• <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
    久久婷婷人人爽人人干人人爱| 亚洲五月天丁香| 亚洲成人精品中文字幕电影| 18禁黄网站禁片免费观看直播| 亚洲国产精品999在线| ponron亚洲| 精品久久久久久久久亚洲 | 中文字幕高清在线视频| 午夜精品一区二区三区免费看| 丰满人妻熟妇乱又伦精品不卡| 成熟少妇高潮喷水视频| 亚洲精品影视一区二区三区av| 亚洲五月天丁香| 国产欧美日韩一区二区精品| 免费看美女性在线毛片视频| 久久久久久久久久黄片| 国产精品乱码一区二三区的特点| 久久久久精品国产欧美久久久| 色av中文字幕| 午夜激情欧美在线| 国产精华一区二区三区| www.www免费av| 男插女下体视频免费在线播放| 男人的好看免费观看在线视频| 最近在线观看免费完整版| 别揉我奶头~嗯~啊~动态视频| 久久午夜亚洲精品久久| 麻豆成人午夜福利视频| 免费看日本二区| 亚洲精品乱码久久久v下载方式| 国产成人a区在线观看| 日韩欧美在线乱码| 内射极品少妇av片p| 波野结衣二区三区在线| 日韩欧美国产一区二区入口| 欧美+亚洲+日韩+国产| 国产一区二区在线观看日韩| 99在线人妻在线中文字幕| 18禁在线播放成人免费| 最近最新免费中文字幕在线| 一级作爱视频免费观看| 国产单亲对白刺激| 成人美女网站在线观看视频| 成人永久免费在线观看视频| 国产精品国产高清国产av| 国产一级毛片七仙女欲春2| 人人妻,人人澡人人爽秒播| eeuss影院久久| 麻豆国产97在线/欧美| 男女床上黄色一级片免费看| 国产黄色小视频在线观看| 性色avwww在线观看| 一区二区三区免费毛片| 国模一区二区三区四区视频| 少妇人妻精品综合一区二区 | 毛片一级片免费看久久久久 | 国产蜜桃级精品一区二区三区| 热99在线观看视频| 亚洲综合色惰| 变态另类丝袜制服| 日日干狠狠操夜夜爽| 在线观看午夜福利视频| 亚洲一区高清亚洲精品| 成人一区二区视频在线观看| 麻豆一二三区av精品| h日本视频在线播放| 国产爱豆传媒在线观看| 午夜免费成人在线视频| av欧美777| 噜噜噜噜噜久久久久久91| 中亚洲国语对白在线视频| 国产成人aa在线观看| 内地一区二区视频在线| 国产综合懂色| 国产 一区 欧美 日韩| 日日摸夜夜添夜夜添小说| 看免费av毛片| 99riav亚洲国产免费| 久久久久性生活片| 亚洲av电影不卡..在线观看| 老师上课跳d突然被开到最大视频 久久午夜综合久久蜜桃 | 真人做人爱边吃奶动态| 伦理电影大哥的女人| 亚洲专区中文字幕在线| 亚洲美女视频黄频| 国产欧美日韩精品亚洲av| 一夜夜www| 久久午夜福利片| 女人被狂操c到高潮| 看免费av毛片| 三级男女做爰猛烈吃奶摸视频| 亚洲国产精品999在线| 国产伦精品一区二区三区视频9| 哪里可以看免费的av片| 午夜福利在线观看吧| 国产成人av教育| 欧美一级a爱片免费观看看| 成人高潮视频无遮挡免费网站| 亚洲成人久久性| 国产午夜精品久久久久久一区二区三区 | 欧美精品啪啪一区二区三区| 九九在线视频观看精品| 亚洲真实伦在线观看| 亚洲,欧美精品.| 亚洲av第一区精品v没综合| 欧美成狂野欧美在线观看| 国产精品久久久久久久久免 | 精品一区二区三区视频在线观看免费| 亚洲欧美日韩无卡精品| 国产精品三级大全| 欧美成人一区二区免费高清观看| 中文字幕精品亚洲无线码一区| 午夜久久久久精精品| 在线观看舔阴道视频| 三级国产精品欧美在线观看| 精华霜和精华液先用哪个| 成人特级av手机在线观看| 欧美成人a在线观看| 首页视频小说图片口味搜索| a级一级毛片免费在线观看| 日韩中文字幕欧美一区二区| 自拍偷自拍亚洲精品老妇| 免费高清视频大片| 日韩av在线大香蕉| 一本一本综合久久| 成熟少妇高潮喷水视频| 中文字幕久久专区| 亚洲va日本ⅴa欧美va伊人久久| 日韩免费av在线播放| 国产主播在线观看一区二区| 制服丝袜大香蕉在线| 久久九九热精品免费| 日本三级黄在线观看| 窝窝影院91人妻| 国产视频一区二区在线看| 精品久久久久久久久亚洲 | 亚洲av熟女| 51国产日韩欧美| 国内久久婷婷六月综合欲色啪| 国产精品电影一区二区三区| 99视频精品全部免费 在线| 久久精品久久久久久噜噜老黄 | 97人妻精品一区二区三区麻豆| 变态另类丝袜制服| 桃红色精品国产亚洲av| 久久久久性生活片| 99精品在免费线老司机午夜| 午夜两性在线视频| 男女下面进入的视频免费午夜| 黄色视频,在线免费观看| 成人无遮挡网站| 最新在线观看一区二区三区| 日韩大尺度精品在线看网址| 亚洲va日本ⅴa欧美va伊人久久| 最好的美女福利视频网| 麻豆国产97在线/欧美| 国模一区二区三区四区视频| 最新在线观看一区二区三区| 日韩有码中文字幕| 欧美丝袜亚洲另类 | 欧美高清性xxxxhd video| 成人精品一区二区免费| 悠悠久久av| 国产私拍福利视频在线观看| 日本免费a在线| 亚洲人成电影免费在线| 精品久久久久久久久久久久久| 婷婷亚洲欧美| 国产真实伦视频高清在线观看 | 舔av片在线| 好男人在线观看高清免费视频| 亚洲精品乱码久久久v下载方式| 国产探花极品一区二区| 人人妻,人人澡人人爽秒播| 天堂√8在线中文| 性色avwww在线观看| 亚洲第一电影网av| 国产黄片美女视频| eeuss影院久久| 欧美成人性av电影在线观看| 日本撒尿小便嘘嘘汇集6| 午夜免费男女啪啪视频观看 | 直男gayav资源| 欧美日韩瑟瑟在线播放| 亚洲真实伦在线观看| 九九久久精品国产亚洲av麻豆| www.999成人在线观看| 欧美性猛交╳xxx乱大交人| 免费搜索国产男女视频| 亚洲欧美精品综合久久99| 午夜免费男女啪啪视频观看 | 美女 人体艺术 gogo| 日日摸夜夜添夜夜添小说| 亚洲成人久久性| 免费高清视频大片| 久久久色成人| 精品人妻一区二区三区麻豆 | 日日干狠狠操夜夜爽| 91麻豆av在线| 婷婷亚洲欧美| 久久人人精品亚洲av| 免费看美女性在线毛片视频| 久久九九热精品免费| 又黄又爽又刺激的免费视频.| 看十八女毛片水多多多| eeuss影院久久| 欧美性猛交╳xxx乱大交人| 18禁黄网站禁片午夜丰满| 亚洲三级黄色毛片| 亚洲国产高清在线一区二区三| 丰满的人妻完整版| 国产成人影院久久av| 久久伊人香网站| 三级男女做爰猛烈吃奶摸视频| 搡女人真爽免费视频火全软件 | 亚洲,欧美,日韩| 亚洲成人中文字幕在线播放| 综合色av麻豆| 亚洲成av人片免费观看| www.熟女人妻精品国产| 人妻制服诱惑在线中文字幕| 神马国产精品三级电影在线观看| 美女高潮喷水抽搐中文字幕| 亚洲国产精品成人综合色| 97超级碰碰碰精品色视频在线观看| 成人美女网站在线观看视频| 国产白丝娇喘喷水9色精品| 国产真实伦视频高清在线观看 | 免费大片18禁| 精品熟女少妇八av免费久了| 国产综合懂色| 亚洲人成网站在线播放欧美日韩| avwww免费| 免费看美女性在线毛片视频| 久99久视频精品免费| 老师上课跳d突然被开到最大视频 久久午夜综合久久蜜桃 | 亚洲中文字幕一区二区三区有码在线看| 午夜福利欧美成人| av在线天堂中文字幕| 久99久视频精品免费| 天堂网av新在线| 夜夜看夜夜爽夜夜摸| 国产欧美日韩精品亚洲av| 国产真实乱freesex| 中出人妻视频一区二区| 脱女人内裤的视频| 精品人妻偷拍中文字幕| 波多野结衣高清作品| 国产精品一区二区免费欧美| 免费观看精品视频网站| 三级国产精品欧美在线观看| 国产蜜桃级精品一区二区三区| 亚洲精品一区av在线观看| 国产精品三级大全| 高清毛片免费观看视频网站| 男女那种视频在线观看| 91午夜精品亚洲一区二区三区 | 欧美日韩黄片免| 亚洲成人中文字幕在线播放| 有码 亚洲区| 国产精品三级大全| 免费观看的影片在线观看| 高清在线国产一区| 国产精品电影一区二区三区| 欧美+日韩+精品| 精品福利观看| 午夜a级毛片| 不卡一级毛片| 久久久久久久久大av| 国产成人aa在线观看| 亚洲va日本ⅴa欧美va伊人久久| 国产伦在线观看视频一区| av国产免费在线观看| 精品久久久久久久久av| 欧美色视频一区免费| 精品国内亚洲2022精品成人| 久久精品人妻少妇| 亚洲精品粉嫩美女一区| 在现免费观看毛片| 久久国产精品人妻蜜桃| 亚洲 国产 在线| 精品久久久久久久久av| 三级毛片av免费| 国产伦精品一区二区三区四那| 人人妻人人澡欧美一区二区| 色哟哟·www| 国产69精品久久久久777片| 老司机深夜福利视频在线观看| 国产三级中文精品| 亚洲av熟女| 精品99又大又爽又粗少妇毛片 | 一进一出抽搐gif免费好疼| 90打野战视频偷拍视频| 欧美3d第一页| 亚洲精品一卡2卡三卡4卡5卡| 国产一区二区亚洲精品在线观看| bbb黄色大片| 99久久成人亚洲精品观看| 国产精品一区二区性色av| 久久久久久久午夜电影| 99riav亚洲国产免费| 久久国产精品人妻蜜桃| 91九色精品人成在线观看| 此物有八面人人有两片| 毛片一级片免费看久久久久 | 亚洲人与动物交配视频| 少妇熟女aⅴ在线视频| 亚洲av.av天堂| 美女cb高潮喷水在线观看| 成人欧美大片| 欧美区成人在线视频| 国产欧美日韩精品亚洲av| 99热这里只有是精品在线观看 | 乱人视频在线观看| 国产美女午夜福利| 99久久99久久久精品蜜桃| 久久国产乱子免费精品| 国产欧美日韩一区二区三| 极品教师在线免费播放| 日本一二三区视频观看| 久久精品国产99精品国产亚洲性色| 欧美最黄视频在线播放免费| 脱女人内裤的视频| 中文亚洲av片在线观看爽| 欧美bdsm另类| 91在线精品国自产拍蜜月| 色精品久久人妻99蜜桃| 99精品久久久久人妻精品| 国产精品久久久久久久久免 | 久久香蕉精品热| 人妻夜夜爽99麻豆av| 亚洲av免费在线观看| 最近中文字幕高清免费大全6 | 免费看美女性在线毛片视频| 高清毛片免费观看视频网站| 狂野欧美白嫩少妇大欣赏| 欧美国产日韩亚洲一区| 亚洲经典国产精华液单 | 亚洲成人久久爱视频| 久久人妻av系列| 成人毛片a级毛片在线播放| 成人特级av手机在线观看| 亚洲五月天丁香| 别揉我奶头~嗯~啊~动态视频| 99热6这里只有精品| 中文字幕精品亚洲无线码一区| 性欧美人与动物交配| av黄色大香蕉| 少妇的逼水好多| 国产一区二区在线观看日韩| 嫁个100分男人电影在线观看| 波多野结衣巨乳人妻| 亚洲国产精品sss在线观看| 午夜老司机福利剧场| 中文字幕久久专区| av国产免费在线观看| 国产黄色小视频在线观看| 最近在线观看免费完整版| 小蜜桃在线观看免费完整版高清| 日韩精品青青久久久久久| 久久久久久国产a免费观看| 一卡2卡三卡四卡精品乱码亚洲| 久久久久免费精品人妻一区二区| 日日摸夜夜添夜夜添av毛片 | 怎么达到女性高潮| 国产精品久久久久久亚洲av鲁大| 精品久久国产蜜桃| 午夜福利成人在线免费观看| 女同久久另类99精品国产91| 亚洲av.av天堂| 久久久久性生活片| 国产毛片a区久久久久| 亚洲,欧美精品.| 99国产精品一区二区蜜桃av| 亚洲人成电影免费在线| 乱码一卡2卡4卡精品| 午夜视频国产福利| 国产精品99久久久久久久久| 1000部很黄的大片| 久久国产乱子免费精品| 国产精品美女特级片免费视频播放器| 9191精品国产免费久久| 国内揄拍国产精品人妻在线| 男人和女人高潮做爰伦理| 蜜桃久久精品国产亚洲av| 三级男女做爰猛烈吃奶摸视频| a级毛片免费高清观看在线播放| av在线天堂中文字幕| 校园春色视频在线观看| 亚洲av二区三区四区| 久久这里只有精品中国| 琪琪午夜伦伦电影理论片6080| 欧美黄色淫秽网站| 精品人妻偷拍中文字幕| 日韩高清综合在线| 天堂网av新在线| 日本与韩国留学比较| 欧美色视频一区免费| 欧美绝顶高潮抽搐喷水| 变态另类丝袜制服| 天堂影院成人在线观看| 国内少妇人妻偷人精品xxx网站| 麻豆国产97在线/欧美| 夜夜爽天天搞| 又黄又爽又刺激的免费视频.| 欧美日本亚洲视频在线播放| 美女cb高潮喷水在线观看| 欧美日本视频| 91麻豆精品激情在线观看国产| 精品无人区乱码1区二区| 国产精品不卡视频一区二区 | 中国美女看黄片| 中文在线观看免费www的网站| 久久久久国产精品人妻aⅴ院| 九九热线精品视视频播放| 九色国产91popny在线| 亚洲一区二区三区色噜噜| 狠狠狠狠99中文字幕| 人人妻人人看人人澡| 深夜精品福利| 成人亚洲精品av一区二区| 婷婷精品国产亚洲av在线| 久久久国产成人免费| 夜夜躁狠狠躁天天躁| 深夜a级毛片| 久久久色成人| 黄色一级大片看看| 中文字幕人成人乱码亚洲影| 欧美一区二区国产精品久久精品| 18美女黄网站色大片免费观看| 99久久成人亚洲精品观看| 天天一区二区日本电影三级| 长腿黑丝高跟| 国产单亲对白刺激| 久久精品91蜜桃| 嫩草影院精品99| 亚洲人成网站高清观看| 97人妻精品一区二区三区麻豆| 成人性生交大片免费视频hd| 三级男女做爰猛烈吃奶摸视频| 国产一区二区三区在线臀色熟女| 老师上课跳d突然被开到最大视频 久久午夜综合久久蜜桃 | 亚洲欧美日韩东京热| 日韩欧美 国产精品| 国产成人福利小说| 欧美高清成人免费视频www| 日本成人三级电影网站| 色综合站精品国产| 亚洲第一电影网av| 免费看a级黄色片| 最近在线观看免费完整版| 亚洲,欧美,日韩| 久久久久久国产a免费观看| 国产大屁股一区二区在线视频| 精品人妻1区二区| 亚洲欧美日韩高清专用| 一个人看的www免费观看视频| 国产一区二区亚洲精品在线观看| 精品99又大又爽又粗少妇毛片 | 欧美日本亚洲视频在线播放| 欧美成人性av电影在线观看| 看黄色毛片网站| 久久精品综合一区二区三区| 淫妇啪啪啪对白视频| 久久精品综合一区二区三区| 黄色配什么色好看| 国产成人影院久久av| 1000部很黄的大片| 国产成人欧美在线观看| 热99在线观看视频| 99热精品在线国产| 国内毛片毛片毛片毛片毛片| 婷婷精品国产亚洲av| 亚洲欧美日韩高清专用| 最好的美女福利视频网| 国产av一区在线观看免费| www.999成人在线观看| 在线国产一区二区在线| 欧美激情在线99| 国产午夜福利久久久久久| 三级毛片av免费| 3wmmmm亚洲av在线观看| 亚洲在线自拍视频| 听说在线观看完整版免费高清| 国产精品一区二区三区四区免费观看 | 久久热精品热| 国产成人欧美在线观看| 宅男免费午夜| 亚洲国产日韩欧美精品在线观看| 最近最新免费中文字幕在线| 大型黄色视频在线免费观看| 亚洲avbb在线观看| 亚洲内射少妇av| 搞女人的毛片| 国产亚洲av嫩草精品影院| 91字幕亚洲| 亚洲av电影不卡..在线观看| 精品午夜福利在线看| 9191精品国产免费久久| 性插视频无遮挡在线免费观看| av在线蜜桃| 两人在一起打扑克的视频| 色综合亚洲欧美另类图片| 精品国内亚洲2022精品成人| 一区二区三区四区激情视频 | 老鸭窝网址在线观看| av福利片在线观看| 狂野欧美白嫩少妇大欣赏| 日本与韩国留学比较| 尤物成人国产欧美一区二区三区| 欧美一区二区国产精品久久精品| 国产亚洲精品av在线| 国产精品自产拍在线观看55亚洲| 9191精品国产免费久久| 国模一区二区三区四区视频| 三级国产精品欧美在线观看| 久久久国产成人免费| 亚洲一区二区三区色噜噜| 久久久久性生活片| 他把我摸到了高潮在线观看| 午夜视频国产福利| 亚洲av熟女| 永久网站在线| 蜜桃亚洲精品一区二区三区| 亚洲av免费在线观看| 亚洲午夜理论影院| 啪啪无遮挡十八禁网站| 大型黄色视频在线免费观看| 极品教师在线视频| 久久精品影院6| 亚洲无线观看免费| 嫁个100分男人电影在线观看| 此物有八面人人有两片| 日本三级黄在线观看| 欧美一区二区国产精品久久精品| 99久久九九国产精品国产免费| 男人和女人高潮做爰伦理| 人人妻人人澡欧美一区二区| 中文字幕人妻熟人妻熟丝袜美| 亚洲无线在线观看| 在线观看66精品国产| 亚洲人与动物交配视频| 亚洲美女视频黄频| 精品久久久久久久人妻蜜臀av| 亚洲经典国产精华液单 | 午夜福利18| 老师上课跳d突然被开到最大视频 久久午夜综合久久蜜桃 | av黄色大香蕉| 噜噜噜噜噜久久久久久91| 亚洲av二区三区四区| 可以在线观看的亚洲视频| 国产不卡一卡二| 国产伦精品一区二区三区四那| 亚洲黑人精品在线| 精品人妻偷拍中文字幕| 国产精品久久电影中文字幕| 国产黄a三级三级三级人| 欧美日韩瑟瑟在线播放| 成人性生交大片免费视频hd| 色噜噜av男人的天堂激情| 窝窝影院91人妻| 亚洲欧美日韩卡通动漫| 久久热精品热| 国产久久久一区二区三区| 淫秽高清视频在线观看| 变态另类成人亚洲欧美熟女| 亚洲成a人片在线一区二区| 黄色女人牲交| 韩国av一区二区三区四区| 性色avwww在线观看| 村上凉子中文字幕在线| 九色国产91popny在线| 搡老妇女老女人老熟妇| 一进一出抽搐gif免费好疼| 色av中文字幕| 午夜福利免费观看在线| 夜夜看夜夜爽夜夜摸| 欧美黄色淫秽网站| 十八禁网站免费在线| 少妇高潮的动态图| 中文字幕精品亚洲无线码一区| 国产欧美日韩一区二区三| 成人永久免费在线观看视频| 久久久色成人| 亚洲美女视频黄频| 欧美高清成人免费视频www| 99riav亚洲国产免费| 久久国产乱子免费精品| 久久人妻av系列| 国产高清有码在线观看视频| 一区二区三区高清视频在线| 色尼玛亚洲综合影院| 久久久精品欧美日韩精品| 夜夜看夜夜爽夜夜摸| 男人狂女人下面高潮的视频| 精品久久久久久久久av| 亚洲精品一卡2卡三卡4卡5卡| 很黄的视频免费| 久久中文看片网| 久久精品国产亚洲av涩爱 | 美女大奶头视频| 精品国内亚洲2022精品成人| 免费搜索国产男女视频| av中文乱码字幕在线| 久久久久久久亚洲中文字幕 | 99久久久亚洲精品蜜臀av| 国产成人啪精品午夜网站| a级毛片a级免费在线| 亚洲人成网站在线播放欧美日韩| 极品教师在线视频| 亚洲av日韩精品久久久久久密| 国产老妇女一区| 亚洲成a人片在线一区二区| 国产欧美日韩精品亚洲av| 国产黄片美女视频|