• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    BIFURCATION CONTROL FOR A FRACTIONAL-ORDER DELAYED SEIR RUMOR SPREADING MODEL WITH INCOMMENSURATE ORDERS?

    2023-12-14 13:07:04葉茂林蔣海軍
    關(guān)鍵詞:葉茂海軍

    (葉茂林) (蔣海軍)

    College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

    E-mail: 1733981567@stu.xju.edu.cn; jianghaijunxju@163.com

    Abstract A fractional-order delayed SEIR rumor spreading model with a nonlinear incidence function is established in this paper,and a novel strategy to control the bifurcation of this model is proposed.First,Hopf bifurcation is investigated by considering time delay as bifurcation parameter for the system without a feedback controller.Then,a state feedback controller is designed to control the occurrence of bifurcation in advance or to delay it by changing the parameters of the controller.Finally,in order to verify the theoretical results,some numerical simulations are given.

    Key words rumor spreading;fractional-order;time delay;bifurcation control

    1 Introduction

    A rumor is an unsubstantiated exposition or interpretation of a matter,event or issue which the public are interested in spreading through various channels.Nowadays,with the rapid development of science and technology,online social media platforms bring great convenience,but also have accelerated the spread of rumors in social media networks.The form of rumor transmission has changed from the traditional word of mouth to various online social platforms such as Twitter,WeChat,Weibo,etc..Rumors can have a great impact on people’s daily life and on social order [1–3].For example,the report that shuanghuanglian oral liquid could effectively suppress the spread of COVID-19 caused a panic buying phenomenon and resulted in the shortage of drugstores.However,shuanghuanglian oral liquid can only be used to clear heat and for detoxify,and further clinical data are needed to determine whether it can actually effectively suppress COVID-19.This event demonstrates that it is of great practical significance to study the dynamics of rumor propagation in social networks.

    Research on the dynamics of rumor propagation dates back to the 1960s.In [4,5],the classic DK model was proposed by Daley and Kendall.It divided the population into three categories–ignorant,spreader and removed– and it used numerical methods similar to those used in the study of infectious diseases to understand the process through which rumors spread.After Daley and Kendall,Maki and Thomson,in 1973,improved the DK model and established the MK model [6].Based on that researches,various compartment models have appeared and been widely used in the study of the rumor propagation process;these include SIR [7,8],SEIR [9],ILSCR [10],etc..In recent years,researchers have continued to improve the rumor propagation model.Wanget al.established a SIR rumor propagation model considering the cross-propagation mechanism in a multi-language environment,then conducted global dynamics analysis and sensitivity analysis on the model [11].Liet al.conducted stability analysis and a sensitivity analysis for the I2S2R rumor propagation model in heterogeneous networks [12].Afassinou analyzed the influence of the education level of social media users on rumor spreading mechanisms [13].

    Time delay is a factor that cannot be ignored with regard to rumor propagation;it can be used to help simulate the process by which rumors spread and the process by which governments and social media platforms educate rumor propagators in order to stop spreading rumors[14–17].Functional differential equations provide a dynamic system with infinite dimensions in essence,and the slight change of system parameters may cause a change of equilibrium stability.When the parameters exceed a certain critical value,a branch of periodic orbits can be separated from the equilibrium point and the Hopf bifurcation phenomenon can be generated.For functional differential equations,the delay is generally taken as a parameter,and the condition of bifurcation is considered when the delay changes.In the field of rumor propagation research,Ankuret al.considered the double time delay caused by expert intervention and government control measures and established the 2SI rumor propagation model,and the critical value of expert intervention delay and Hopf bifurcation conditions were obtained through calculation[18].Wanget al.established the I2S2R rumor propagation model with a time delay in a multi-language environment,and obtained the bifurcation conditions of the system [19].

    Bifurcation control has become more and more popular in recent years.The purpose of a bifurcation controller is to change the dynamic properties of the original solution of the system by considering a control criterion.Various controllers are designed to control the dynamic behavior of systems;these include the Proportional-Derivative feedback controller [20],the state feedback controller [21]and so on.Chenget al.established a complex network model with time delays,then performed bifurcation analysis on the model and proposed a hybrid control strategy in [22].

    Fractional calculus after referred to as generalized calculus or arbitrary calculus is a generalization of integral calculus and has a short-term memory effect and a genetic effect.Fractional differential equations have been used in a wide range of fields[23–26],and mainly includes analysis and synthesis of fractional dynamic systems.Wanget al.established a fraction-order delay SIR epidemic model with saturation incidence and recovery functions and then analyzed the stability of the disease-free equilibrium and disease-prevalence equilibrium of the system[27].In recent years,some scholars considered adding controllers to control the bifurcation that occurrs in fractional systems with a time delay;these include the state feedback controller [28]and the fractional PD controller [29].The influence of a memory effect on information transmission process was considered in [30,37],where it can be seen that multiple redundant contacts of the same rumor will change people’s initial thoughts regarding it,and the cumulative effect will impact upon the behavior of individuals in social networks.Due to the memory effect of fractional calculus,the rumor propagation process can be analyzed accurately by studying the rumor propagation process with fractional calculus.In [38],Singh established a SIR rumor spreading model in a social network with the Atangana-Baleanu derivative:

    At the same time,the dynamic behavior of the model was studied.Renet al.investigated a stochastic SIR model in the sense of Caputo’s fractional derivative for rumor spreading in social networks:

    The stability of the given model’s equilibrium point was studied in [39].However,fractional differential equations have rarely been used in the study of rumor propagation,which has fended to mainly focus on the application of the SIR rumor model.Based on the above work,combining the nonlinear function and time delay,we establish a fraction-order SEIR rumor model in a homogeneous network and add a feedback controller to control the occurrence of bifurcation in advance or delay by changing the parameters of the controller.

    The rest of this article is arranged as follows: in Section 2,some preparations related to fractional equations are introduced.In Section 3,a fraction-order SEIR rumor spreading model is proposed.In Section 4,bifurcation conditions for the uncontrolled system and the controlled system are obtained.Two numerical simulation examples are illustrated in Section 5 in order to verify the theoretical results.In Section 6,a brief summary of the whole paper is presented.

    2 Preliminaries

    In this section,some preparations related to fractional differential equations are given.

    Definition 2.1([31]) The Caputo fractional derivative of orderαof a functionf(x) is defined as

    wherenis the positive integer andn-1<α

    Lemma 2.2([32]) Consider the followingn-dimensional linear fractional differential system with multiple time delays:

    Hereqiis real and lies in(0,1),the initial valuesxi=φi(t)are given for-maxi,j τij=-τmax≤t ≤0 andi=1,2,···,n.In this system,the time-delay matrix isT=(τij)n×n ∈(R+)n×n,the coefficient matrix isA=(aij)n×n,the state variables arexi(t),xi(t-τij)∈R,and initial values areφi(t)∈C0[-τmax,0].Then,the characteristic matrix of system (2.1) can be presented:

    Lemma 2.3([32]) Suppose thatτij=0 and that allqisare rational numbers between 0 and 1 fori,j=1,···,n.LetMbe the lowest common multiple of the denominatorsuisofqis,where,(ui,vi)=1,ui,vi ∈Z+,i=1,···,n.Then the zero solution of system (2.1)is Lyapunov globally asymptotically stable if of all the rootsλsof the equation

    3 Description of the Rumor Spreading Model

    We establish the following fractional-order (0<α1,α2,α3,α4<1) rumor propagation model and study the Hopf bifurcation of the model with time delay:

    Here we have the initial conditions

    (H1)〈k〉A(chǔ)βδ-u(δ+u)(θ+u)>0.

    By using (H1),system (3.1) has a rumor-spreading equilibriumP?(S?,E?,I?,R?),where

    4 Main Result

    In this section,the Hopf bifurcation of system (3.1) atP?is studied;hereτis chosen as a bifurcation parameter,and the bifurcation point is identified.Then,after the controller is added to the original system,τis still selected as the bifurcation parameter and a new bifurcation point is obtained.

    4.1 Bifurcation Induced by τ of the Uncontrolled System (3.1)

    In this subsection,τis chosen as the bifurcation parameter that is used to find the bifurcation point of system (3.1) and to get the conditions of the Hopf bifurcation.

    By applying Lemma 2.2,the characteristic matrix of system (4.2) at (0,0,0) is expressed as follows:

    Whenτ=0,the characteristic equation of system (4.2) can be written as

    From Lemma 2.3,the zero solution of system (4.2) is asymptotically stable when all of solutionλiof the above equation satisfies the condition

    Whenτ>0,the characteristic equation that corresponds to system (4.2) is

    Multiplying both sides of eq.(4.3) by eλτ,we have that

    where Re[mj(iω1)]is the real part ofmj(iω1) and Im[mj(iω1)]is the imaginary part ofmj(iω1)(j=1,2,3).Furthermore,

    By eq.(4.5),we obtain

    Squaring both sides of eq.(4.6),we have that

    Define the bifurcation point

    whereτ0(k)is defined by eq.(4.8).

    Remark 4.1In the same way with eq.(4.8),we can also choose the second equation of eq.(4.7) to calculateτ0(k).

    The following assumption is made to obtain our main results:

    Here Re[ρ1(iω0)],Im[ρ1(iω0)],Re[υ1(iω0)],Im[υ1(iω0)]are mentioned in eq.(4.10).

    Lemma 4.2Letλ(τ)=ξ(τ)+iω1(τ) be the root of eq.(4.4) nearτ=τ0satisfying thatξ(τ0)=0,ω1(τ0)=ω0.Then the following condition holds:

    ProofDifferentiate both sides of eq.(4.4) with respect toτ.Then we get that

    As a result,

    where

    It is deduced from eq.(4.9) that

    where Re[ρ1(iω0)],Im[ρ1(iω0)]are the real and imaginary parts,respectively,ofρ1(iω0).Re[υ1(iω0)],Im[υ1(iω0)]are the real and imaginary parts,respectively,ofυ1(iω0).Furthermore,

    Applying (H2) completes the proof.

    According to Lemmas 2.2 and 2.3,the following results can be obtained:

    Theorem 4.3Under (H1) and (H2),we have that

    (1) the equilibriumP?of system (3.1) is asymptotically stable whenτ ∈[0,τ0);

    (2) the equilibriumP?of system (3.1) is unstable whenτ ∈[τ0,+∞).Furthermore,the uncontrolled system (3.1) undergoes a Hopf bifurcation atP?whenτ=τ0.

    Remark 4.4The stability definition in the Lyapunov sense shows that for linear systems,if the equilibrium is asymptotically stable,then it must be globally asymptotically stable.However,asymptotic stability is not globally asymptotic stability for nonlinear systems.Furthermore,it can be seen that system (3.1) is locally asymptotically stable in the Lyapunov sense whenτ ∈[0,τ0).

    4.2 Bifurcation Induced by τ of the Controlled System

    In recent years,bifurcation control has become a hot research topic.Feedback controllers are used in many fractional systems.In [28],Wanget al.established the following fractional order eco-epidemiological system

    Here,a feedback controllerμ(t)=h[I(t)-I(t-ν)]is designed to control the bifurcation behavior of the system with time delayτas the bifurcation parameter.Similarly,Huanget al.established the fractional predator-prey system in [40]:

    At the same time,an effective extended feedback controlleru(t)=K[x1(t)-x1(t-σ)]can be designed for the above system for controlling the creation of bifurcation.In the field of rumor propagation,Huanget al.established following delay reaction-diffusion malware propagation model in [41]

    Furthermore,a state feedback controlleru(t)=-k1(I-I?)-k2(I-I?)2-k3(I-I?)3is designed to control the creation of the Hopf bifurcation for the given system.However,there has been little research on the application of the state feedback controller to the fractional rumor propagation model.In this subsection,we design the feedback controller

    whereσrepresents the feedback controller delay andKdenotes the feedback gain.

    Remark 4.5In this paper,we only take-1≤K ≤1.It is clear thatη(t)=0 whenK=0 orσ=0.

    In order to control the bifurcation and make the bifurcation advance or delay,τis chosen as the bifurcation parameter to get the conditions of Hopf bifurcation.The model with a feedback controller is as follows:

    Here we have the initial conditions

    By the same linearization treatment as the one mentioned above,the characteristic equation of the controlled system (4.11) can be obtained.

    Remark 4.6For the sake of the simplicity of the derivation,we linearize the controller separately when linearizing the controlled system (4.11).

    For convenience,we still just deal with the first three equations.The characteristic matrix of system (4.12) at (0,0,0) is expressed as follows:

    Hence,the characteristic equation that corresponds to system (4.12) is

    Multiplying both sides of eq.(4.13) by eλτ,we have that

    where Re[nj(iω2)]is the real part ofnj(iω2) and Im[nj(iω2)]is the imaginary part ofnj(iω2)(j=1,2,3).Furthermore,

    By eq.(4.15),we obtain that

    Squaring both sides of eq.(4.16),we have that

    Define the bifurcation point

    whereτ01(k) is defined by eq.(4.18).

    The following assumption is made to obtain our main results:

    Lemma 4.7Letλ(τ)=ξ(τ)+iω2(τ)be the root of eq.(4.14)nearτ=satisfying thatThen the following condition holds:

    ProofDifferentiating both sides of eq.(4.14) with respect toτ,we get

    As a result,

    It can be deduced from eq.(4.19) that

    applying (H3) completes the proof.

    According to Lemmas 2.2 and 4.7,the following results can be concluded:

    Theorem 4.8Under (H1) and (H3),we have that

    (1) the equilibriumP?of system (4.11) is asymptotically stable whenτ ∈[0,);

    (2) the equilibriumP?of system (4.11) is unstable whenτ ∈[,+∞).Furthermore,the controlled system (4.11) undergoes a Hopf bifurcation atP?whenτ=.

    Remark 4.9Compared with the results of [19],the ordinary differential equation theory is replaced by the fractional differential equation theory to study the Hopf bifurcation of the rumor spreading model;this is consistent with the memory effect of the rumor spreading process.

    Remark 4.10Apart from the results of the bifurcation condition in [19],this paper not only conducts a bifurcation analysis for the fractional order rumor propagation model with a time delay,but also adds a state feedback controller and controls the advance or delay of bifurcation by adjusting the controller parameters.

    Remark 4.11For the selection of a bifurcation controller,differently from the fractional PD controller used in[29],the state feedback controller is simple in design and is easy to operate.More importantly,the state feedback controller has a wide range of applications,including in neural networks [21],eco-epidemiological systems [28]and predator-prey systems [34].

    5 Numerical Simulations

    In this section,some numerical simulation examples are given to verify the correctness of the theoretical results.To solve the fractional differential equations,we mainly use a predictorcorrector method which is described in [35,36].

    Case 1A specific example for the uncontrolled system (3.1).

    In this case,A=0.01,〈k〉=3,β=0.09,u=0.01,α=0.1,δ=0.85,θ=0.006,α1=0.99,α2=0.98,α3=0.98,α4=0.95 are selected,and then the specific system is as follows:

    By a simple calculation,we can obtain the rumor-spreading equilibrium point

    It is calculated thatω0=0.130,τ0=11.27.Based on Theorem 4.1,theP?of system (5.1)is asymptotically stable whenτ ∈[0,τ0),however,Hopf bifurcation occurs atP?whenτ=τ0.We chooseτ=11.26<11.27 andτ=12.05>11.27.The results are shown in Figures 1 and 2.

    Case 2A specific example for the controlled system (4.11).

    In this case,A=0.01,〈k〉=3,β=0.09,u=0.01,α=0.1,δ=0.85,θ=0.006,α1=0.99,α2=0.98,α3=0.98,α4=0.95 are chosen,and then the specific system is as follows:

    By a simple calculation,we can obtain the rumor-spreading equilibrium

    Remark 5.1In the process of calculating the equilibrium point of the controlled system(5.2),the added controller does not affect the original state of system (5.1),which means that whateverKandσare,they do not affect the equilibrium of the original system (5.1).

    Now the influence of controller parameters on bifurcation points studied based on the original uncontrolled system (5.1).

    First,letK=0.2,σ=1.It is calculated that=0.166,=8.94.Based on Theorem 4.2,theP?of system (5.2) is asymptotically stable whenτ ∈[0,),however,Hopf bifurcation occurs atP?whenτ=.Letτ=8.93<8.94 andτ=9.04>8.94.The results are shown in Figures 3 and 4.

    Then,letK=-0.21,σ=1.By a simple calculation,we get that=0.105,=13.91.We chooseτ=13.9<13.91 andτ=15.01>13.91.The results are shown in Figures 5 and 6.

    Furthermore,in order to study the effect of the feedback gainKon the bifurcation pointτ0for system (5.2),we selectσ=1 and takeK ∈[-1,0],K ∈[0,1].We can note that the value of the bifurcation point decreases with the increase ofKwhenK ∈[-1,0],and the value of the bifurcation point decreases first and then increases whenK ∈[0,1].These results are shown in Figures 7 and 8.

    Remark 5.2When the feedback gain isK<0,the controllerη(t)contributes to the system stability,while the controller has the opposite effect on system stability when the feedback gain isK>0.

    Next,the influence of a controller delay on the bifurcation points is discussed.We chooseK=0.2 andK=-0.2.Figure 9 shows that the stability of the controlled system(5.2)increases as the feedback delayσincreases whenK=0.2,however,the stability of the controlled system(5.2) decreases as the feedback delayσincreases whenK=-0.2,which can be seen in Figure 10.

    Figure 1 State trajectories of system (5.1) when τ=11.26<τ0

    Figure 2 State trajectories of system (5.1) when τ=12.05>τ0ν

    Figure 3 State trajectories of system (5.2) when K=0.2, σ=1, τ=8.93<

    Figure 4 State trajectories of system (5.2) when K=0.2, σ=1, τ=9.04>

    Figure 5 State trajectories of system (5.2) when K=-0.21, σ=1, τ=13.9<

    Figure 6 State trajectories of system (5.2) when K=-0.21, σ=1, τ=15.01>

    Figure 7 Effect of feedback gain K (K ∈[-1,0]) on bifurcation point τ0 for system (5.2)

    Figure 8 Effect of feedback gain K (K ∈[0,1]) on bifurcation point τ0 for system (5.2)

    Figure 10 Phase diagrams of system (5.2) with K=-0.2 and τ=15.7

    Finally,the effect of the fractional order on system stability is discussed.α2=0.97,α3=0.98 andα4=0.95 are selected.For convenience,only the effect of fractional orderα1is studied.From Figure 11 and Figure 12,it can be seen that the solution of system(5.2)changes from stable to unstable as the fractional order increases,and this is the case whetherK ∈[0,1]orK ∈[-1,0].

    Figure 12 State trajectories of system (5.2) with K=-0.15,τ=15,σ=1

    Remark 5.3It can be seen from Figure 11 and Figure 12 that the stability of system(5.2)changes while changing the order.Also,the solution of system(5.2)changes from unstable to asymptotically stable as the fractional order decreases.Thus,the order of the fractional differential system can be considered in order to discuss Hopf bifurcation under certain conditions,and these bifurcation conditions will be studied in the future.

    6 Conclusions

    In this paper,a fractional-order delayed SEIR rumor spreading model with a nonlinear incidence function was analyzed.Fractional order theory was used so as to make the rumor propagation process more accurate.First,we studied the asymptotically stable condition of rumor-spreading equilibrium and a time delay was chosen as a bifurcation parameter to discuss the bifurcation induced by a time delay.Then,we added feedback controllers to the original system in order to control the occurrence of bifurcation in advance or delay.In the numerical simulation,we discussed the influence of the controller parameters and the fractional order on the value of the bifurcation point.With the increase of the fractional order,the value of the bifurcation point increased continuously.Finally,the effect of controller parameters on the value of bifurcation points was discussed.It was shown that the feedback gain and delay of the controller has a great influence on the stability of the controlled system.

    Conflict of InterestThe authors declare no conflict of interest.

    猜你喜歡
    葉茂海軍
    Dynamic range and linearity improvement for zero-field single-beam atomic magnetometer
    楊國(guó)珍
    曉褐蜻
    綠色天府(2022年6期)2022-07-14 11:59:42
    A套餐
    我的海軍之夢(mèng)
    軍事文摘(2020年22期)2021-01-04 02:17:24
    相信愛
    根深才會(huì)葉茂源遠(yuǎn)方能流長(zhǎng)
    尋根(2020年1期)2020-04-07 03:44:34
    遼寧法庫(kù)葉茂臺(tái)七號(hào)遼墓的年代及墓主身份
    絲瓜
    封面人物·楊海軍
    新聞愛好者(2016年3期)2016-12-01 06:04:24
    亚洲精品粉嫩美女一区| 亚洲激情五月婷婷啪啪| 日本vs欧美在线观看视频| 男人爽女人下面视频在线观看| 精品久久蜜臀av无| 男女下面插进去视频免费观看| 国产av又大| 亚洲av欧美aⅴ国产| 日韩,欧美,国产一区二区三区| 十八禁网站免费在线| h视频一区二区三区| 99精品欧美一区二区三区四区| 亚洲成av片中文字幕在线观看| 国产一区二区三区在线臀色熟女 | tube8黄色片| 午夜免费鲁丝| 在线av久久热| 91字幕亚洲| 女性被躁到高潮视频| 国产免费视频播放在线视频| 国产精品免费大片| 亚洲欧洲日产国产| 男人操女人黄网站| 亚洲全国av大片| 99久久人妻综合| 日本五十路高清| av在线播放精品| 精品卡一卡二卡四卡免费| 女性被躁到高潮视频| 999久久久精品免费观看国产| 午夜久久久在线观看| 亚洲情色 制服丝袜| 免费久久久久久久精品成人欧美视频| 老司机午夜福利在线观看视频 | 在线永久观看黄色视频| 又紧又爽又黄一区二区| 9热在线视频观看99| 久久青草综合色| 亚洲成人手机| 777米奇影视久久| 老司机福利观看| 极品人妻少妇av视频| 日韩制服骚丝袜av| 亚洲国产av影院在线观看| 超色免费av| 亚洲精品日韩在线中文字幕| 国产伦人伦偷精品视频| 日韩三级视频一区二区三区| 欧美激情极品国产一区二区三区| 成人亚洲精品一区在线观看| 俄罗斯特黄特色一大片| 免费观看人在逋| 免费观看av网站的网址| 国产精品 国内视频| 成年人黄色毛片网站| 久久香蕉激情| 最新在线观看一区二区三区| 国产片内射在线| 两个人看的免费小视频| 啦啦啦视频在线资源免费观看| 99国产精品99久久久久| 老熟妇乱子伦视频在线观看 | 日本欧美视频一区| 久久精品国产a三级三级三级| 男人舔女人的私密视频| 99国产极品粉嫩在线观看| 伊人久久大香线蕉亚洲五| 国产亚洲精品第一综合不卡| 在线看a的网站| 一本—道久久a久久精品蜜桃钙片| 热re99久久国产66热| 日韩欧美国产一区二区入口| 免费女性裸体啪啪无遮挡网站| 爱豆传媒免费全集在线观看| 国产精品1区2区在线观看. | 97精品久久久久久久久久精品| 亚洲少妇的诱惑av| 永久免费av网站大全| 久久久国产一区二区| a级毛片黄视频| h视频一区二区三区| 亚洲精华国产精华精| 欧美另类一区| 国产高清视频在线播放一区 | 久久精品亚洲熟妇少妇任你| 国产不卡av网站在线观看| 欧美亚洲 丝袜 人妻 在线| 久久午夜综合久久蜜桃| 另类精品久久| 黑丝袜美女国产一区| 精品熟女少妇八av免费久了| 国产亚洲精品一区二区www | 国产亚洲av片在线观看秒播厂| 最近中文字幕2019免费版| 成人国产av品久久久| 久久久精品区二区三区| 亚洲人成77777在线视频| 99热网站在线观看| 欧美日韩av久久| 久久精品国产亚洲av香蕉五月 | 少妇裸体淫交视频免费看高清 | videos熟女内射| 国内毛片毛片毛片毛片毛片| av一本久久久久| 亚洲欧美激情在线| 女性生殖器流出的白浆| 大片电影免费在线观看免费| 精品国产超薄肉色丝袜足j| 91精品三级在线观看| 最黄视频免费看| 亚洲欧美一区二区三区久久| 成年动漫av网址| 亚洲一区二区三区欧美精品| 精品高清国产在线一区| 精品国产一区二区三区四区第35| 国产精品成人在线| 亚洲成人手机| 丁香六月天网| av有码第一页| 在线av久久热| av不卡在线播放| 老汉色∧v一级毛片| 国产一区二区三区综合在线观看| 最新在线观看一区二区三区| 亚洲成人免费电影在线观看| 精品久久久久久久毛片微露脸 | 黑人巨大精品欧美一区二区蜜桃| 啦啦啦免费观看视频1| 久久精品国产亚洲av香蕉五月 | 午夜日韩欧美国产| 国产99久久九九免费精品| 日日摸夜夜添夜夜添小说| a 毛片基地| 新久久久久国产一级毛片| 蜜桃在线观看..| 亚洲性夜色夜夜综合| 欧美精品人与动牲交sv欧美| 日本一区二区免费在线视频| 多毛熟女@视频| 搡老熟女国产l中国老女人| 动漫黄色视频在线观看| 欧美av亚洲av综合av国产av| 精品人妻熟女毛片av久久网站| 久久人妻熟女aⅴ| 99精品久久久久人妻精品| 777米奇影视久久| 777久久人妻少妇嫩草av网站| av欧美777| 人人妻人人澡人人爽人人夜夜| 一本—道久久a久久精品蜜桃钙片| av免费在线观看网站| 一本一本久久a久久精品综合妖精| 欧美日韩福利视频一区二区| 欧美日韩av久久| 91精品三级在线观看| 99热网站在线观看| 999久久久精品免费观看国产| 91成年电影在线观看| 精品久久蜜臀av无| 午夜两性在线视频| 欧美xxⅹ黑人| av在线播放精品| 丝瓜视频免费看黄片| 精品人妻一区二区三区麻豆| av在线播放精品| 一本综合久久免费| 久久国产精品影院| 亚洲三区欧美一区| 日韩制服丝袜自拍偷拍| av在线app专区| 美女中出高潮动态图| 国产一区二区三区在线臀色熟女 | 香蕉丝袜av| 久久久国产成人免费| av又黄又爽大尺度在线免费看| 欧美久久黑人一区二区| 欧美少妇被猛烈插入视频| 伦理电影免费视频| 下体分泌物呈黄色| 日本欧美视频一区| 香蕉丝袜av| 免费人妻精品一区二区三区视频| 国产区一区二久久| 丰满人妻熟妇乱又伦精品不卡| 亚洲av男天堂| 国产男人的电影天堂91| 99国产精品99久久久久| 国产欧美日韩一区二区三区在线| 男女无遮挡免费网站观看| 国产精品av久久久久免费| 国产精品成人在线| 777米奇影视久久| 无限看片的www在线观看| 久久久精品区二区三区| 久久天堂一区二区三区四区| 美女大奶头黄色视频| 狂野欧美激情性xxxx| 高清在线国产一区| 我要看黄色一级片免费的| 嫩草影视91久久| 亚洲av电影在线进入| 久久天躁狠狠躁夜夜2o2o| 免费久久久久久久精品成人欧美视频| 亚洲精品久久午夜乱码| 老司机亚洲免费影院| 999精品在线视频| 午夜福利免费观看在线| 国产一区二区三区av在线| 性少妇av在线| 国产精品欧美亚洲77777| 热re99久久国产66热| 亚洲精品国产av蜜桃| 亚洲精品美女久久av网站| 国产一区二区在线观看av| 两个人看的免费小视频| 亚洲一区中文字幕在线| 日本五十路高清| 久热这里只有精品99| 黄片小视频在线播放| 久久人妻熟女aⅴ| 亚洲国产av新网站| 欧美一级毛片孕妇| 夜夜骑夜夜射夜夜干| 涩涩av久久男人的天堂| 一本一本久久a久久精品综合妖精| 男女下面插进去视频免费观看| 欧美黄色淫秽网站| 午夜福利视频精品| 精品福利观看| 国产视频一区二区在线看| 天天躁狠狠躁夜夜躁狠狠躁| 狠狠精品人妻久久久久久综合| 少妇精品久久久久久久| 国产有黄有色有爽视频| 国产成人av教育| 啦啦啦视频在线资源免费观看| 国产精品99久久99久久久不卡| 天天影视国产精品| 欧美黑人欧美精品刺激| 视频区图区小说| 50天的宝宝边吃奶边哭怎么回事| 国产精品熟女久久久久浪| 丝瓜视频免费看黄片| 国产亚洲欧美在线一区二区| 最新的欧美精品一区二区| 操出白浆在线播放| 9色porny在线观看| 热re99久久精品国产66热6| 亚洲男人天堂网一区| 人人妻人人爽人人添夜夜欢视频| 18在线观看网站| 91老司机精品| 啦啦啦啦在线视频资源| 考比视频在线观看| 欧美在线一区亚洲| 老司机午夜十八禁免费视频| 我要看黄色一级片免费的| 久久狼人影院| 黄色视频,在线免费观看| 一级,二级,三级黄色视频| 亚洲第一欧美日韩一区二区三区 | 美女主播在线视频| 美女扒开内裤让男人捅视频| 国产精品1区2区在线观看. | 亚洲欧美成人综合另类久久久| 国产成人av激情在线播放| 男人舔女人的私密视频| 女人久久www免费人成看片| 操美女的视频在线观看| 午夜福利视频精品| 嫩草影视91久久| 久久久水蜜桃国产精品网| 亚洲激情五月婷婷啪啪| 国产高清videossex| 下体分泌物呈黄色| 美女中出高潮动态图| 久久人妻福利社区极品人妻图片| 成年美女黄网站色视频大全免费| 91麻豆av在线| 一二三四在线观看免费中文在| 一本一本久久a久久精品综合妖精| 亚洲人成电影观看| 9色porny在线观看| 日韩中文字幕欧美一区二区| 亚洲视频免费观看视频| 十八禁人妻一区二区| 亚洲欧洲精品一区二区精品久久久| 久久久久网色| 久久精品熟女亚洲av麻豆精品| 久久性视频一级片| 午夜日韩欧美国产| av在线老鸭窝| 99国产精品一区二区蜜桃av | 欧美激情 高清一区二区三区| 国产av又大| 啦啦啦视频在线资源免费观看| 叶爱在线成人免费视频播放| 成年人黄色毛片网站| 午夜91福利影院| 午夜精品国产一区二区电影| 99re6热这里在线精品视频| 99久久精品国产亚洲精品| netflix在线观看网站| 欧美在线黄色| 91成年电影在线观看| 91国产中文字幕| 91老司机精品| 嫩草影视91久久| 在线看a的网站| www.熟女人妻精品国产| 侵犯人妻中文字幕一二三四区| 99国产精品免费福利视频| 国产黄色免费在线视频| 成年女人毛片免费观看观看9 | 考比视频在线观看| 性少妇av在线| av免费在线观看网站| 一个人免费看片子| 久久精品成人免费网站| 天堂8中文在线网| 亚洲熟女毛片儿| 天堂中文最新版在线下载| 精品一区二区三区四区五区乱码| 黄色视频在线播放观看不卡| 亚洲,欧美精品.| 一进一出抽搐动态| 中文字幕精品免费在线观看视频| 极品少妇高潮喷水抽搐| 欧美精品人与动牲交sv欧美| 亚洲 欧美一区二区三区| 涩涩av久久男人的天堂| 肉色欧美久久久久久久蜜桃| 久久久国产成人免费| 国产成人a∨麻豆精品| 男男h啪啪无遮挡| 欧美变态另类bdsm刘玥| 热99re8久久精品国产| 国产老妇伦熟女老妇高清| 大香蕉久久网| 可以免费在线观看a视频的电影网站| 亚洲欧美激情在线| 亚洲av美国av| 精品国产乱码久久久久久小说| 夜夜骑夜夜射夜夜干| 少妇被粗大的猛进出69影院| 伊人亚洲综合成人网| 99精品久久久久人妻精品| 国产精品一区二区免费欧美 | 久久九九热精品免费| svipshipincom国产片| videos熟女内射| 精品国产乱码久久久久久男人| 久久精品久久久久久噜噜老黄| 午夜日韩欧美国产| 国产真人三级小视频在线观看| 国产精品久久久久久人妻精品电影 | 欧美日韩av久久| 国产免费av片在线观看野外av| 亚洲av国产av综合av卡| 啦啦啦视频在线资源免费观看| 国产日韩欧美在线精品| 美女大奶头黄色视频| 亚洲国产欧美网| 久久中文看片网| 国产区一区二久久| 亚洲中文字幕日韩| 成在线人永久免费视频| 国产97色在线日韩免费| 热re99久久精品国产66热6| 男女免费视频国产| 国产人伦9x9x在线观看| 精品人妻一区二区三区麻豆| 18在线观看网站| 亚洲欧美一区二区三区久久| 午夜福利一区二区在线看| 国产熟女午夜一区二区三区| 超碰成人久久| 久热这里只有精品99| 中文字幕人妻熟女乱码| 免费日韩欧美在线观看| 91精品三级在线观看| 色综合欧美亚洲国产小说| 少妇粗大呻吟视频| 90打野战视频偷拍视频| 亚洲精品国产色婷婷电影| videos熟女内射| 亚洲精华国产精华精| 午夜91福利影院| 18禁国产床啪视频网站| 久久久久视频综合| 欧美日韩亚洲高清精品| 久久久久国内视频| 一本久久精品| 在线永久观看黄色视频| 国产精品秋霞免费鲁丝片| av免费在线观看网站| 亚洲国产日韩一区二区| 人人妻人人澡人人看| 啦啦啦在线免费观看视频4| 黑人巨大精品欧美一区二区mp4| 美女高潮到喷水免费观看| 日本猛色少妇xxxxx猛交久久| 999久久久国产精品视频| 久久天躁狠狠躁夜夜2o2o| 91九色精品人成在线观看| 一区福利在线观看| 亚洲美女黄色视频免费看| 亚洲av成人一区二区三| 国产91精品成人一区二区三区 | 久久久久精品国产欧美久久久 | 麻豆av在线久日| 国产1区2区3区精品| 久久国产精品男人的天堂亚洲| 欧美日韩视频精品一区| 国产精品久久久久久精品电影小说| 欧美激情极品国产一区二区三区| 色综合欧美亚洲国产小说| av线在线观看网站| 桃红色精品国产亚洲av| 女性被躁到高潮视频| 国产亚洲欧美在线一区二区| 成人影院久久| 亚洲第一av免费看| 精品熟女少妇八av免费久了| 精品少妇一区二区三区视频日本电影| 日韩视频一区二区在线观看| 精品少妇久久久久久888优播| 久久精品国产亚洲av香蕉五月 | 在线天堂中文资源库| 国产亚洲欧美精品永久| 在线观看人妻少妇| 人人妻人人澡人人爽人人夜夜| 窝窝影院91人妻| 久久毛片免费看一区二区三区| 国产精品秋霞免费鲁丝片| 男人爽女人下面视频在线观看| 欧美xxⅹ黑人| av电影中文网址| 国产精品九九99| 91大片在线观看| 飞空精品影院首页| 18在线观看网站| 欧美国产精品va在线观看不卡| 午夜福利在线免费观看网站| 久久综合国产亚洲精品| 精品一品国产午夜福利视频| 成人国语在线视频| 亚洲专区中文字幕在线| 色精品久久人妻99蜜桃| 中文字幕精品免费在线观看视频| 欧美日韩中文字幕国产精品一区二区三区 | 丝瓜视频免费看黄片| 欧美日韩成人在线一区二区| 欧美日韩精品网址| 国产有黄有色有爽视频| 少妇猛男粗大的猛烈进出视频| a级片在线免费高清观看视频| 久久久久国产一级毛片高清牌| 天天躁夜夜躁狠狠躁躁| 亚洲天堂av无毛| 色综合欧美亚洲国产小说| 精品久久久久久久毛片微露脸 | 亚洲av电影在线进入| 在线十欧美十亚洲十日本专区| 亚洲av国产av综合av卡| 91字幕亚洲| 精品欧美一区二区三区在线| 久久亚洲国产成人精品v| 在线观看免费高清a一片| 天堂中文最新版在线下载| 亚洲欧美日韩另类电影网站| 人成视频在线观看免费观看| 精品久久久精品久久久| 免费看十八禁软件| 91字幕亚洲| 自拍欧美九色日韩亚洲蝌蚪91| 精品国产一区二区三区四区第35| 国产精品一区二区免费欧美 | av免费在线观看网站| 亚洲熟女精品中文字幕| 少妇精品久久久久久久| 亚洲av电影在线观看一区二区三区| 成人免费观看视频高清| 丝袜脚勾引网站| 成人手机av| 十八禁高潮呻吟视频| 少妇人妻久久综合中文| 午夜福利乱码中文字幕| 久久青草综合色| 少妇裸体淫交视频免费看高清 | 美女大奶头黄色视频| 欧美亚洲日本最大视频资源| 女人精品久久久久毛片| av欧美777| 免费在线观看日本一区| 日本a在线网址| 日韩中文字幕欧美一区二区| 日本wwww免费看| 成年av动漫网址| 国产亚洲欧美精品永久| 亚洲精品美女久久av网站| 国产精品久久久久成人av| 免费人妻精品一区二区三区视频| 亚洲少妇的诱惑av| 国产高清国产精品国产三级| 免费高清在线观看日韩| 九色亚洲精品在线播放| 热99re8久久精品国产| 国产淫语在线视频| 亚洲欧美日韩高清在线视频 | 亚洲综合色网址| 99国产精品免费福利视频| 一本大道久久a久久精品| 超色免费av| 97在线人人人人妻| 制服人妻中文乱码| 在线观看免费视频网站a站| 亚洲 欧美一区二区三区| 亚洲国产精品一区二区三区在线| 美女视频免费永久观看网站| 国产精品亚洲av一区麻豆| 亚洲第一青青草原| 一本久久精品| 成人手机av| 少妇精品久久久久久久| 我的亚洲天堂| 亚洲国产精品一区二区三区在线| 亚洲国产欧美日韩在线播放| 2018国产大陆天天弄谢| av有码第一页| 久热爱精品视频在线9| 精品国产一区二区久久| 热99久久久久精品小说推荐| 丝袜美足系列| a在线观看视频网站| 中文字幕制服av| 国产男女内射视频| 欧美激情极品国产一区二区三区| 欧美日韩av久久| 18禁黄网站禁片午夜丰满| 一区二区日韩欧美中文字幕| 丝袜美足系列| 国产成人影院久久av| 女人高潮潮喷娇喘18禁视频| 成人影院久久| 中文字幕人妻丝袜制服| 美女视频免费永久观看网站| 熟女少妇亚洲综合色aaa.| 亚洲中文av在线| 9热在线视频观看99| 黄色片一级片一级黄色片| 欧美变态另类bdsm刘玥| 丰满少妇做爰视频| 久久精品亚洲av国产电影网| 午夜影院在线不卡| 亚洲精品av麻豆狂野| 老鸭窝网址在线观看| 99久久人妻综合| 伦理电影免费视频| 欧美激情高清一区二区三区| 欧美亚洲日本最大视频资源| 亚洲精品国产一区二区精华液| 性高湖久久久久久久久免费观看| 侵犯人妻中文字幕一二三四区| 国产成人免费观看mmmm| 91精品三级在线观看| 久久久久网色| 午夜老司机福利片| 精品视频人人做人人爽| 真人做人爱边吃奶动态| 亚洲精品久久久久久婷婷小说| 丁香六月天网| 看免费av毛片| 中亚洲国语对白在线视频| 亚洲少妇的诱惑av| 午夜免费鲁丝| 69av精品久久久久久 | 99精国产麻豆久久婷婷| 久久久久久免费高清国产稀缺| 777米奇影视久久| 国产精品.久久久| 麻豆乱淫一区二区| av又黄又爽大尺度在线免费看| 亚洲精品国产精品久久久不卡| 777久久人妻少妇嫩草av网站| 亚洲伊人久久精品综合| 男女床上黄色一级片免费看| 曰老女人黄片| 极品人妻少妇av视频| 男女床上黄色一级片免费看| 国产精品久久久久久精品电影小说| 精品一区二区三区四区五区乱码| 午夜福利,免费看| 美女国产高潮福利片在线看| 亚洲欧美成人综合另类久久久| 久久中文看片网| 成年人午夜在线观看视频| 国产一区二区在线观看av| 国产精品二区激情视频| a 毛片基地| 国产97色在线日韩免费| 成年人免费黄色播放视频| 97精品久久久久久久久久精品| 一本色道久久久久久精品综合| 国产精品九九99| 亚洲成人免费电影在线观看| 午夜激情久久久久久久| 日韩视频一区二区在线观看| 老汉色∧v一级毛片| 黑人猛操日本美女一级片| 中文字幕人妻丝袜制服| 免费观看a级毛片全部| 欧美97在线视频| 精品国产国语对白av| 一区在线观看完整版|