CN104809303A - Limit cycle amplitude control method in rumor propagation model - Google Patents

Abstract

The invention discloses a limit cycle amplitude control method in a rumor propagation model. A new rumor propagation model is built on the basis of an SIR model, in consideration that the infection rate is reduced along with the increasing of the rumor propagation nodes, a piecewise function is introduced to describe the infection rate, and rumor propagation behaviors in a complex network can be more accurately described; on the basis of the new model, a control function is introduced and is added into the model, and limit cycle amplitude appearing in the rumor propagation model is controlled, the fluctuation range of number of people who propagate a rumor is controlled.

Description

Limit cycle amplitude control method in a kind of gossip propagation model

Technical field

The invention belongs to complex network Transmission dynamic field, more specifically say, relate to the limit cycle amplitude control method in a kind of gossip propagation model.

Background technology

The basic research of Transmission dynamic to as if the contacting of static statistics character of kinetic model character over different networks and corresponding network, comprise known and unknown static geometric sense.And can not as some other subject as the research of the communication process of infectious disease, rumour, data are obtained by doing the mode of testing in crowd, related data, data can only obtain from existing report and record, and these data are often not comprehensively with abundant, be difficult to determine some parameter exactly according to these data, carry out forecasting and control work.Therefore producing data by rational network model and carry out theory and numerically modeling on this basis, is the dynamic (dynamical) important research method of current propagation.

The gossip propagation model of current foundation is mostly the differential equation.Found by theoretical analysis and simulation study, in the iterative process of model, there will be the dynamic phenomena of many complexity.Hopf fork and the Limit Cycle Phenomena produced by fork are exactly one of them.When generation Hopf fork, that just means that the quantity of the people spread rumors in network tends towards stability never, and along with the increase of time, the number spread rumors in network there will be periodic concussion, and concussion is gone down always.This just means that rumour is difficult to be controlled, and will constantly propagate in a network.In reality, we by means such as communication and education, will intervene the key parameter affecting gossip propagation, make system not meet the condition that Hopf fork occurs, avoid system to occur that Hopf diverges as far as possible.If really can cannot avoid the generation of Hopf bifurcation, I just should take control measure, and the amplitude of the limit cycle that Hopf fork is produced reduces, and the concussion scope of such gossip propagation can reduce, and also effectively can suppress the propagation of rumour.

Summary of the invention

The object of the invention is to overcome the deficiencies in the prior art, the control method of the limit cycle amplitude in a kind of gossip propagation model is provided, by controlling the limit cycle amplitude occurred in gossip propagation model, thus controlling the fluctuation range size of the number spread rumors.

For achieving the above object, the limit cycle amplitude control method in a kind of gossip propagation model of the present invention, is characterized in that, comprise the following steps:

(S1) gossip propagation model, is set up

Logistic increasing law is incorporated on SIR gossip propagation model, sets up gossip propagation model:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}=\mathrm{\γS}(1-\frac{S}{k})-\mathrm{\αSI}-\mathrm{\μS}\\ \frac{\mathrm{dI}}{\mathrm{dt}}=\mathrm{\αSI}-\mathrm{\βI}-\mathrm{\μI}\\ \frac{\mathrm{dR}}{\mathrm{dt}}=\mathrm{\βI}-\mathrm{\μR}\end{array}\right.---\left(1\right)$

Wherein, S represents healthy node, I represents propagation node, R represents immune node, α represents that healthy node becomes propagation node with probability α when gossip propagation node and healthy node contact, β represents that propagating node when gossip propagation node and immune node contact becomes immune node with probability β, the μ representation unit time shifts out the number of users of existing network, nodes all in network is regarded as a colony, then γ is the intrinsic growth rate of this colony and γ > μ, k are environmental capacity;

If α is fixing, then along with the increase of propagating node I, and the quantity that healthy node S changes propagation node I into reaches capacity, and is expressed as with piecewise function T (I):

$T\left(I\right)=\left\{\begin{array}{cc}\mathrm{\&alpha;I}& I<I_0\\ \mathrm{\&alpha;}{I}_0& I>=I_0\end{array}\right.,$ Wherein, I ₀represent the number of the propagation node I when the quantity that healthy node S changes propagation node I into reaches capacity, carry out the α I in alternative gossip propagation model with T (I), then gossip propagation model can be expressed as:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}=\mathrm{\&gamma;S}(1-\frac{S}{k})-\mathrm{ST}-\left(I\right)\mathrm{\&mu;S}\\ \frac{\mathrm{dI}}{\mathrm{dt}}=\mathrm{ST}\left(I\right)-\mathrm{\&beta;I}-\mathrm{\&mu;I}\\ \frac{\mathrm{dR}}{\mathrm{dt}}=\mathrm{\&beta;I}-\mathrm{\&mu;R}\end{array}\right.---\left(2\right)$

(S2), gossip propagation model is emulated

By the parameter k chosen in advance ^*, γ ^*, β ^*, μ ^*, α ₀gossip propagation model (2) is emulated, makes gossip propagation model that Hopf fork occur, near equilibrium point, produce limit cycle, obtain I < I ₀time not controlled gossip propagation model, that is:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}={\mathrm{\&gamma;}}^{*}S(1-\frac{S}{k^{*}})-{\mathrm{\&alpha;}}_0\mathrm{SI}-{\mathrm{\&mu;}}^{*}S\\ \frac{\mathrm{dI}}{\mathrm{dt}}={\mathrm{\&alpha;}}_0\mathrm{ST}-{\mathrm{\&beta;}}^{*}I-{\mathrm{\&mu;}}^{*}I\\ \frac{\mathrm{dR}}{\mathrm{dt}}={\mathrm{\&beta;}}^{*}I-{\mathrm{\&mu;}}^{*}R\end{array}\right.---\left(3\right)$

When fork occurs gossip propagation model, if the Jacobi matrix at equilibrium point place is T ^*, T ^*three eigenwerts be respectively λ ₁(α ₀), λ ₂(α ₀) and λ ₃(α ₀), and λ ₁(α ₀) and λ ₂(α ₀) be conjugate complex number, λ ₃(α ₀) be real number;

Eigenvalue λ ₁(α ₀) characteristic of correspondence vector is v ₁, Imv ₁represent v ₁imaginary part, Rev ₁represent v ₁real part, eigenvalue λ ₃(α ₀) characteristic of correspondence vector is v ₃; Thus define matrix T ₀, T ₀=(Imv ₁, Rev ₁, v ₃);

(S3) the limit cycle coefficient of curvature σ of not controlled gossip propagation model, is calculated ₁

(S3.1), linear transformation is carried out to not controlled gossip propagation model

Linear transformation is carried out to the state variable in formula (3), that is:

$\left[\begin{array}{c}S\\ I\\ R\end{array}\right]=\left[\begin{array}{c}{S}^{*}\\ I^{*}\\ R^{*}\end{array}\right]+T_0\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]---\left(4\right)$

Wherein, (S ^*, I ^*, R ^*) ^tthe equilibrium point place of gossip propagation model, the vector of the value composition of S, I, R tri-state variables, () ^trepresent transposition; Matrix T ₀=(Imv ₁, Rev ₁, v ₃) obtain hereinbefore; y ₁, y ₂, y ₃represent three state variables that S, I, R obtain after linear transformation respectively;

Formula (3), after linear change, obtains Jordan standard form, that is:

$\left\{\begin{array}{c}\frac{{\mathrm{dy}}_1}{\mathrm{dt}}=-w_3{y}_2+Q_1(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)\\ \frac{{\mathrm{dy}}_2}{\mathrm{dt}}=w_3{y}_1+Q_2(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)\\ \frac{{\mathrm{dy}}_3}{\mathrm{dt}}=w_4{y}_3+Q_3(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)\end{array}\right.---\left(5\right)$

Wherein w ₃, w ₄be constant;

(S3.2), calculating limit ring coefficient of curvature σ ₁

${\mathrm{\&sigma;}}_1=\mathrm{Re}\{\frac{g_{20}{g}_{11}}{2w_3}i+G_{110}{w}_{11}+\frac{G_{21}+G_{101}{w}_{20}}{2}\}---\left(6\right)$

Wherein, Re{} is for getting real part; I represents imaginary part;

$g_{20}=\frac{1}{4}(\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_1}^2}-\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_2}^2}+2\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_1{\&PartialD;y}_2}+i(\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_1}^2}-\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_2}^2}-2\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_1{\&PartialD;y}_2}))---\left(7\right)$

$g_{11}=\frac{1}{4}(\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_1}^2}+\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_2}^2}+i(\frac{{\&PartialD;}^2{Q}_2}{\&PartialD;{y_1}^2}+\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_2}^2}))---\left(8\right)$

$G_{110}=\frac{1}{2}(\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_1{\&PartialD;y}_3}+\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_2{\&PartialD;y}_3}+i(\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_1{\&PartialD;y}_3}-\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_2{\&PartialD;y}_3}))---\left(9\right)$

$G_{101}=\frac{1}{2}(\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_1{\&PartialD;y}_3}+\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_2{\&PartialD;y}_3}+i(\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_1{\&PartialD;y}_3}-\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_2{\&PartialD;y}_3}))---\left(10\right)$

$w_{11}=-\frac{1}{4{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)}(\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_1}^2}+\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_2}^2})---\left(11\right)$

$w_{20}=-\frac{1}{4(2i{w}_3-{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right))}(\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_1}^2}-\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_2}^2}-2i\frac{{\&PartialD;}^2{Q}_3}{{\&PartialD;y}_1\&PartialD;y_2})---\left(12\right)$

$G_{21}=\frac{1}{8}(\frac{{\&PartialD;}^3{Q}_1}{{{\&PartialD;y}_1}^3}+\frac{{\&PartialD;}^3{Q}_1}{{\&PartialD;y}_1{{\&PartialD;y}_2}^2}+\frac{{\&PartialD;}^3{Q}_2}{{{\&PartialD;y}_1}^2{\&PartialD;y}_2}+\frac{{\&PartialD;}^3{Q}_2}{{{\&PartialD;y}_2}^3}+i(\frac{{\&PartialD;}^3{Q}_2}{{{\&PartialD;y}_1}^3}+\frac{{\&PartialD;}^3{Q}_2}{{\&PartialD;y}_1{{\&PartialD;y}_2}^2}-\frac{{\&PartialD;}^3{Q}_1}{{{\&PartialD;y}_1}^2\&PartialD;y_2}-\frac{{\&PartialD;}^3{Q}_1}{{{\&PartialD;y}_2}^2}))---\left(13\right)$

In above-mentioned formula (7) ~ (13), Q ₁, Q ₂and Q ₃q in formula (5) respectively ₁(y ₁, y ₂, y ₃, α ₀), Q ₂(y ₁, y ₂, y ₃, α ₀) and Q ₃(y ₁, y ₂, y ₃, α ₀) abbreviation;

(S4) be, controlled gossip propagation model by not controlled gossip propagation model conversation

Design a square of FEEDBACK CONTROL function wherein, k ₂for controling parameters, for constant;

Square FEEDBACK CONTROL function U is joined second equation in formula (3), obtain controlled gossip propagation model

(S5) the limit cycle coefficient of curvature of controlled gossip propagation model, is calculated

(S5.1), linear transformation is carried out to controlled gossip propagation model

Linear transformation is carried out to the state variable in formula (14), that is:

$\left[\begin{array}{c}S\\ I\\ R\end{array}\right]=\left[\begin{array}{c}{S}^{*}\\ I^{*}\\ R^{*}\end{array}\right]+T_0\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]---\left(15\right)$

Linear transformation is carried out to a square FEEDBACK CONTROL function U, namely obtains:

$\left\{\begin{array}{c}{\tilde{U}}_1=w_1{k}_2{y_1}^2\\ {\tilde{U}}_2=0\\ {\tilde{U}}_3=w_2{k}_2{y_1}^2\end{array}\right.---\left(16\right)$

Wherein, w ₁and w ₂it is constant;

Formula (14), after linear change, obtains Jordan standard form, that is:

$\left\{\begin{array}{c}\frac{{\mathrm{dy}}_1}{\mathrm{dt}}=-w_3{y}_2+Q_1(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)+{\tilde{U}}_1\\ \frac{{\mathrm{dy}}_2}{\mathrm{dt}}=w_3{y}_1+Q_2(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)+{\tilde{U}}_2\\ \frac{{\mathrm{dy}}_3}{\mathrm{dt}}=w_4{y}_3+Q_3(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)+{\tilde{U}}_3\end{array}\right.---\left(17\right)$

Wherein, w ₃, w ₄be constant, can be expressed as:

${\tilde{U}}_q=\underset{m,n=1}{\overset{3}{\mathrm{\&Sigma;}}}{C}_{\mathrm{mn}}^q{y}_m{y}_n+\underset{m,n,l=1}{\overset{3}{\mathrm{\&Sigma;}}}{D}_{\mathrm{mnl}}^q{y}_m{y}_n{y}_l(q=\mathrm{1,2,3})---\left(18\right)$

the ride gain of the quadratic term obtained after representing square FEEDBACK CONTROL function linear transformation, the ride gain of the cube item obtained after representing square FEEDBACK CONTROL function linear transformation;

(S5.2), calculating limit ring coefficient of curvature

${\widetilde{\mathrm{\&sigma;}}}_1={\mathrm{\&sigma;}}_1+\mathrm{Re}\left\{\mathrm{\&psi;}\right\}+\mathrm{\&phi;}---\left(19\right)$

Wherein, σ ₁be the limit cycle coefficient of curvature of not controlled gossip propagation model, ψ and φ is respectively:

$\begin{array}{c}\mathrm{\&psi;}=\frac{w_{20}}{4}(C_{13}^1-C_{23}^2)+\frac{w_{20}}{4}(C_{13}^2+C_{23}^1)i+\frac{g_{11}}{4w_3}(C_{12}^1+i{C}_{12}^2)\\ -\frac{g_{11}+g_{20}}{4w_3}(C_{11}^2-i{C}_{11}^1)+\frac{g_{11}-g_{20}}{4w_3}(C_{22}^2-i{C}_{22}^1)-\frac{G_{110}}{2{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)}(C_{11}^3+C_{22}^3)\\ +\frac{G_{101}({\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)+2w_{3}i)(C_{22}^3-C_{11}^3)+G_{101}({\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)i-2w_3)C_{12}^3}{4({{4w}_3}^2+{{\mathrm{\&lambda;}}_3}^2\left({\mathrm{\&alpha;}}_0\right))}\\ =-\frac{g_{11}+g_{20}}{4w_3}(C_{11}^2-i{C}_{11}^1)\end{array}---\left(20\right)$

$\begin{array}{c}\mathrm{\&phi;}=\frac{1}{8}(3D_{111}^1+D_{122}^1+D_{112}^2+3D_{222}^2)+\frac{w_{11}}{2}(C_{13}^1+C_{23}^2)+\frac{1}{4w_3}(C_{22}^1{C}_{22}^2-C_{11}^1{C}_{11}^2)\\ +\frac{1}{8w_3}(C_{12}^1(C_{11}^1+C_{22}^1)-C_{12}^2(C_{11}^2+C_{22}^2))-\frac{1}{4{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)}(C_{11}^3+C_{22}^3)(C_{13}^1+C_{23}^2)\\ -\frac{{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)C_{12}^3+2w_3(C_{22}^3-C_{11}^3)}{8({{4w}_3}^2+{{\mathrm{\&lambda;}}_3}^2\left({\mathrm{\&alpha;}}_0\right))}(C_{13}^2+C_{23}^1)+\frac{C_{12}^3+{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)(C_{22}^3-C_{11}^3)}{8({{4w}_3}^2+{{\mathrm{\&lambda;}}_3}^2\left({\mathrm{\&alpha;}}_0\right))}(C_{13}^1-C_{23}^2)\end{array}---\left(21\right)$

(S6), the amplitude of control limit ring

As α < α ₀and | α-α ₀| during < < 1, the limit cycle amplitude of formula (17) can be expressed as approx:

${\tilde{R}}_a=\sqrt{-\frac{{\mathrm{\&epsiv;}}^{\&prime;}\left({\mathrm{\&alpha;}}_0\right)}{{\widetilde{\mathrm{\&sigma;}}}_1}(\mathrm{\&alpha;}-{\mathrm{\&alpha;}}_0)}=\sqrt{-\frac{{\mathrm{\&epsiv;}}^{\&prime;}\left({\mathrm{\&alpha;}}_0\right)}{a^{*}{k}_2+b^{*}}(\mathrm{\&alpha;}-{\mathrm{\&alpha;}}_0)}---\left(22\right)$

Wherein, a ^*and b ^*for constant, ε ' (α ₀) be eigenvalue λ ₁(α ₀) real part to α at α ₀differentiate in place, its expression formula is:

${\mathrm{\&epsiv;}}^{\&prime;}\left({\mathrm{\&alpha;}}_0\right)=\frac{\&PartialD;\mathrm{\&epsiv;}\left(\mathrm{\&alpha;}\right)}{\&PartialD;\mathrm{\&alpha;}}{|}_{\mathrm{\&alpha;}={\mathrm{\&alpha;}}_0}$

Again by adjustment controling parameters k ₂carry out the amplitude size of control limit ring.

Goal of the invention of the present invention is achieved in that

Limit cycle amplitude control method in a kind of gossip propagation model of the present invention, a new gossip propagation model is set up on based on SIR model, consider that infection rate declines on the contrary along with the number of nodes increase that spreads rumors, introduce a piecewise function and describe infection rate, so just more accurately can must describe the gossip propagation behavior in complex network.Secondly, on the basis of this new model, introduce a control function, joined by control function in this model, the limit cycle amplitude realized occurring in gossip propagation model controls, thus controls the fluctuation range size of the number spread rumors.

Limit cycle amplitude control method in a kind of gossip propagation of the present invention model also has following beneficial effect:

(1) by the improvement to classical SIR gossip propagation model, the gossip propagation behavior in complex network is described more accurately, for the analysis and control of gossip propagation provides theoretical foundation.

(2) by introducing control function, the limit cycle amplitude realized occurring in gossip propagation model controls, thus controls the fluctuation range size of the number spread rumors.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of the limit cycle amplitude control method in a kind of gossip propagation model of the present invention;

Fig. 2 is not controlled gossip propagation model state variable variation diagram in time when there is limit cycle;

Fig. 3 is the changing trend diagram of not controlled gossip propagation model three dimensions track when there is limit cycle;

Fig. 4 is the phasor of not controlled gossip propagation model I-S plane when there is limit cycle;

Fig. 5 is under different controling parameters, the state variable S variation diagram in time of controlled gossip propagation model;

Fig. 6 is under different controling parameters, amplitude comparison diagram when limit cycle appears in controlled gossip propagation model.

Embodiment

Below in conjunction with accompanying drawing, the specific embodiment of the present invention is described, so that those skilled in the art understands the present invention better.Requiring particular attention is that, in the following description, when perhaps the detailed description of known function and design can desalinate main contents of the present invention, these are described in and will be left in the basket here.

Embodiment

Fig. 1 is the process flow diagram of the limit cycle amplitude control method in a kind of gossip propagation model of the present invention.

In the present embodiment, as shown in Figure 1, the gossip propagation control method in a kind of complex network of the present invention, mainly comprises the following steps:

S1, set up gossip propagation model

In actual life, the number of users adding network in unit interval is unfixed, research is had to show, in the similar biotic population of growth of number of users, population increases, meet logistic increasing law, therefore logistic increasing law is incorporated on SIR gossip propagation model, sets up gossip propagation model:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}=\mathrm{\&gamma;S}(1-\frac{S}{k})-\mathrm{\&alpha;SI}-\mathrm{\&mu;S}\\ \frac{\mathrm{dI}}{\mathrm{dt}}=\mathrm{\&alpha;SI}-\mathrm{\&beta;I}-\mathrm{\&mu;I}\\ \frac{\mathrm{dR}}{\mathrm{dt}}=\mathrm{\&beta;I}-\mathrm{\&mu;R}\end{array}\right.---\left(1\right)$

Wherein, S represents healthy node, I represents propagation node, R represents immune node, α to represent that when gossip propagation node and healthy node contact healthy node becomes with probability α and propagates node I, β represents that propagating node when gossip propagation node and immune node contact becomes immune node with probability β, the μ representation unit time shifts out the number of users of existing network, nodes all in network is regarded as a colony, then γ is the intrinsic growth rate of this colony and γ > μ, k are environmental capacity;

On this basis, consider that α is fixing, so along with the increase of propagating node I, be propagate the number of users of node not meet the linear rule increased from healthy node transitions, because along with the increase of I, healthy node transitions is that the quantity propagating node will slowly reach capacity, and can not a linear increase, so carry out the α I in alternative gossip propagation model with a piecewise function T (I), then gossip propagation model can be expressed as:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}=\mathrm{\&gamma;S}(1-\frac{S}{k})-\mathrm{ST}-\left(I\right)\mathrm{\&mu;S}\\ \frac{\mathrm{dI}}{\mathrm{dt}}=\mathrm{ST}\left(I\right)-\mathrm{\&beta;I}-\mathrm{\&mu;I}\\ \frac{\mathrm{dR}}{\mathrm{dt}}=\mathrm{\&beta;I}-\mathrm{\&mu;R}\end{array}\right.---\left(2\right)$

Wherein, $T\left(I\right)=\left\{\begin{array}{cc}\mathrm{\&alpha;I}& I<I_0\\ \mathrm{\&alpha;}{I}_0& I>=I_0\end{array}\right.,$ I ₀represent the number of the propagation node I when the quantity that healthy node S changes propagation node I into reaches capacity;

S2, gossip propagation model to be emulated

In the present embodiment, following parameters k is chosen in advance ^*=100, γ ^*=0.099, β ^*=0.05, μ ^*=0.05, α ₀=0.1055, I ₀=2; Being emulated by MATLAB software, can there is Hopf fork in gossip propagation model again, and then at equilibrium point E ^*limit cycle is produced near (0.9945,0.4784,0.4784).Substitute into above-mentioned parameter, an E can be balanced ^*the Jacobi matrix at (0.9945,0.4784,0.4784) place can approximate expression be:

$T^{*}\&ap;\left[\begin{array}{ccc}0& -0.1& 0\\ 0.048& 0& 0\\ 0& 0.05& -0.05\end{array}\right]$

Its characteristic of correspondence polynomial expression is: λ ³+ 0.05 λ ²+ 0.0048 λ+0.0024=0;

Thus obtain three eigenwert: λ _1,2≈ ± 0.0693i, λ ₃≈-0.05.;

Wherein, λ ₁characteristic of correspondence vector is v ₁=(1 ,-0.0693i ,-0.3287-0.2372i) ^t, λ ₃characteristic of correspondence vector is v ₃=(0,0,1) ^t.Thus define matrix T ₀, T ₀=(Imv ₁, Rev ₁, v ₃);

Can obtain $T_0=\left[\begin{array}{ccc}0& 1& 0\\ -0.693& 0& 0\\ -0.2372& -0.3287& 1\end{array}\right].$

As shown in Figure 2, when limit cycle appears in gossip propagation model, along with time variations, all there is periodic solution in its state variable S, I and R, and wherein, state variable S, I and R are corresponding to Fig. 2 (a) ~ Fig. 2 (c) respectively.As shown in Figure 3, along with time variations, gossip propagation model is constantly repeat around a ring at the movement locus of three dimensions (S, I and R) all the time, the three-dimensional plot in Fig. 3 is projected to I-S phase plane, then obtains limit cycle as shown in Figure 4.After substituting into the parameter of preliminary election, now gossip propagation model can be expressed as:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}=0.049S-0.00099S^2-0.10055\mathrm{SI}\\ \frac{\mathrm{dI}}{\mathrm{dt}}=0.10055\mathrm{SI}-0.1I\\ \frac{\mathrm{dR}}{\mathrm{dt}}=0.05I-0.05R\end{array},\mathrm{if\; I}<I_0\right.---\left(3\right)$

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}=-0.1521S-0.00099S^2\\ \frac{\mathrm{dI}}{\mathrm{dt}}=0.2011S-0.1I\\ \frac{\mathrm{dR}}{\mathrm{dt}}=0.05I-0.05R\end{array}\right.,\mathrm{if\; I}\&GreaterEqual;I_0$ (giving up)

Because fork is at equilibrium point E ^*(0.9945,0.4784,0.4784) place produces, and equilibrium point E ^*the generation of (0.9945,0.4784,0.4784) is at I < I ₀condition under obtain, therefore, by I>=I ₀time equation give up.

(S3) the limit cycle coefficient of curvature σ of not controlled gossip propagation model, is calculated ₁

(S3.1), linear transformation is carried out to not controlled gossip propagation model

Linear transformation is carried out to the state variable in formula (3), that is:

$\left[\begin{array}{c}S\\ I\\ R\end{array}\right]=\left[\begin{array}{c}{S}^{*}\\ I^{*}\\ R^{*}\end{array}\right]+T_0\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]---\left(4\right)$

Wherein, (S ^*, I ^*, R ^*) ^tthe vector of the value composition of S, I, R tri-state variables at the equilibrium point place of gossip propagation model, () ^trepresent transposition; y ₁, y ₂, y ₃represent three state variables that S, I, R obtain after linear transformation respectively; In the present embodiment, (S ^*, I ^*, R ^*) ^t=(0.9945,0.4784,0.4784) ^t; $T_0=\left[\begin{array}{ccc}0& 1& 0\\ -0.693& 0& 0\\ -0.2372& -0.3287& 1\end{array}\right].$

Formula (3), after linear change, obtains Jordan standard form, that is:

$\left\{\begin{array}{c}\frac{{\mathrm{dy}}_1}{\mathrm{dt}}=-0.0693y_2+Q_1(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)\\ \frac{{\mathrm{dy}}_2}{\mathrm{dt}}=0.0693y_1+Q_2(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)\\ \frac{{\mathrm{dy}}_3}{\mathrm{dt}}=0.05y_3+Q_3(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)\end{array}\right.---\left(5\right)$

Wherein, $\left\{\begin{array}{c}{Q}_1=0.10055y_1{y}_2\\ Q_2=-0.00099{y_2}^2+0.06968y_1{y}_2\\ Q_3=-0.0003254{y_2}^2+0.04712y_1{y}_2\end{array}\right.;$ As can be seen from formula (5), w ₃, w ₄value is respectively: w ₃=0.0693, w ₄=-0.05;

(S3.2), calculating limit ring coefficient of curvature σ ₁

${\mathrm{\&sigma;}}_1=\mathrm{Re}\{\frac{g_{20}{g}_{11}}{2w_3}i+G_{110}{w}_{11}+\frac{G_{21}+G_{101}{w}_{20}}{2}\}---\left(6\right)$

Wherein, Re{} is for getting real part; I represents imaginary part;

$\begin{array}{c}{g}_{20}=\frac{1}{4}(\frac{{\&PartialD;}^2{Q}_1}{\&PartialD;{y_1}^2}-\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_2}^2}+2\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_1\&PartialD;y_2}+i(\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_1}^2}-\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_2}^2}-2\frac{{\&PartialD;}^2{Q}_1}{\&PartialD;y_1{\&PartialD;y}_1}))\\ =0.03484-0.0500275i\end{array}---\left(7\right)$

$g_{11}=\frac{1}{4}(\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_1}^2}+\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_2}^2}+i(\frac{{\&PartialD;}^2{Q}_2}{\&PartialD;{y_1}^2}+\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_2}^2}))=-0.0002475i---\left(8\right)$

$G_{110}=\frac{1}{2}(\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_1{\&PartialD;y}_3}+\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_2{\&PartialD;y}_3}+i(\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_1{\&PartialD;y}_3}-\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_2{\&PartialD;y}_3}))=0---\left(9\right)$

$G_{101}=\frac{1}{2}(\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_1{\&PartialD;y}_3}+\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_2{\&PartialD;y}_3}+i(\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_1{\&PartialD;y}_3}-\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_2{\&PartialD;y}_3}))=0---\left(10\right)$

$w_{11}=-\frac{1}{4{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)}(\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_1}^2}+\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_2}^2})=-0.001627---\left(11\right)$

$\begin{array}{c}{w}_{20}=-\frac{1}{4(2i{w}_3-{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right))}(\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_1}^2}-\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_2}^2}-2i\frac{{\&PartialD;}^2{Q}_3}{{\&PartialD;y}_1\&PartialD;y_2})\\ =-0.150224-0.05478i\end{array}---\left(12\right)$

$G_{21}=\frac{1}{8}(\frac{{\&PartialD;}^3{Q}_1}{{{\&PartialD;y}_1}^3}+\frac{{\&PartialD;}^3{Q}_1}{{\&PartialD;y}_1{{\&PartialD;y}_2}^2}+\frac{{\&PartialD;}^3{Q}_2}{{{\&PartialD;y}_1}^2{\&PartialD;y}_2}+\frac{{\&PartialD;}^3{Q}_2}{{{\&PartialD;y}_2}^3}+i(\frac{{\&PartialD;}^3{Q}_2}{{{\&PartialD;y}_1}^3}+\frac{{\&PartialD;}^3{Q}_2}{{\&PartialD;y}_1{{\&PartialD;y}_2}^2}-\frac{{\&PartialD;}^3{Q}_1}{{{\&PartialD;y}_1}^2\&PartialD;y_2}-\frac{{\&PartialD;}^3{Q}_1}{{{\&PartialD;y}_2}^2}))=0---\left(13\right)$

Following calculating limit ring coefficient of curvature σ ₁formula (7) ~ (13) are substituted in formula (6), obtain by (Lyapunov exponent):

${\mathrm{\&sigma;}}_1=\mathrm{Re}\{\frac{g_{20}{g}_{11}}{2w_3}i+G_{110}{w}_{11}+\frac{G_{21}+G_{101}{w}_{20}}{2}\}=2.1557\&times;{10}^{-6}$

S4, be controlled gossip propagation model by not controlled gossip propagation model conversation

The gossip propagation model that formula (3) represents is not controlled gossip propagation model, therefore, when being converted into controlled gossip propagation model, needs first to design a square of FEEDBACK CONTROL function U=k ₂(I-0.4784) ², join second equation in not controlled gossip propagation model (3), thus be converted into controlled gossip propagation model.In this step, after adding square FEEDBACK CONTROL function, ensure that the balance point position of gossip propagation model and bifurcation point position do not change.

Controlled gossip propagation model can be expressed as:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}=0.049S-0.00099S^2-0.10055\mathrm{SI}\\ \frac{\mathrm{dI}}{\mathrm{dt}}=0.10055\mathrm{SI}-0.1I+k_2{(I-0.4784)}^2\\ \frac{\mathrm{dR}}{\mathrm{dt}}=0.05I-0.05R\end{array}\right.---\left(14\right)$

(S5) the limit cycle coefficient of curvature of controlled gossip propagation model, is calculated

(S5.1), linear transformation is carried out to controlled gossip propagation model

Linear transformation is carried out to the state variable in formula (14), that is:

$\left[\begin{array}{c}S\\ I\\ R\end{array}\right]=\left[\begin{array}{c}{S}^{*}\\ I^{*}\\ R^{*}\end{array}\right]+T_0\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]---\left(15\right)$

Linear transformation is carried out to a square FEEDBACK CONTROL function U, namely obtains:

$\left\{\begin{array}{c}{\tilde{U}}_1=-0.693w{k}_2{y_1}^2\\ {\tilde{U}}_2=0\\ {\tilde{U}}_3=-0.16438k_2{y_1}^2\end{array}\right.---\left(16\right)$

Wherein, w ₁, w ₂value is respectively: w ₁=-0.693, w ₂=-0.16438;

Formula (14), after linear change, obtains Jordan standard form, that is:

$\left\{\begin{array}{c}\frac{{\mathrm{dy}}_1}{\mathrm{dt}}=-0.0693y_2+Q_1(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)+{\tilde{U}}_1\\ \frac{{\mathrm{dy}}_2}{\mathrm{dt}}=0.0693y_1+Q_2(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)+{\tilde{U}}_2\\ \frac{{\mathrm{dy}}_3}{\mathrm{dt}}=-0.05y_3+Q_3(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)+{\tilde{U}}_3\end{array}\right.---\left(17\right)$

Wherein, can be expressed as:

${\tilde{U}}_q=\underset{m,n=1}{\overset{3}{\mathrm{\&Sigma;}}}{C}_{\mathrm{mn}}^q{y}_m{y}_n+\underset{m,n,l=1}{\overset{3}{\mathrm{\&Sigma;}}}{D}_{\mathrm{mnl}}^q{y}_m{y}_n{y}_l(q=\mathrm{1,2,3})---\left(18\right)$

the ride gain of the quadratic term obtained after representing square FEEDBACK CONTROL function linear transformation, the ride gain of the cube item obtained after representing square FEEDBACK CONTROL function linear transformation;

(S5.2), calculating limit ring coefficient of curvature

${\widetilde{\mathrm{\&sigma;}}}_1={\mathrm{\&sigma;}}_1+\mathrm{Re}\left\{\mathrm{\&psi;}\right\}+\mathrm{\&phi;}---\left(19\right)$

Wherein, σ ₁be the limit cycle coefficient of curvature of not controlled gossip propagation model, ψ and φ is respectively:

$\begin{array}{c}\mathrm{\&psi;}=\frac{w_{20}}{4}(C_{13}^1-C_{23}^2)+\frac{w_{20}}{4}(C_{13}^2+C_{23}^1)i+\frac{g_{11}}{4w_3}(C_{12}^1+i{C}_{12}^2)\\ -\frac{g_{11}+g_{20}}{4w_3}(C_{11}^2-i{C}_{11}^1)+\frac{g_{11}-g_{20}}{4w_3}(C_{22}^2-i{C}_{22}^1)-\frac{G_{110}}{2{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)}(C_{11}^3+C_{22}^3)\\ +\frac{G_{101}({\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)+2w_{3}i)(C_{22}^3-C_{11}^3)+G_{101}({\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)i-2w_3)C_{12}^3}{4({{4w}_3}^2+{{\mathrm{\&lambda;}}_3}^2\left({\mathrm{\&alpha;}}_0\right))}\\ =-\frac{g_{11}+g_{20}}{4w_3}(C_{11}^2-i{C}_{11}^1)=0.1256875k_2+0.0871k_{2}i\end{array}---\left(20\right)$

$\begin{array}{c}\mathrm{\&phi;}=\frac{1}{8}(3D_{111}^1+D_{122}^1+D_{112}^2+3D_{222}^2)+\frac{w_{11}}{2}(C_{13}^1+C_{23}^2)+\frac{1}{4w_3}(C_{22}^1{C}_{22}^2-C_{11}^1{C}_{11}^2)\\ +\frac{1}{8w_3}(C_{12}^1(C_{11}^1+C_{22}^1)-C_{12}^2(C_{11}^2+C_{22}^2))-\frac{1}{4{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)}(C_{11}^3+C_{22}^3)(C_{13}^1+C_{23}^2)\\ -\frac{{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)C_{12}^3+2w_3(C_{22}^3-C_{11}^3)}{8({{4w}_3}^2+{{\mathrm{\&lambda;}}_3}^2\left({\mathrm{\&alpha;}}_0\right))}(C_{13}^2+C_{23}^1)+\frac{C_{12}^3+{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)(C_{22}^3-C_{11}^3)}{8({{4w}_3}^2+{{\mathrm{\&lambda;}}_3}^2\left({\mathrm{\&alpha;}}_0\right))}(C_{13}^1-C_{23}^2)\\ =0\end{array}---\left(21\right)$

Formula (20) and (21) are substituted into formula (19), the limit cycle coefficient of curvature of controlled system can be obtained:

${\widetilde{\mathrm{\&sigma;}}}_1={\mathrm{\&sigma;}}_1+\mathrm{Re}\left\{\mathrm{\&psi;}\right\}+\mathrm{\&phi;}=2.1557\&times;{10}^{-6}+0.1256875k_2$

(S6), the amplitude of control limit ring

As α < α ₀and | α-α ₀| during < < 1, the limit cycle amplitude of formula (17) can be expressed as approx:

${\tilde{R}}_a=\sqrt{-\frac{{\mathrm{\&epsiv;}}^{\&prime;}\left({\mathrm{\&alpha;}}_0\right)}{{\widetilde{\mathrm{\&sigma;}}}_1}(\mathrm{\&alpha;}-{\mathrm{\&alpha;}}_0)}=\sqrt{-\frac{{\mathrm{\&epsiv;}}^{\&prime;}\left({\mathrm{\&alpha;}}_0\right)}{a^{*}{k}_2+b^{*}}(\mathrm{\&alpha;}-{\mathrm{\&alpha;}}_0)}---\left(22\right)$

Wherein, ε ' (α ₀) be eigenvalue λ ₁(α ₀) real part to α at equilibrium point α ₀differentiate in place, its expression formula is:

${\mathrm{\&epsiv;}}^{\&prime;}\left({\mathrm{\&alpha;}}_0\right)=\frac{\&PartialD;\mathrm{\&epsiv;}\left(\mathrm{\&alpha;}\right)}{\&PartialD;\mathrm{\&alpha;}}{|}_{\mathrm{\&alpha;}={\mathrm{\&alpha;}}_0}$

Again by adjustment controling parameters k ₂carry out the amplitude size of control limit ring.

Below by choosing different controling parameters k ₂value, to emulation result analyze as follows:

Choose following key parameter k=100, α=0.1, γ=0.099, μ=0.05, β=0.05, I ₀=2, as controling parameters k ₂when=0, the state variable S of system over time trend as shown in Fig. 5 (a); As controling parameters k ₂when=-0.000006, the state variable S of system over time trend as shown in Fig. 5 (b); As controling parameters k ₂when=0.00004, the state variable S of system over time trend as shown in Fig. 5 (c).

When α=0.1, k ₂when=0, the amplitude being calculated limit cycle by formula (22) is 1.0652, and be 1.1045 by emulating Fig. 5 (a) numerical solution obtained, error is 3%; When α=0.1, k ₂when=-0.000006, the amplitude being calculated limit cycle by formula (22) is 1.3210, and be 1.3155 by emulating Fig. 5 (b) numerical solution obtained, error is 0.4%; When α=0.1, k ₂when=0.00004, the amplitude being calculated limit cycle by formula (22) is 0.5835, and be 0.5855 by emulating Fig. 5 numerical solution obtained, error is 0.35%.

In sum, access control function U=k ₂(I-0.4784) ²can the amplitude of limit cycle in control system effectively, and the limit cycle amplitude after controlling can be estimated by formula (22), and error is not too large.We also depict k ₂when getting different value, the phasor of track in S-I plane of system, as shown in Figure 6.

From Fig. 6 can intuitively to, add controller U=k ₂(I-0.4784) ²afterwards, by regulable control parameter k ₂value, just can increase or in reduction system due to the amplitude size of limit cycle that Hopf fork produces.

Although be described the illustrative embodiment of the present invention above; so that those skilled in the art understand the present invention; but should be clear; the invention is not restricted to the scope of embodiment; to those skilled in the art; as long as various change to limit and in the spirit and scope of the present invention determined, these changes are apparent, and all innovation and creation utilizing the present invention to conceive are all at the row of protection in appended claim.

Claims (1)

1. the limit cycle amplitude control method in gossip propagation model, is characterized in that, comprise the following steps:

(1) gossip propagation model, is set up

Logistic increasing law is incorporated on SIR gossip propagation model, sets up gossip propagation model:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}=\mathrm{\&gamma;S}(1-\frac{S}{k})-\mathrm{\&alpha;SI}-\mathrm{\&mu;S}\\ \frac{\mathrm{dI}}{\mathrm{dt}}=\mathrm{\&alpha;SI}-\mathrm{\&beta;I}-\mathrm{\&mu;I}\\ \frac{\mathrm{dR}}{\mathrm{dt}}=\mathrm{\&beta;I}-\mathrm{\&mu;R}\end{array}\right.---\left(1\right)$

Wherein, S represents healthy node, I represents propagation node, R represents immune node, α represents that healthy node becomes propagation node with probability α when gossip propagation node and healthy node contact, β represents that propagating node when gossip propagation node and immune node contact becomes immune node with probability β, the μ representation unit time shifts out the number of users of existing network, nodes all in network is regarded as a colony, then γ is the intrinsic growth rate of this colony and γ > μ, k are environmental capacity;

If α is fixing, then along with the increase of propagating node I, and the quantity that healthy node S changes propagation node I into reaches capacity, and is expressed as with piecewise function T (I):

$T\left(I\right)=\left\{\begin{array}{cc}\mathrm{\&alpha;I}& I<I_0\\ \mathrm{\&alpha;}{I}_0& I>=I_0\end{array}\right.,$ Wherein, I ₀represent the number of the propagation node I when the quantity that healthy node S changes propagation node I into reaches capacity, carry out the α I in alternative gossip propagation model with T (I), then gossip propagation model can be expressed as:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}=\mathrm{\&gamma;S}(1-\frac{S}{k})-\mathrm{ST}\left(I\right)-\mathrm{\&mu;S}\\ \frac{\mathrm{dI}}{\mathrm{dt}}=\mathrm{ST}\left(I\right)-\mathrm{\&beta;I}-\mathrm{\&mu;I}\\ \frac{\mathrm{dR}}{\mathrm{dt}}=\mathrm{\&beta;I}-\mathrm{\&mu;R}\end{array}\right.---\left(2\right)$

(2), gossip propagation model is emulated

By the parameter k chosen in advance ^*, γ ^*, β ^*, μ ^*, α ₀gossip propagation model (2) is emulated, makes gossip propagation model that Hopf fork occur, near equilibrium point, produce limit cycle, obtain I < I ₀time not controlled gossip propagation model, that is:

$\left\{\begin{array}{c}\frac{\mathrm{dS}}{\mathrm{dt}}={\mathrm{\&gamma;}}^{*}S(1-\frac{S}{k^{*}})-{\mathrm{\&alpha;}}_0\mathrm{SI}-{\mathrm{\&mu;}}^{*}S\\ \frac{\mathrm{dI}}{\mathrm{dt}}={\mathrm{\&alpha;}}_0\mathrm{SI}-{\mathrm{\&beta;}}^{*}I-{\mathrm{\&mu;}}^{*}I\\ \frac{\mathrm{dR}}{\mathrm{dt}}={\mathrm{\&beta;}}^{*}I-{\mathrm{\&mu;}}^{*}R\end{array}\right.---\left(3\right)$

When fork occurs gossip propagation model, if the Jacobi matrix at equilibrium point place is T ^*, T ^*three or three eigenwerts be respectively λ ₁(α ₀), λ ₂(α ₀) and λ ₃(α ₀), and λ ₁(α ₀) and λ ₂(α ₀) be conjugate complex number, λ ₃(α ₀) be real number;

Eigenvalue λ ₁(α ₀) characteristic of correspondence vector is v ₁, Imv ₁represent v ₁imaginary part, Rev ₁represent v ₁real part, eigenvalue λ ₃(α ₀) characteristic of correspondence vector is v ₃; Thus define matrix T ₀, T ₀=(Imv ₁, Rev ₁, v ₃);

(3) the limit cycle coefficient of curvature σ of not controlled gossip propagation model, is calculated ₁

(3.1), linear transformation is carried out to not controlled gossip propagation model

Linear transformation is carried out to the state variable in formula (3), that is:

$\left[\begin{array}{c}S\\ I\\ R\end{array}\right]=\left[\begin{array}{c}{S}^{*}\\ I^{*}\\ R^{*}\end{array}\right]+T_0\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]---\left(4\right)$

Wherein, (S ^*, I ^*, R ^*) ^tthe equilibrium point place of gossip propagation model, the vector of the value composition of S, I, R tri-state variables, () ^trepresent transposition; y ₁, y ₂, y ₃represent three state variables that S, I, R obtain after linear transformation respectively;

Formula (3), after linear change, obtains Jordan standard form, that is:

$\left\{\begin{array}{c}\frac{{\mathrm{dy}}_1}{\mathrm{dt}}=-w_3{y}_2+Q_1(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)\\ \frac{{\mathrm{dy}}_2}{\mathrm{dt}}=w_3{y}_1+Q_2(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)\\ \frac{{\mathrm{dy}}_3}{\mathrm{dt}}=w_4{y}_3+Q_3(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)\end{array}\right.---\left(5\right)$

Wherein w ₃, w ₄be constant;

(3.2), calculating limit ring coefficient of curvature σ ₁

${\mathrm{\&sigma;}}_1=\mathrm{Re}\{\frac{g_{20}{g}_{11}}{{2w}_3}i+G_{110}{w}_{11}+\frac{G_{21}+G_{101}{w}_{20}}{2}\}---\left(6\right)$

Wherein, Re{} is for getting real part; I represents imaginary part;

$g_{20}=\frac{1}{4}(\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_1}^2}-\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_2}^2}+2\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_1{\&PartialD;y}_2}+i(\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_1}^2}-\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_2}^2}-2\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_1{\&PartialD;y}_2}))---\left(7\right)$

$g_{11}=\frac{1}{4}(\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_1}^2}+\frac{{\&PartialD;}^2{Q}_1}{{{\&PartialD;y}_2}^2}+i(\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_1}^2}+\frac{{\&PartialD;}^2{Q}_2}{{{\&PartialD;y}_2}^2}))---\left(8\right)$

$G_{110}=\frac{1}{2}(\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_1{\&PartialD;y}_3}+\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_2{\&PartialD;y}_3}+i(\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_1{\&PartialD;y}_3}-\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_2{\&PartialD;y}_3}))---\left(9\right)$

$G_{101}=\frac{1}{2}(\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_1{\&PartialD;y}_3}+\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_2{\&PartialD;y}_3}+i(\frac{{\&PartialD;}^2{Q}_2}{{\&PartialD;y}_1{\&PartialD;y}_3}-\frac{{\&PartialD;}^2{Q}_1}{{\&PartialD;y}_2{\&PartialD;y}_3}))---\left(10\right)$

$w_{11}=-\frac{1}{{4\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)}(\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_1}^2}+\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_2}^2})---\left(11\right)$

$w_{20}=-\frac{1}{4({2\mathrm{iw}}_3-{\mathrm{\&lambda;}}_3\left(a_0\right))}(\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_1}^2}-\frac{{\&PartialD;}^2{Q}_3}{{{\&PartialD;y}_2}^2}-2i\frac{{\&PartialD;}^2{Q}_3}{{\&PartialD;y}_1{\&PartialD;y}_2})---\left(12\right)$

$G_{21}=\frac{1}{8}(\frac{{\&PartialD;}^3{Q}_1}{{{\&PartialD;y}_1}^3}+\frac{{\&PartialD;}^3{Q}_1}{{\&PartialD;y}_1{{\&PartialD;y}_2}^2}+\frac{{\&PartialD;}^3{Q}_2}{{{\&PartialD;y}_1}^2{\&PartialD;y}_2}+\frac{{\&PartialD;}^3{Q}_2}{{{\&PartialD;y}_2}^3}+i(\frac{{\&PartialD;}^3{Q}_2}{{{\&PartialD;y}_1}^3}+\frac{{\&PartialD;}^3{Q}_2}{{\&PartialD;y}_1{{\&PartialD;y}_2}^2}-\frac{{\&PartialD;}^3{Q}_1}{{{\&PartialD;y}_1}^2{\&PartialD;y}_2}-\frac{{\&PartialD;}^3{Q}_1}{{{\&PartialD;y}_2}^3}))---\left(13\right)$

In above-mentioned formula (7) ~ (13), Q ₁, Q ₂and Q ₃q in formula (5) respectively ₁(y ₁, y ₂, y ₃, α ₀), Q ₂(y ₁, y ₂, y ₃, α ₀) and Q ₃(y ₁, y ₂, y ₃, α ₀) abbreviation;

(4) be, controlled gossip propagation model by not controlled gossip propagation model conversation

Design a square of FEEDBACK CONTROL function wherein, k ₂for controling parameters, for constant;

Square FEEDBACK CONTROL function U is joined first equation in formula (3), obtain controlled gossip propagation model

(5) the limit cycle coefficient of curvature of controlled gossip propagation model, is calculated

(5.1), linear transformation is carried out to controlled gossip propagation model

Linear transformation is carried out to the state variable in formula (14), that is:

$\left[\begin{array}{c}S\\ I\\ R\end{array}\right]=\left[\begin{array}{c}{S}^{*}\\ I^{*}\\ R^{*}\end{array}\right]+T_0\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]---\left(15\right)$

Linear transformation is carried out to a square FEEDBACK CONTROL function U, namely obtains:

$\left\{\begin{array}{c}{\tilde{U}}_1=w_1{k}_2{y_1}^2\\ {\tilde{U}}_2=0\\ {\tilde{U}}_3=w_2{k}_2{y_1}^2\end{array}\right.---\left(16\right)$

Wherein, w ₁and w ₂it is constant;

Formula (14), after linear change, obtains Jordan standard form, that is:

$\left\{\begin{array}{c}\frac{{\mathrm{dy}}_1}{\mathrm{dt}}=-w_3{y}_2+Q_1(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)+{\tilde{U}}_1\\ \frac{{\mathrm{dy}}_2}{\mathrm{dt}}=w_3{y}_1+Q_2(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)+{\tilde{U}}_2\\ \frac{{\mathrm{dy}}_3}{\mathrm{dt}}=w_4{y}_3+Q_3(y_1,y_2,y_3,{\mathrm{\&alpha;}}_0)+{\tilde{U}}_3\end{array}\right.---\left(17\right)$

Wherein, w ₃, w ₄be constant, can be expressed as:

${\tilde{U}}_q=\underset{m,n=1}{\overset{3}{\mathrm{\&Sigma;}}}{C}_{\mathrm{mn}}^q{y}_m{y}_n+\underset{m,n,l=1}{\overset{3}{\mathrm{\&Sigma;}}}{D}_{\mathrm{mnl}}^q{y}_m{y}_n{y}_l(q=\mathrm{1,2,3})---\left(18\right)$

the ride gain of the quadratic term obtained after representing square FEEDBACK CONTROL function linear transformation, the ride gain of the cube item obtained after representing square FEEDBACK CONTROL function linear transformation;

(5.2), calculating limit ring coefficient of curvature

${\widetilde{\mathrm{\&sigma;}}}_1={\mathrm{\&sigma;}}_1+\mathrm{Re}\left\{\mathrm{\&psi;}\right\}+\mathrm{\&phi;}---\left(\text{19}\right)$

Wherein, σ ₁be the limit cycle coefficient of curvature of not controlled gossip propagation model, ψ and φ is respectively:

$\begin{array}{c}\mathrm{\&psi;}=\frac{w_{20}}{4}(C_{13}^1-C_{23}^2)+\frac{w_{20}}{4}(C_{13}^2+C_{23}^1)i+\frac{g_{11}}{{4w}_3}(C_{12}^1+i{C}_{12}^2)\\ -\frac{g_{11}+g_{20}}{4w_3}(C_{11}^2-{\mathrm{iC}}_{11}^1)+\frac{g_{11}-g_{20}}{{4w}_3}(C_{22}^2-i{C}_{22}^1)-\frac{G_{110}}{{2\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)}(C_{11}^3+C_{22}^3)\\ +\frac{G_{101}({\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)+{2w}_{3}i)(C_{22}^3-C_{11}^3)+G_{101}({\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)i-{2w}_3)C_{12}^3}{4({{4w}_3}^2+{{\mathrm{\&lambda;}}_3}^2\left({\mathrm{\&alpha;}}_0\right))}\\ =-\frac{g_{11}+g_{20}}{4w_3}(C_{11}^2-i{C}_{11}^1)\end{array}---\left(20\right)$

$\begin{array}{c}\mathrm{\&phi;}=\frac{1}{8}({3D}_{111}^1+D_{122}^1+D_{112}^2+{3D}_{222}^2)+\frac{w_{11}}{2}(C_{13}^1+C_{23}^2)+\frac{1}{{4w}_3}(C_{22}^1{C}_{22}^2-C_{11}^1{C}_{11}^2)\\ +\frac{1}{{8w}_3}(C_{12}^1(C_{11}^1+C_{22}^1)-C_{12}^2(C_{11}^2+C_{22}^2))-\frac{1}{{4\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)}(C_{11}^3+C_{22}^3)(C_{13}^1+C_{23}^2)\\ -\frac{{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)C_{12}^3+{2w}_3(C_{22}^3-C_{11}^3)}{8({{4w}_3}^2+{{\mathrm{\&lambda;}}_3}^2\left({\mathrm{\&alpha;}}_0\right))}(C_{13}^2+C_{23}^1)+\frac{C_{12}^3+{\mathrm{\&lambda;}}_3\left({\mathrm{\&alpha;}}_0\right)(C_{22}^3-C_{11}^3)}{8({{4w}_3}^2+{{\mathrm{\&lambda;}}_3}^2\left({\mathrm{\&alpha;}}_0\right))}(C_{13}^1-C_{23}^2)\end{array}---\left(21\right)$

(6), the amplitude of control limit ring

As α < α ₀and | α-α ₀| during < < 1, the limit cycle amplitude of formula (17) can be expressed as approx:

${\tilde{R}}_a=\sqrt{-\frac{{\mathrm{\&epsiv;}}^{\&prime;}\left({\mathrm{\&alpha;}}_0\right)}{{\widetilde{\mathrm{\&sigma;}}}_1}(\mathrm{\&alpha;}-{\mathrm{\&alpha;}}_0)}=\sqrt{-\frac{{\mathrm{\&epsiv;}}^{\&prime;}\left({\mathrm{\&alpha;}}_0\right)}{a^{*}{k}_2+b^{*}}(\mathrm{\&alpha;}-{\mathrm{\&alpha;}}_0)}---\left(22\right)$

Wherein, a ^*and b ^*for constant, ε ' (α ₀) be eigenvalue λ ₁(α ₀) real part to α at α ₀differentiate in place, its expression formula is:

${\mathrm{\&epsiv;}}^{\text{\&prime;}}\left({\mathrm{\&alpha;}}_0\right)=\frac{\&PartialD;\mathrm{\&epsiv;}\left(\mathrm{\&alpha;}\right)}{\text{\&PartialD;\&alpha;}}{|}_{\mathrm{\&alpha;}={\mathrm{\&alpha;}}_0}$

Again by adjustment controling parameters k ₂carry out the amplitude size of control limit ring.