CN113703317A - Design method of bifurcation delay controller based on improved predation and predation model - Google Patents

Design method of bifurcation delay controller based on improved predation and predation model Download PDF

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CN113703317A
CN113703317A CN202110860396.7A CN202110860396A CN113703317A CN 113703317 A CN113703317 A CN 113703317A CN 202110860396 A CN202110860396 A CN 202110860396A CN 113703317 A CN113703317 A CN 113703317A
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陈功
肖敏
陆云翔
陈实
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses a bifurcation delay controller design method based on an improved predation and predation model, which comprises the following steps of improving on the basis of a traditional predation and predation model, establishing a partial differential predation and predation model containing fear time lag and diffusion, and obtaining balance point information; applying a bifurcation delay controller to an uncontrolled partial differential predation model containing fear time lag and diffusion, and adding the bifurcation delay controller at a balance point to obtain a predation model added with the bifurcation delay controller; linearizing the predation and predation model at a balance point to obtain a feature equation of a linearized controlled network; selecting time lag, and selecting appropriate controller parameters by performing stability analysis and bifurcation analysis on the linearized characteristic equation of the controlled network to ensure that the network is locally stable near a balance point; the method solves the problem that the traditional predation model is low in fitting degree with the change of the actual population quantity, and improves the accuracy of the model.

Description

Design method of bifurcation delay controller based on improved predation and predation model
Technical Field
The invention relates to the technical field of controllers, in particular to a design method of a bifurcation delay controller based on an improved predation and prey model.
Background
Leslie first proposed a predator model in 1958, opening the hot tide for studying the ecological model. In practical ecoenvironments, different location factors and predator phobic effects have a significant impact on predation behavior. Therefore, fear factors and spatial effects of diffusion are essential to establish a predation model. Time delays often lead to sudden changes in the dynamic behavior of the system, making the system unstable. Therefore, the influence of time lag must be considered when exploring the dynamic model performance. Predator models with diffusion and fear time lags can better fit population quantity changes in the actual ecological environment. The method introduces fear time lag and reaction diffusion on the basis of a classical model, and can better accord with the development condition of a population in an actual environment. The prior predation and predation model has the problems of low fitting degree with the reproduction rule of the actual nature and inaccurate reaction rule. Compared with the traditional predation model, the improved predation model is closer to the reproduction rule of the population in the actual nature, better reflects the population quantity change in the nature and has better fitting degree. The improved novel predation and predation model solves the problem that the traditional predation and predation model is low in fitting degree with the change of the actual population, and improves the accuracy of model description.
Disclosure of Invention
The purpose of the invention is as follows: in order to overcome the defects of the prior art, the invention provides a design method of a bifurcation delay controller based on an improved predator model, which solves the problem of overall simulation precision in the combination of a controller and a predator model in the prior art. By applying the controller design method provided by the invention, the effect of expanding the effective stable region can be realized by setting two parameters, and the simulation effect is better.
The technical scheme is as follows: the invention discloses a design method of a bifurcation delay controller based on an improved predation and prey model, which comprises the following steps:
improving on the basis of a traditional predatory model, and establishing a partial differential predatory model containing fear time lag and diffusion to obtain balance point information;
applying a bifurcation delay controller to an uncontrolled partial differential predation model containing fear time lag and diffusion, and adding the bifurcation delay controller at a balance point to obtain a predation model added with the bifurcation delay controller;
linearizing the predation and predation model acted by the bifurcation delay controller at a balance point to obtain a feature equation of a linearized controlled network;
selecting time lag, and selecting proper controller parameters by performing stability analysis and bifurcation analysis on the linearized characteristic equation of the controlled network so that the network is locally stable near a balance point.
Optionally, the partial differential predation model containing fear time lag and spread is expressed as:
Figure BDA0003183454850000031
Figure BDA0003183454850000032
where u (t, x) and v (t, x) represent the population density at time t and location x of the predator and predator, respectively. d1And d2Is the diffusion coefficient of predators and predators, Δ is the Laplace operator, and the characteristic root of Δ is-k2And k is equal to N. Both populations follow a logistic growth, r/1+ Cv (t- τ, x) is the modified natural growth rate of the predator, r is the prey intrinsic growth rate, C is the fear parameter, τ is the fear time lag, K is the environmental load rate of the predator, s is the natural growth rate of the predator, u (t, x)/h is the modified environmental load rate of the predator, representing that the environmental load capacity of the predator is directly proportional to the predator density, h is the ratio of the environmental load rate to u (t, x). mu (t, x) v (t, x)/[ u (t, x) + Av (t, x)]Is a functional response function of the Holling-type II, m and A are parameters of the Holling-type II function;
balance point E*(u*,v*) Expressed as:
Figure BDA0003183454850000033
u*＝hv*
u*and v*Respectively represent the system at an equilibrium point E*Here, predators and predators take on values.
Optionally, the expression of adding the bifurcation delay controller at the balance point is as follows:
Figure BDA0003183454850000034
wherein, α is a regulation parameter, u is a component u in the equilibrium point, and β is a state feedback parameter.
Optionally, the predation model incorporating the bifurcation delay controller is as follows:
Figure BDA0003183454850000041
Figure BDA0003183454850000042
optionally, the neuron model acted on by the bifurcation delay controller is at equilibrium point E*(u*,v*) And (4) carrying out linearization to obtain:
Figure BDA0003183454850000043
Figure BDA0003183454850000044
the characteristic equation of the linearized controlled network is expressed as:
Figure BDA0003183454850000045
namely:
λ2+(d1k2+d2k2-a11-a22)λ+(d1k2-a11)(d2k2-a22)-a21(a12+a13e-λτ)＝0 (7)
wherein:
Figure BDA0003183454850000046
Figure BDA0003183454850000047
Figure BDA0003183454850000048
Figure BDA0003183454850000049
Figure BDA00031834548500000410
optionally, the model is at equilibrium point E*(u*,v*) The condition of near local asymptotic stability is that the root of the characteristic equation of the model has a negative real part, and therefore, a condition of critical stability is found, that is, a condition that the characteristic equation has a pure virtual root.
Optionally, the step of making the root of the characteristic equation have a negative real part specifically includes: when the system has no time lag τ equal to 0, the characteristic equation is:
λ2+(d1k2+d2k2-a11-a22)λ+(d1k2-a11)(d2k2-a22)-a21(a12+a13)＝0 (9)
the essential condition for the root of the above equation to have a negative real part is that the following Laus-Helverz Routeth-Hurwitz criterion is satisfied:
d1k2+d2k2-a11-a22＞0 (10)
(d1k2-a11)(d2k2-a22)-a21(a12+a13)＞0 (11)
thus, when the controller parameters satisfy the two inequalities, the model is stable without time lag;
when the system has time lag (tau is more than 0), the lambda is equal to i omega and is put into the characteristic equation, and the real part and the imaginary part are separated to obtain:
Figure BDA0003183454850000051
Figure BDA0003183454850000052
wherein
Figure BDA0003183454850000053
In this case, let h (ω) be ω4+[p1 2(k2)-2q1(k2)+2a21a12]ω2+[q1(k2)-a21a12]2-a21 2a13 2When [ q ] is1(k2)-a21a12]2-a21 2a13 2< 0, the above equation has at least one positive root ω0Correspondingly, the time lag at this time can be solved:
Figure BDA0003183454850000054
bifurcation lag is a critical threshold for the system to go from stable to unstable, then the root of the corresponding feature equation crosses the imaginary axis from the left half plane to the right half plane of the imaginary axis, so the crossing condition at this point: the derivative of the characteristic root with respect to the bifurcation parameter τ is at τ0The real part of (a) is greater than zero, the feature root can cross from the imaginary axis left half-plane to the right half-plane, thus obtaining:
selecting the time lag to satisfy tau epsilon [0, tau ∈)0) At equilibrium point E for the controlled model*(u*,v*) The part is asymptotically stable;
when the time lag satisfies tau0While the system is at equilibrium point E*(u*,v*) To generate Hopf bifurcation when tau passes through tau0The system generates a set of periodic solutions.
Has the advantages that: the improved predation and predation model fully considers the influence of time delay, fear factors and diffusion space effect on the model, and better fits the population quantity change in the actual ecological environment.
Drawings
FIG. 1 is a flow chart of a method according to the present invention;
fig. 2 is a waveform diagram showing the stability of a predator when τ of the uncontrolled model (19) is 10;
fig. 3 is a waveform diagram showing the predator stability when τ of the uncontrolled model (19) is 10;
fig. 4 is a waveform diagram showing that the predator is unstable when τ of the uncontrolled model (19) is 24;
fig. 5 is a waveform diagram showing that the predator is unstable when τ of the uncontrolled model (19) is 24;
fig. 6 is a waveform diagram of the controlled model (20) in which the predator returns to a steady state under the conditions of the controller parameters α being 1.4, β being-0.03 and τ being 24;
fig. 7 is a waveform diagram of the controlled model (20) showing that the predator returns to a steady state with the controller parameters α -1.4, β -0.03, and τ -24;
fig. 8 is a waveform diagram of the controlled model (20) in which the predator is unstable when the controller parameters α is 1.4, β is-0.03, and τ is 33;
fig. 9 is a waveform diagram of the controlled model (20) showing the instability of the predator when the controller parameters α is 1.4, β is-0.03, and τ is 33.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.
The invention relates to a design method of a bifurcation delay controller based on an improved predation and predation model, which comprises the following steps:
the method is improved on the basis of the traditional predation model, and a partial differential predation model containing fear time lag and diffusion is established to obtain balance point information.
The classical predation model is expressed as:
Figure BDA0003183454850000071
where u and v represent the population density of the predators and predators, respectively. ru (1-u/K) represents the predator following logistic growth, with an intrinsic prey growth rate of r and an environmental load-bearing capacity of K. muv/u + Av is a classical Holling-type II coupling function representing the predator's predation behavior on the predator, and m and A are parameters of the Holling-type II function. sv (1-hv/u) represents the growth of the predator following logistic growth, with the natural growth rate of s, the environmental bearing capacity of u/h, and h represents the ratio of the environmental bearing capacity to u.
The diffusion predation model with uncontrolled fear time lag is obtained by introducing fear time lag and reaction diffusion on the basis of the classical predation model.
The uncontrolled fear-time-lapse diffusion predation prey model is expressed as:
Figure BDA0003183454850000072
where u (t, x) and v (t, x) represent the population density at time t and location x of the predator and predator, respectively. d1And d2Is the diffusion coefficient of predators and predators, Δ is the Laplace operator, and the characteristic root of Δ is-k2And k is equal to N. Both populations follow a logistic growth, r/1+ Cv (t- τ, x) is the modified natural growth rate of the predator, r is the prey intrinsic growth rate, C is the fear parameter, τ is the fear time lag, K is the environmental load rate of the predator, s is the natural growth rate of the predator, u (t, x)/h is the modified environmental load rate of the predator, representing that the environmental load capacity of the predator is directly proportional to the predator density, h is the ratio of the environmental load rate to u (t, x). mu (t, x) v (t, x)/[ u (t, x) + Av (t, x)]Is a functional response function of the Holling-II type, the coupling function of the Holling-II type represents the predation behavior of the predator to the predator, m and A are parameters of the Holling-II type function;
neumann boundary conditions of the predation model are as follows:
Figure BDA0003183454850000081
wherein n is in
Figure BDA0003183454850000082
The outer unit normal vector above, Ω ═ 0, π, is the smooth boundary
Figure BDA0003183454850000083
Is provided. Neumann boundary conditions mean that no population traverses a bounded area
Figure BDA0003183454850000084
At this point, the only positive balance point of the model is E*(u*,v*)，
Figure BDA0003183454850000085
And applying a bifurcation delay controller to the uncontrolled partial differential predation model containing fear time lag and diffusion, and adding the bifurcation delay controller at a balance point to obtain the predation model added with the bifurcation delay controller.
At equilibrium point E*(u*,v*) The expression of the added bifurcation delay controller is as follows:
Figure BDA0003183454850000091
wherein, α is a regulation parameter, u is a component of u in the equilibrium point, and β is a state feedback parameter.
The mathematical expression of the predation model incorporating the bifurcation delay controller is as follows:
Figure BDA0003183454850000092
and (3) linearizing the predation and predation model acted by the bifurcation delay controller at a balance point to obtain a linear characteristic equation of the controlled network.
The predation prey model acted on by the bifurcation delay controller was linearized at the equilibrium point, yielding:
Figure BDA0003183454850000093
wherein
Figure BDA0003183454850000094
Figure BDA0003183454850000095
Figure BDA0003183454850000096
Figure BDA0003183454850000097
Figure BDA0003183454850000098
The characteristic equation of the linearized controlled network is expressed as:
Figure BDA0003183454850000101
namely:
λ2+(d1k2+d2k2-a11-a22)λ+(d1k2-a11)(d2k2-a22)-a21(a12+a13e-λτ)＝0 (10)
selecting time lag, and selecting proper controller parameters by performing stability analysis and bifurcation analysis on the linearized characteristic equation of the controlled network so that the network is locally stable near a balance point.
The controlled model is at the equilibrium point E*(u*,v*) The condition of near local asymptotic stability is that the root of the characteristic equation of the model has a negative real part, and therefore, a condition of critical stability is found, that is, a condition that the characteristic equation has a pure virtual root.
The case where the root of the characteristic equation has a negative real part specifically includes:
(1) when the system no-time lag τ is 0, the characteristic equation (10) becomes:
λ2+(d1k2+d2k2-a11-a22)λ+(d1k2-a11)(d2k2-a22)-a21(a12+a13)＝0 (11)
the essential condition for the root of the characteristic equation (11) to have a negative real part is that the following Laus-Helverz Routeh-Hurwitz criterion is satisfied:
Figure BDA0003183454850000102
wherein the content of the first and second substances,
Figure BDA0003183454850000111
Figure BDA0003183454850000112
Figure BDA0003183454850000113
Figure BDA0003183454850000114
Figure BDA0003183454850000115
therefore, when the controller parameters are such that the controlled system satisfies the above inequality (12), the controlled model without time lag is at the equilibrium point E*(u*,v*) The site is locally asymptotically stable;
(2) when the system has time lag (tau is more than 0), a characteristic equation of the controlled system is assumed to have a pair of pure imaginary roots +/-i omega, and lambda is taken into the characteristic equation, and the real part and the imaginary part are separated to obtain:
Figure BDA0003183454850000116
wherein
Figure BDA0003183454850000117
Equation obtained by squaring and summing the above trigonometric functions (13)
ω4+[p1 2(k2)-2q1(k2)+2a21a12]ω2+[q1(k2)-a21a12]2-a21 2a13 2＝0 (14)
At this time, let
h(ω)＝ω4+[p1 2(k2)-2q1(k2)+2a21a12]ω2+[q1(k2)-a21a12]2-a21 2a13 2When is coming into contact with
[q1(k2)-a21a12]2-a21 2a13 2< 0, the above equation has at least one positive root ω0Corresponds to ω0Can solve the time tau at this moment0：
Figure BDA0003183454850000118
Bifurcation lag is a critical threshold for the system to go from stable to unstable, then the root of the corresponding feature equation crosses the imaginary axis from the left half plane to the right half plane of the imaginary axis, so the crossing condition at this point: the derivative of the characteristic root with respect to the bifurcation parameter τ is at τ0The real part of (b) is greater than zero, the feature root can cross from the imaginary axis left half plane to the right half plane.
The characteristic equation is derived from tau at both sides and the reciprocal is taken to obtain:
Figure BDA0003183454850000121
paired (16) type solid-taking part
Figure BDA0003183454850000122
It is clear that,
Figure BDA0003183454850000123
from the crossing condition (18), the root of the feature equation (10) crosses the imaginary axis from the imaginary axis left half plane to the imaginary axis right half plane, and the Hopf bifurcation occurs.
The above results can be seen at τ0Satisfies a crossing condition, and thus τ0Is the branch point of the original controlled system. We can conclude that:
A. selecting the time lag to satisfy tau epsilon [0, tau ∈)0) At equilibrium point E for the controlled model*(u*,v*) The part is asymptotically stable;
B. when the time lag satisfies tau0While the system is at equilibrium point E*(u*,v*) To generate Hopf bifurcation when tau passes through tau0The system generates a set of periodic solutions.
The invention is further illustrated by the following examples. The invention was verified using Matlab simulation examples.
The first step is as follows: selecting an uncontrolled predatory model containing fear time lag and diffusion:
Figure BDA0003183454850000131
the bifurcation time lag of an uncontrolled system can be obtained by calculating Hopf bifurcation
Figure BDA0003183454850000132
As shown in fig. 2 and 3, when choosing the time lag
Figure BDA0003183454850000133
When the u (t, x) and v (t, x) of the uncontrolled system (1) are at the equilibrium point E*(u*,v*) The part is asymptotically stable.
As shown in fig. 4 and 5, when choosing the time lag
Figure BDA0003183454850000134
When the u (t, x) and v (t, x) of the uncontrolled system (1) are at the equilibrium point E*(u*,v*) Is in oscillation state due to stability loss, and the uncontrolled system (1) is in E state at the moment*(u*,v*) Hopf bifurcation occurred.
The second step is that: a bifurcation delay controller is added to a predation model containing fear lag and diffusion, and the controller parameters alpha are 1.4, and beta is-0.03. The mathematical expression for the controlled system is as follows:
Figure BDA0003183454850000135
through calculation of Hopf bifurcation, bifurcation time lag of the controlled system is
Figure BDA0003183454850000136
When the time lag is selected, as shown in FIGS. 6 and 7
Figure BDA0003183454850000137
The controlled system (3) is under the action of the bifurcation delay controller (2) at the balance point E*(u*,v*) And then returns to be stable.
When the time lag is selected, as shown in FIGS. 8 and 9
Figure BDA0003183454850000138
U (t, x) and v (t, x) of the controlled system (3) are at an equilibrium point E*(u*,v*) Is out of stability and is in oscillation state, the controlled system (3) is at E*(u*,v*) Hopf bifurcation occurred.
For the system/apparatus embodiments, since they are substantially similar to the method embodiments, the description is relatively simple, and reference may be made to some descriptions of the method embodiments for relevant points.
It is to be noted that, in this document, relational terms such as first and second, and the like are used solely to distinguish one entity or operation from another entity or operation without necessarily requiring or implying any actual such relationship or order between such entities or operations.
As will be appreciated by one skilled in the art, embodiments of the present application may be provided as a method, system, or computer program product. Accordingly, the present application may take the form of an entirely hardware embodiment, an entirely application embodiment, or an embodiment combining application and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flow diagrams and/or block diagrams, and combinations of flows and/or blocks in the flow diagrams and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.
These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. Therefore, it is intended that the appended claims be interpreted as including preferred embodiments and all such alterations and modifications as fall within the scope of the invention.
It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.

Claims (7)

1. A method for designing a bifurcation delay controller based on an improved predation and prey model, which is characterized by comprising the following steps:
improving on the basis of a traditional predatory model, and establishing a partial differential predatory model containing fear time lag and diffusion to obtain balance point information;
applying a bifurcation delay controller to an uncontrolled partial differential predation model containing fear time lag and diffusion, and adding the bifurcation delay controller at a balance point to obtain a predation model added with the bifurcation delay controller;
linearizing the predation and predation model acted by the bifurcation delay controller at a balance point to obtain a feature equation of a linearized controlled network;
selecting time lag, and selecting proper controller parameters by performing stability analysis and bifurcation analysis on the linearized characteristic equation of the controlled network so that the network is locally stable near a balance point.
2. A bifurcation delay controller design method based on an improved predation model as claimed in claim 1, wherein the partial differential predation model containing fear lag and diffusion is expressed as:
Figure FDA0003183454840000011
Figure FDA0003183454840000012
where u (t, x) and v (t, x) represent the population density at time t and location x of the predator and predator, respectively. d1And d2Is the diffusion coefficient of predators and predators, Δ is the Laplace operator, and the characteristic root of Δ is-k2And k is equal to N. Both populations follow a logistic growth, r/1+ Cv (t- τ, x) is the modified natural growth rate of the predator, r is the prey intrinsic growth rate, C is the fear parameter, τ is the fear time lag, K is the environmental load rate of the predator, s is the natural growth rate of the predator, u (t, x)/h is the modified environmental load rate of the predator, representing that the environmental load capacity of the predator is directly proportional to the predator density, h is the ratio of the environmental load rate to u (t, x). mu (t, x) v (t, x)/[ u (t, x) + Av (t, x)]Is a Holling-type II coupling function, and m and A are parameters of the Holling-type II function;
balance point E*(u*,v*) Expressed as:
Figure FDA0003183454840000021
u*＝hv*
u*and v*Respectively represent the system at an equilibrium point E*To be caughtValues for predators and predators.
3. The design method of the bifurcation delay controller based on the improved predation prey model as claimed in claim 2, wherein the expression of adding the bifurcation delay controller at the balance point is as follows:
Figure FDA0003183454840000022
wherein, α is a regulation parameter, u is a component u in the equilibrium point, and β is a state feedback parameter.
4. A method for designing a bifurcation delay controller based on an improved predation model as recited in claim 3, wherein the predation model added to the bifurcation delay controller is as follows:
Figure FDA0003183454840000023
Figure FDA0003183454840000024
5. the design method of the bifurcation delay controller based on the improved predation and prey model as claimed in claim 4, wherein the neuron model acted by the bifurcation delay controller is at an equilibrium point E*(u*,v*) And (4) carrying out linearization to obtain:
Figure FDA0003183454840000031
Figure FDA0003183454840000032
the characteristic equation of the linearized controlled network is expressed as:
Figure FDA0003183454840000033
namely:
λ2+(d1k2+d2k2-a11-a22)λ+(d1k2-a11)(d2k2-a22)-a21(a12+a13e-λτ)＝0 (7)
wherein:
Figure FDA0003183454840000034
Figure FDA0003183454840000035
Figure FDA0003183454840000036
Figure FDA0003183454840000037
Figure FDA0003183454840000038
6. the design method of branched delay controller based on improved predation and prey model of claim 5, wherein the model is at equilibrium point E*(u*,v*) The condition that the local asymptotic in the vicinity is stable is that the modelThe root of the characteristic equation of (a) has a negative real part, and therefore, a critically stable condition is found, i.e., a case where the characteristic equation has a pure virtual root.
7. The design method of the bifurcation delay controller based on the improved predation prey model as claimed in claim 6, wherein the case that the root of the characteristic equation has a negative real part specifically comprises:
when the system has no time lag τ equal to 0, the characteristic equation is:
λ2+(d1k2+d2k2-a11-a22)λ+(d1k2-a11)(d2k2-a22)-a21(a12+a13)＝0 (9)
the essential condition for the root of the above equation to have a negative real part is that the following Laus-Helverz Routeth-Hurwitz criterion is satisfied:
d1k2+d2k2-a11-a22＞0 (10)
(d1k2-a11)(d2k2-a22)-a21(a12+a13)＞0 (11)
thus, when the controller parameters satisfy the two inequalities, the model is stable without time lag;
when the system has time lag (tau is more than 0), the lambda is equal to i omega and is put into the characteristic equation, and the real part and the imaginary part are separated to obtain:
Figure FDA0003183454840000041
Figure FDA0003183454840000042
wherein
Figure FDA0003183454840000043
In this case, let h (ω) be ω4+[p1 2(k2)-2q1(k2)+2a21a12]ω2+[q1(k2)-a21a12]2-a21 2a13 2When [ q ] is1(k2)-a21a12]2-a21 2a13 2< 0, the above equation has at least one positive root ω0Correspondingly, the time lag at this time can be solved:
Figure FDA0003183454840000044
bifurcation lag is a critical threshold for the system to go from stable to unstable, then the root of the corresponding feature equation crosses the imaginary axis from the left half plane to the right half plane of the imaginary axis, so the crossing condition at this point: the derivative of the characteristic root with respect to the bifurcation parameter τ is at τ0The real part of (a) is greater than zero, the feature root can cross from the imaginary axis left half-plane to the right half-plane, thus obtaining:
selecting the time lag to satisfy tau epsilon [0, tau ∈)0) At equilibrium point E for the controlled model*(u*,v*) The part is asymptotically stable;
when the time lag satisfies tau0While the system is at equilibrium point E*(u*,v*) To generate Hopf bifurcation when tau passes through tau0The system generates a set of periodic solutions.
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