CN108932214A - Dynamic behavior analysis method based on small-world network model - Google Patents

Dynamic behavior analysis method based on small-world network model Download PDF

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CN108932214A
CN108932214A CN201810495793.7A CN201810495793A CN108932214A CN 108932214 A CN108932214 A CN 108932214A CN 201810495793 A CN201810495793 A CN 201810495793A CN 108932214 A CN108932214 A CN 108932214A
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朱凡超
肖敏
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Nanjing Post and Telecommunication University
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Nanjing Post and Telecommunication University
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Abstract

Dynamic behavior analysis method based on small-world network model.Present invention discloses a kind of design methods of the fractional order of small-world network model, include the following steps：The small-world network model of the integer rank of script is changed into the small-world network model of fractional order by the theory and method of fractional order by the small-world network model for providing controlled device integer rank；It selects Time Delay of Systems parameter as fork parameter, stability and Bifurcation is carried out to controlled system by corresponding theory；Some corresponding conclusions are obtained by the analysis of the characteristic equation to controlled system, and the correctness of proof theory and the characteristic of dynamic behavior are come finally by MATLAB emulation.Original integer model is transitioned into fractional model and studied by this method, by selecting suitable delay parameter and fractional order index, the stability of change system can control increasing or decreasing for stable region by specific parameter setting, controlled system is allowed to have better operability.

Description

Dynamic behavior analysis method based on small world network model
Technical Field
The invention relates to a fractional order dynamic behavior analysis method based on a small-world network model, and belongs to the technical field of controllers.
Background
In many fields, the small-world network is taken as a mathematical graph type, and the graph has the characteristic that most nodes in the graph are not adjacent to other nodes, but most nodes can be reached from other nodes through a few steps. In addition, if points in a small world network are represented as a person and connected lines are represented as being known to each other, the small world network can represent a small world phenomenon in which strangers are connected by persons known to each other. In 1998, Watts and Strogatz proposed a small-world network model network that was used to describe a gradual process from a regular network to a random network. The N-W model was also proposed at the same year, but it does not take into account the communication delay of the system. Later, the small world networks were continuously perfected by others. In 2004, a more perfect and more adaptable non-linear small world network was proposed.
1695, fractional calculus and classical calculus occurred almost at the same time. In recent years, fractional calculus has been used in a variety of fields as a necessary branch of mathematics, and people have also slowly found that fractional calculus can accurately describe some unusual phenomena in natural science and engineering application fields. Compared with the integral calculus, the fractional calculus has better performance in the aspects of application range, accuracy of the characterization problem and the like than the integral calculus.
Disclosure of Invention
The invention aims to provide a dynamic behavior analysis method based on a small-world network model, which solves the problem of perfecting dynamic analysis based on the introduction of a fractional order theory.
The technical solution for realizing the above object is a dynamic behavior analysis method based on a small world network model, which is characterized by comprising the following steps: giving a controlled object integer order small-world network model, and analyzing the dynamic behavior of the model; converting the integer-order small-world network model into a fractional-order small-world network model by using a fractional-order theory and method, and analyzing the stability characteristics and balance point information of the fractional-order small-world network model; selecting system time delay as a bifurcation parameter, performing stability analysis and bifurcation analysis on a characteristic equation of a controlled object, selecting more than two fractional order indexes, and analyzing the influence degree of the indexes on a stable domain and bifurcation of a controlled object system.
Further, the integer-order small-world network model is as follows:
wherein,is the total influence quantity at the time x, tau is the system time delay (tau is more than or equal to 0), k is the system parameter (k is more than or equal to 0 and less than or equal to 1) determining the topological structure of the system, η is a positive real number, and the whole is obtainedPositive balance point of several orders of the model
Further, the fractional order small-world network model is as follows:
DθU(x)＝1+2kU(x-τ)-η(1+2k)U2(x-τ)，
where θ ∈ (0,1) and the remaining parameters have the same meaning as the corresponding parameters of the small-world network model of integer order.
Further, the linearized small-world network model of the fractional order is:
DθZ(x)＝1+2k{Z(x-τ)+U*}-η(1+2k){Z(x-τ)+U*}2，
wherein Z (x) is set to be Z (x) ═ U (x) -U*。
Further, after the analysis was completed, the theoretical correctness and the characteristics of the dynamic behavior were verified by MATLAB simulation.
The dynamic behavior analysis scheme of the small-world network model has the prominent substantive characteristics and remarkable progressiveness: the method is characterized in that an original integer order model is transferred to a fractional order model for research, the stability of the system is changed by selecting a proper time delay parameter and a proper fractional order index, and the increase or decrease of a control stable domain can be set through specific parameters, so that the controlled system has better operability.
Drawings
FIG. 1 shows the fractional index θ of 0.85 and τ of 0.62<τ0In the case of (2), a waveform diagram of the model.
FIG. 2 shows the fractional index θ of 0.85 and τ of 0.65>τ0In the case of (2), a waveform diagram of the model.
FIG. 3 shows the fractional index θ of 0.85 and τ of 0.55<τ0In the case of (2), a waveform diagram of the model.
Fig. 4 shows the fractional index θ is 1 and τ is 0.62<τ0In the case of (2), a waveform diagram of the model.
FIG. 5 shows the fractional index θ is 1 and τ is 0.55<τ0In the case of (2), a waveform diagram of the model.
FIG. 6 shows the fractional index θ is 1 and τ is 0.65>τ0In the case of (2), a waveform diagram of the model.
FIG. 7 is a schematic diagram of a fractional order kinetic behavior analysis model framework of the small-world network model of the present invention.
Detailed Description
The following detailed description of the embodiments of the present invention is provided in connection with the accompanying drawings for the purpose of understanding and supporting the present invention, and it will be understood by those skilled in the art that the present invention is not limited to the embodiments shown.
In summary, the invention innovatively provides a dynamic behavior analysis method based on a small-world network model, and the analysis is performed based on fractional order theory. The method comprises the following main steps: giving a controlled object integer order small-world network model, and analyzing the dynamic behavior of the model; converting the integer-order small-world network model into a fractional-order small-world network model by using a fractional-order theory and method, and analyzing the stability characteristics and balance point information of the fractional-order small-world network model; selecting system time delay as a bifurcation parameter, performing stability analysis and bifurcation analysis on a characteristic equation of a controlled object, selecting more than two fractional order indexes, and analyzing the influence degree of the indexes on a stable domain and bifurcation of a controlled object system.
In a more specific embodiment, step one: and analyzing the stability characteristic and balance point information of the original small-world network model.
The small world network model is as follows:
wherein,is the total influence quantity at the time x, tau is the system time delay (tau is more than or equal to 0), k is the system parameter (k is more than or equal to 0 and less than or equal to 1), which determines the topological structure of the system, η is a positive real number
Step two: and (4) carrying out fractional order on the original small world network model.
The Caputo fractional calculus is:
wherein w-1 is not more than theta<w∈Z+；
Specifically, when 0< θ <1,
here, by calculation, the only positive balance point of the model is easily obtained:
next, assume that Z (x) ═ U (x) -U*And linearizing the model to obtain:
DθZ(x)＝1+2k{Z(x-τ)+U*}-η(1+2k){Z(x-τ)+U*}2，
when Z (x) is equal to U (x) -U*Then Z2(x) Is a high order infinite term, the linearized model can be reduced as follows:
and the characteristic equation can be obtained as follows:
step three: and (4) taking the time delay link as a bifurcation parameter to discuss the bifurcation distribution of the system.
When τ is 0, the characteristic equation changes to:
according to the Laus criterion of fractional order, the time when tau is 0 and k can be obtained2+η(1+2k)>At 0, the above fractional order model becomes progressively stable.
When τ ≠ 0, orderAnd e-iωτSubstituting cos ω τ -isin ω τ into the characteristic equation, and separating the real part and the imaginary part to obtain:
wherein
From the above equation, it can be obtained by calculation and simplification:
and
further analysis of the above formula we can obtain:
next, the distribution of the root is discussed,
(1) if there is k2+η(1+2k)>0, then the characteristic equation has a pair of pure virtual roots.
(2) Let s (τ) be l (τ) + i ω (τ) as the root of the characteristic equation, and l (τ) can be satisfied0)＝0,ω(τ0) ω, then we have:
order to
According to the characteristic equation, the following results are obtained:
therefore, there are:
analyzing the model, the following three conclusions can be drawn, as shown in fig. 1 to 6:
1. if we have k2+η(1+2k)>0, when satisfying tau epsilon (0, tau)0) Then there will be progressive stabilization of the model at the equilibrium point.
2. If the above conditions 1 are all satisfied, then the model will be τ ═ τ0When the HOPF split is generated.
3. If the above conditions 1 are all satisfied, when τ is satisfied>τ0In time, instability occurs at the equilibrium point of the model.
In conclusion, the detailed description combined with the drawings shows that the dynamic behavior analysis scheme applying the small-world network model of the invention has prominent substantive features and remarkable progressiveness: the method is characterized in that an original integer order model is transferred to a fractional order model for research, the stability of the system is changed by selecting a proper time delay parameter and a proper fractional order index, and the increase or decrease of a control stable domain can be set through specific parameters, so that the controlled system has better operability.
Although the preferred embodiments of the present invention have been described in detail, the present invention is not limited to the specific embodiments, and modifications and equivalents within the scope of the claims may be made by those skilled in the art and are included in the scope of the present invention.

Claims (5)

1. The dynamic behavior analysis method based on the small world network model is characterized by comprising the following steps of:
giving a controlled object integer order small-world network model, and analyzing the dynamic behavior of the model;
converting the integer-order small-world network model into a fractional-order small-world network model by using a fractional-order theory and method, and analyzing the stability characteristics and balance point information of the fractional-order small-world network model;
selecting system time delay as a bifurcation parameter, performing stability analysis and bifurcation analysis on a characteristic equation of a controlled object, selecting more than two fractional order indexes, and analyzing the influence degree of the indexes on a stable domain and bifurcation of a controlled object system.
2. The small-world network model-based dynamic behavior analysis method according to claim 1, wherein: the integer order small world network model is as follows:
wherein,is the total influence quantity at the time x, tau is the system time delay (tau is more than or equal to 0), k is the system parameter (k is more than or equal to 0 and less than or equal to 1) determining the topological structure of the system, η is a positive real number, and the positive balance point of the integral order model is obtained
3. The small-world network model-based dynamic behavior analysis method according to claim 1, wherein: the fractional order small world network model is as follows:
DθU(x)＝1+2kU(x-τ)-η(1+2k)U2(x-τ)，
where θ ∈ (0,1) and the remaining parameters have the same meaning as the corresponding parameters of the small-world network model of integer order.
4. The small-world-network-model-based dynamic behavior analysis method according to claim 3, wherein: the linearized small-world network model of the fractional order is as follows:
DθZ(x)＝1+2k{Z(x-τ)+U*}-η(1+2k){Z(x-τ)+U*}2，
wherein Z (x) is set to be Z (x) ═ U (x) -U*。
5. The small-world network model-based dynamic behavior analysis method according to claim 1, wherein: after the analysis is completed, the theoretical correctness and the characteristics of the dynamic behavior are verified through MATLAB simulation.
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Cited By (1)

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CN109932897A (en) * 2019-03-28 2019-06-25 南京邮电大学 A method of small-world network model bifurcation point is adjusted with PD control device

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US8831864B1 (en) * 2010-06-30 2014-09-09 Purdue Research Foundation Interactive conflict detection and resolution for air and air-ground traffic control
CN106325068A (en) * 2016-08-22 2017-01-11 哈尔滨理工大学 Improved function projection method of complicated dynamic network

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CN109932897A (en) * 2019-03-28 2019-06-25 南京邮电大学 A method of small-world network model bifurcation point is adjusted with PD control device
CN109932897B (en) * 2019-03-28 2022-09-23 南京邮电大学 Method for adjusting bifurcation point of small-world network model by PD controller

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Application publication date: 20181204