization can reduce blood flow velocity and induce thrombosis in aneurysms, which is consistent with the numerical simulation results of Fig. 8(a). It can be seen from the bifurcation diagram in Fig. 8(b) that the system (14) gradually changes from chaotic state to steady as the reciprocal of heart rate increases (the heart rate decreases). Liu et al. in Ref. [42] explained that drugs that can stabilize or reduce the heart rate have auxiliary effects on the treatment of cerebral aneurysms. This result is consistent with the numerical simulation results of Fig. 8(b). Therefore, it is important for clinical diagnosis to study the effects of blood flow damping coefficient and heart rate derivative on cerebral aneurysms using ABWAS.
4. Control of the chaotic ABWAS and FWASThe chaotic state of ABWAS is characterized by unstable blood flow. Furthermore, cerebral aneurysms are more likely to rupture in extremely chaotic speed state, which can lead to serious consequences such as headache, vomiting, and disturbance of consciousness. Therefore, the system (14) should be controlled as much as possible to avoid the occurrence of chaos. By designing a suitable controller, the chaotic phenomenon of ABWAS is controlled, where the parameters are set according to the stability conditions of the system.
With the same parameters as those in Section 3, the ABWAS with the initial value condition is obtained as follow:
Theorem 2 The AB fractional system (26) with initial conditions is not stable.
Proof According to Theorem 1, we have
Since the value of the function
P(
x,
y) cannot be estimated, the stability of the system (
26) becomes difficult to determine. However, according to the simulation results of Fig.
1, we can conclude that ABWAS (
26) with initial conditions is not stable. Proof is completed.
In the quest to stabilize the system (26), a controller system is introduced as follow:
Theorem 3 The controller-controlled AB fractional system (28) is stable if
,
, and
.
Proof According to Theorem 1, the function P(x,y) is obtained as follow:
The function
for
and
. Therefore, according to Theorem 1, the controller-controlled AB fractional system (
28) is stable. Theorem 3 is verified.
With
and
, the simulation results of system (28) are shown in Fig. 9. From the results of the Poincaré section and phase diagram in Fig. 9, it can be seen that the controlled ABWAS (28) is quickly limited to small-scale range periodic fluctuations and amplitude.
The bifurcation diagrams of the controller-controlled ABWAS (28) with the fractional order q1 and q2 are presented in Fig. 10. For the Caputo fractional Willis aneurysm system (13), we design the same controller as system (28). Furthermore, we set the same parameters as those of system (28) and k1 =0.3, k2 =1.4. The results of the numerical simulation are shown in Fig. 10. Under the influence of q1, the steady blood flow velocity of the controller-controlled ABWAS varies from 0.1024 to 0.0858 and the steady blood flow velocity of the controller-controlled FWAS varies from 0.1025 to 0.0906 (Fig. 10(a)). In addition, when q2 is in the range of 0.9 to 0,99, the maximum steady blood flow velocity is 0.0966 for the controller-controlled ABWAS and equals to 0.0994 for the controller-controlled FWAS. Furthermore, the minimum steady blood flow velocity is 0.0955 and 0.0973 for the controller-controlled ABWAS and the controller-controlled FWAS, respectively. From Fig. 10, we can observe that after introducing the controller, the chaotic interval of system (28) also becomes a stable interval.
Figure 10 present the comparative study between system (28) and the Caputo fractional Willis aneurysm system with same controller. From Fig. 10, we can see that the blood flow velocity (x) of system (28) changes in a smaller steady value compared to the Caputo fractional Willis aneurysm system with the same controller.
5. ConclusionIn view of the fact that the Atangana–Baleanu fractional derivative can better describe the complex dynamics of blood flow, we establish a Willis aneurysm system based on the Atangana–Baleanu fractional derivative. The research results show that ABWAS is chaotic under the certain parameter values. Comparing to the bifurcation diagrams of order and pulse pressure of FWAS, the bifurcation diagrams of ABWAS show that less chaos occurs in the same ranges of order and pulse pressure. Therefore, we conclude that the order and pulse pressure have a significant effect on ABWAS.
With the increase of blood flow damping coefficient μ or reciprocal heart rate ω, the system tends to be stable. These results accord with the clinical treatment of cerebral aneurysms using the method of drugs or coils embolization. Therefore, the study of blood flow damping coefficient and reciprocal heart rate has certain significance for clinical diagnosis and treatment of cerebral aneurysms.
By designing a suitable controller based on the stability theorem, the blood flow velocity of the ABWAS can be stabilized in a short time. In addition, compared with the bifurcation diagrams of the controller-controlled Caputo fractional order Willis aneurysm system, under the same parameters, the blood flow velocity of the controller-controlled ABWAS changes within a smaller numerical interval. Therefore, the control effect on the ABWAS is better. This paper provides a theoretical basis for the study on the chaotic control and treatment of brain aneurysms.
