Distinguishing method and system of phase state of oscillator system based on density peak clustering and poincare section

Abstract

The application discloses a kind of based on density peak clustering and poincare section vibrator system phase state discrimination method and system, constructs standard Duffing vibrator system, and determines the state of standard Duffing vibrator system;Then determine the poincare section corresponding to different states of standard Duffing vibrator system;Second, the poincare section corresponding to different states of standard Duffing vibrator system is input into density peak clustering algorithm, and density peak clustering algorithm outputs clustering center;Finally, according to the number of clustering center, determine the state of standard Duffing vibrator system.The application adopts the method that density peak clustering and poincare section are combined to determine the state of Duffing vibrator system, overcome the shortcoming that qualitative discrimination method is not high in accuracy and quantitative discrimination method is long in time, calculation is complicated, realize the state of fast and accurate judgment chaotic system.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of nonlinear system and weak signal detection, and particularly relates to a method and system for determining the phase state of a Duffing oscillator system based on density peak clustering and Poincare section. BACKGROUND

[0002] Weak signal detection has a wide range of applications in the field of engineering, and plays a key role in the fields of military, communication and electronic reconnaissance. The detection of weak signals has been a focus and hotspot of research by scholars at home and abroad. Traditional weak signal detection methods include correlation analysis, high-order statistics and stochastic resonance. In recent years, chaotic systems have been increasingly applied in the field of weak signal detection due to their high sensitivity and strong noise resistance.

[0003] When using a Duffing oscillator system to detect weak signals, it is crucial to accurately and quickly determine the chaotic state. Generally, the methods for determining the state of a system are mainly divided into qualitative determination methods and quantitative determination methods. The qualitative determination method has the defects of no quantitative standard, existence of a fuzzy area and low accuracy. The quantitative determination method has the defects of a complicated process, large amount of calculation and inability to meet engineering requirements. SUMMARY

[0004] In view of the problems in the prior art, the present application provides a method and system for determining the phase state of a Duffing oscillator system based on density peak clustering and Poincare section, which can quickly and accurately determine the chaotic state of the system.

[0005] The present application is achieved by the following technical solutions:

[0006] A method for determining the phase state of a Duffing oscillator system based on density peak clustering and Poincare section, comprising the following steps:

[0007] Step 1: constructing a standard Duffing oscillator system and determining the state of the standard Duffing oscillator system;

[0008] Step 2: determining the Poincare section corresponding to different states of the standard Duffing oscillator system;

[0009] Step 3: inputting the Poincare section corresponding to different states of the standard Duffing oscillator system into a density peak clustering algorithm, and outputting the clustering center from the density peak clustering algorithm;

[0010] Step 4: determining the state of the standard Duffing oscillator system according to the number of clustering centers.

[0011] Preferably, the equation of the standard Duffing oscillator system in step 1 is as follows:

[0012] x”+kx'-ax+bx3 = A cos ωt

[0013] where k is the damping ratio, A is the amplitude of the internal periodic force, -ax + bx 3 is the nonlinear restoring force term, k, a, b are the parameters of the standard Duffing oscillator system.

[0014] Preferably, in step 1, when the amplitude of the internal periodic force A = 0.20, the standard Duffing oscillator system is in the homoclinic orbit state;

[0015] When A = 0.39, the standard Duffing oscillator system is in the period-doubling bifurcation state;

[0016] When A = 0.75, the standard Duffing oscillator system is in the chaotic state;

[0017] When A = 0.8255, the standard Duffing oscillator system is in the chaotic critical state;

[0018] When A = 0.8256, the standard Duffing oscillator system is in the large-scale periodic state.

[0019] Preferably, in step 3, the density peak clustering algorithm determines the clustering center output by the density peak clustering algorithm by introducing the local density p i of sample i, and the distance δ i of sample i to the nearest sample j with a larger local density.

[0020] Preferably, the expression of the local density p i is as follows:

[0021]

[0022] where d ij is the Euclidean distance between samples i and j, and d c is the truncation distance.

[0023] Preferably, the expression of the distance δ i is as follows:

[0024]

[0025] Preferably, the sample point density is determined using a Gaussian kernel, and the expression is as follows:

[0026]

[0027] Step 3: input the Poincare section corresponding to different states of the standard Duffing oscillator system into the density peak clustering algorithm, and the density peak clustering algorithm outputs the clustering center.

[0028] Preferably, the method for determining the state of the standard Duffing oscillator system according to the number of cluster centers in step 4 is as follows:

[0029] The Poincare section in different states of the chaotic system is input into the density peak value algorithm, and when there are multiple cluster centers in the Poincare section, the standard Duffing oscillator system is in a chaotic state.

[0030] When there is only one cluster center in the Poincare section, the standard Duffing oscillator system is in a large-scale periodic state.

[0031] A system of a method for determining the state of an oscillator system based on density peak clustering and Poincare section, comprising,

[0032] An oscillator system module for constructing a standard Duffing oscillator system and determining the state of the standard Duffing oscillator system;

[0033] A Poincare section identification module for determining the Poincare section corresponding to different states of the standard Duffing oscillator system;

[0034] A clustering algorithm module for inputting the Poincare section corresponding to different states of the standard Duffing oscillator system into a density peak clustering algorithm, and the density peak clustering algorithm outputs cluster centers;

[0035] An identification module for determining the state of the Duffing oscillator system according to the number of cluster centers.

[0036] Compared with the prior art, the present application has the following beneficial technical effects:

[0037] The method for determining the state of an oscillator system based on density peak clustering and Poincare section provided by the present application has different Poincare sections corresponding to different states of the Duffing oscillator system, that is, the points on the Poincare section corresponding to the homoclinic orbit state, the bifurcation state, the chaotic state and the large-scale periodic state have different aggregation states, which are quantified by the density peak clustering algorithm, and different numbers of categories correspond to different states of the Duffing oscillator system. BRIEF DESCRIPTION OF DRAWINGS

[0038] Figure 1is a step diagram of the Duffing oscillator system phase state discrimination method based on density peak clustering and Poincare section of the application;

[0039] Figure 2 is a phase diagram of the Duffing oscillator system chaotic state of the application;

[0040] Figure 3 is a phase diagram of the Duffing oscillator system large-scale periodic state of the application;

[0041] Figure 4 is a Poincare section diagram of the Duffing oscillator system chaotic state of the application;

[0042] Figure 5 is a Poincare section diagram of the Duffing oscillator system large-scale periodic state of the application. DETAILED DESCRIPTION

[0043] The application will be further described in detail below in conjunction with the accompanying drawings, which are an explanation of the application rather than a limitation.

[0044] Reference Figure 1 A Duffing oscillator system phase state discrimination method based on density peak clustering and Poincare section, comprising the following steps:

[0045] Step 1, constructing a standard Duffing oscillator system, determining the initial parameters of the standard Duffing oscillator system, observing the phase diagram of the chaotic state and the large-scale periodic state of the standard Duffing oscillator system.

[0046] The equation of the standard Duffing oscillator system is as follows:

[0047] x”+kx'-ax+bx 3 =Acosωt (1)

[0048] Wherein, k is the damping ratio, A is the amplitude of the built-in periodic strategy driving force, -ax+bx 3 is a nonlinear restoring force term, k, a, and b are parameters of the standard Duffing oscillator system, respectively set to k=0.5, a=1, and b=1. Under the condition of keeping other parameters unchanged, the chaotic state changes with the continuous increase of the amplitude of the built-in periodic strategy driving force.

[0049] When A=0.39, the standard Duffing oscillator system is in a period-doubling bifurcation state;

[0050] When A=0.75, the standard Duffing oscillator system is in a chaotic state, and the chaotic state phase diagram is as shown in Figure 2 ;

[0051] When A = 0.8255, the standard Duffing oscillator system is in a chaotic critical state;

[0052] When A = 0.8256, the standard Duffing oscillator system is in a large-scale periodic state, and the large-scale periodic phase diagram is as follows: Figure 3 Show.

[0053] Step 2: Determine the Poincaré cross section corresponding to different states of the standard Duffing oscillator system;

[0054] In the late 19th century, Poincaré proposed the concept of an n-dimensional phase space (x1, dx1 / dt, x2, dx2 / dt, ..., x...). n ,dx n In the equation / dt), a suitable cross section is selected, and a pair of conjugate variables are set to fixed values ​​on this cross section. This cross section is defined as the Poincare cross section. The Poincare cross section intersects the motion trajectory of the standard Duffing oscillator system, mapping the continuous trajectory in the original phase space to a series of discrete points on the cross section.

[0055] When a standard Duffing oscillator system is in a periodic state, the position of each phase point within the period is fixed. Sampling is performed with the driving period as the sampling period, and the points on the Poincaré section of the phase plane should be distributed within a very small neighborhood. The Poincaré section diagram in a chaotic state is shown below. Figure 4 As shown;

[0056] When the standard Duffing oscillator system is in a chaotic state, sampling is performed with the motive period as the sampling period. The points on the Poincaré section of the phase plane are widely distributed across the entire plane. The Poincaré section diagram under large-scale periodic conditions is shown below. Figure 5 As shown.

[0057] Step 3: Input the Poincaré cross sections corresponding to different states of the standard Duffing oscillator system into the density peak clustering algorithm, and the density peak clustering algorithm outputs the cluster centers.

[0058] The basic principle of density peak clustering is based on two assumptions: first, cluster centers are surrounded by neighboring data points with lower local density; second, any cluster center is relatively far from data points with higher density. Density peak clustering introduces the local density ρ of sample i. i The distance δ from sample i to the nearest sample j with a local density greater than it. i Its expression is as follows:

[0059]

[0060] In (2), d ijis the Euclidean distance between samples i, j, d c is the truncated distance, χ(x) = 1 when x < 0, otherwise χ(x) = 0, and it can be seen that ρ i is equal to the number of data points distributed in the d c neighborhood of sample i.

[0061]

[0062] For the local density ρ i maximal sample i, its δ i = max j (d ij ).

[0063] The sample point density can also be calculated by using a Gaussian kernel, and the expression is as follows:

[0064]

[0065] The density peak clustering algorithm divides the sample points into three different types according to ρ i and δ i , which are density peak points, normal points and outliers. These points are density peak points and can be used as clustering centers.

[0066] Step 4, according to the number of clustering centers output by the density peak clustering algorithm, the state of the Duffing oscillator system is quickly judged.

[0067] The Poincare section under different states of the chaotic system is input into the density peak algorithm. In the chaotic state, the points of the Poincare section are widely distributed, and there are several density peak points; in the large-scale periodic state, the points of the Poincare section are distributed in a very small neighborhood, and there is only one density peak point. By clustering analysis on the points on the Poincare section, different states of the system correspond to different numbers of clustering centers, and the application solves the problems of low accuracy and complicated calculation in state discrimination of the chaotic system, and realizes fast and accurate judgment of the state of the chaotic system.

[0068] The application also provides a Duffing oscillator system phase state discrimination method system based on density peak clustering and Poincare section, which comprises an oscillator system module, a Poincare section identification module, a clustering algorithm module and an identification module.

[0069] The oscillator system module is used for constructing a standard Duffing oscillator system and determining the state of the standard Duffing oscillator system.

[0070] The Poincare section identification module is used for determining the Poincare section corresponding to different states of the standard Duffing oscillator system.

[0071] The clustering algorithm module is used for inputting the Poincare section corresponding to different states of the standard Duffing oscillator system into a density peak clustering algorithm, and the density peak clustering algorithm outputs clustering centers;

[0072] The identification module determines the state of the Duffing oscillator system according to the number of the clustering centers.

[0073] The method combining the density peak clustering and the Poincare section is used to determine the state of the Duffing oscillator system, which overcomes the shortcomings of low accuracy of the qualitative discrimination method and long time consumption and complicated calculation of the quantitative discrimination method, and realizes the fast and accurate determination of the state of the chaotic system.

[0074] The above content only illustrates the technical idea of the present application, and cannot limit the protection scope of the present application, and any modification made according to the technical idea of the present application on the basis of the technical scheme falls within the protection scope of the claims of the present application.

Claims

1. A method for identifying the phase of a system of oscillators based on density peak clustering and Poincare sections, characterized in that, The method comprises the following steps: Step 1, constructing a standard Duffing oscillator system, using the Duffing oscillator system to detect weak signals, and determining the state of the standard Duffing oscillator system; Step 2, determining the Poincare section corresponding to different states of the standard Duffing oscillator system; Step 3, inputting the Poincare section corresponding to different states of the standard Duffing oscillator system into a density peak clustering algorithm, and outputting a clustering center by the density peak clustering algorithm; The density peak clustering algorithm determines a density peak clustering algorithm output clustering center by introducing a local density of a sample i , a distance of the sample i to a sample j with a local density greater than that of the sample , and a distance of the sample i to the nearest sample j . The local density The expression is as follows: wherein, is the sample i , j Euclidean distance between, is the cutoff distance; The distance The expression is as follows: ； Step 4, determining the state of the standard Duffing oscillator system according to the number of clustering centers, and the method is as follows: When the Poincare section has multiple clustering centers, the standard Duffing oscillator system is in a chaotic state; When the Poincare section has only one clustering center, the standard Duffing oscillator system is in a large-scale periodic state.

2. The method of claim 1, wherein the method is characterized by: The equation of the standard Duffing oscillator system in step 1 is as follows: where, k is the damping ratio, A is the amplitude of the built-in periodic driving force, is the nonlinear restoring force term, k, a, b are the parameters of the standard Duffing oscillator system.

3. The method of claim 2, wherein the method is characterized by: In step 1, when the amplitude of the built-in periodic strategy power A =0.20, the standard Duffing oscillator system is in the homoclinic orbit state; When A = 0.39, the standard Duffing oscillator system is in a period-doubling bifurcation state; When A = 0.75, the standard Duffing oscillator system is in a chaotic state; When A = 0.8255, the standard Duffing oscillator system is in a chaotic critical state; When A = 0.8256, the standard Duffing oscillator system is in a large-scale periodic state.

4. The method of claim 1, wherein the method is characterized by: The local density is determined by using a Gaussian kernel, and the expression is as follows: 。 5. A system for performing the method of identifying the phase of a system of oscillators based on density peak clustering and Poincare sections of any of claims 1-4, characterized in that, The method comprises the following steps: A Duffing oscillator system module is configured to construct a standard Duffing oscillator system and determine the state of the standard Duffing oscillator system; A Poincare section identification module is configured to determine the Poincare section corresponding to different states of the standard Duffing oscillator system; A clustering algorithm module is configured to input the Poincare section corresponding to different states of the standard Duffing oscillator system into a density peak clustering algorithm, and output a clustering center by the density peak clustering algorithm; An identification module is configured to determine the state of the Duffing oscillator system according to the number of clustering centers.

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