Weak signal detection method based on multi-attractor Lorenz-like system containing cubic terms

Abstract

The invention discloses a weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms, and relates to the technical field of mechanical engineering. The method comprises the following steps: respectively replacing a cubic term of a state variable x and a cubic cross term of x and y in a first equation and a second equation of a Lorenz system, and constructing a multi-attractor Lorenz system containing cubic terms; chaos characteristics and dynamic behaviors of the Lorenz-like system are analyzed, and chaos attractors in different shapes are obtained by changing control parameters of the system; according to the method, the sensitivity of a Lorenz-like system to parameters in the critical state of transient chaos and steady chaos is utilized, the phase change from the transient chaos state to the steady chaos state is used as a judgment basis, a weak signal detection system is constructed, and weak signal detection is carried out. According to the method, the weak signals can be effectively detected by constructing the multi-attractor Lorenz-like system containing the cubic terms.

Description

Technical Field

[0001] The invention relates to the technical field of mechanical engineering, and in particular to a weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms. Background Art

[0002] In recent years, the research on Lorenz-like chaotic systems has focused on two aspects: one is the understanding and control of the complex dynamic behavior mechanism of the discovered chaotic system, and the other is the exploration of new chaotic systems and further research. Lorenz-like chaotic systems are often based on the Lorenz system, and the internal structure of the chaotic system is changed by adding or changing the structure of nonlinear terms or linear terms in the polynomial. The new chaotic system is topologically not equivalent to the Lorenz system. Most of the existing research is based on the Lorenz system, and the original chaotic system is improved by adding or changing the first-order linear terms and second-order nonlinear terms of the equation polynomial, and it is unable to effectively detect weak periodic signals. Summary of the invention

[0003] The purpose of the present invention is to provide a weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms, which can effectively detect weak signals by constructing a multi-attractor Lorenz-like system containing cubic terms.

[0004] To achieve the above object, the present invention provides the following solutions:

[0005] A weak signal detection method based on a multi-attractor Lorenz-like system with cubic terms, comprising:

[0006] On the basis of the Lorenz system, the cubic term of the state variable x and the cubic cross terms of x and y are substituted in the first equation and the second equation of the Lorenz system respectively, and a multi-attractor-like Lorenz system containing cubic terms is constructed.

[0007] The chaotic characteristics and dynamic behaviors of the Lorenz-like system are analyzed, and by changing the control parameters of the system, chaotic attractors of different shapes are obtained; the shapes include single wing, double wings and superimposed double wings;

[0008] Based on each chaotic attractor and utilizing the sensitivity of the Lorenz-like system to parameters in the critical state between transient chaos and steady-state chaos, the phase transition from transient chaos to steady-state chaos is used as the judgment basis to construct a weak signal detection system and perform weak signal detection.

[0009] Optionally, the construction process of the multi-attractor Lorenz-like system containing cubic terms specifically includes:

[0010] Replace the x term of the first equation in the Lorenz system with x 3 and replace the y term of the second equation with the cross term (1-x 2 )y, and adding two control parameters k and e, the multi-attractor Lorenz-like system with cubic terms is obtained:

[0011]

[0012] In the formula, x, y, and z represent system state variables. They represent the first-order derivatives of the state variables, a, b, c represent constant system parameters, and e and k represent the introduced system control parameters.

[0013] Optionally, the analysis of chaotic characteristics and dynamic behaviors includes symmetry and dissipation, equilibrium point characteristics, bifurcation characteristics, attractors and transient chaotic phenomena.

[0014] Optionally, the symmetry and dissipation analysis steps include:

[0015] Perform symmetry analysis on the Lorenz-like system under the coordinate transformation (x, y, z) → (-x, -y, z) to determine the rotational symmetry of the system around the z-axis;

[0016] The divergence of the Lorenz-like system is defined as:

[0017] When x is satisfied 2 <(e+b) / (-3ak+e), and ▽V<0, the system is in a dissipative state.

[0018] Optionally, the step of analyzing the equilibrium point characteristics includes:

[0019] According to the equilibrium condition, Bring-in get:

[0020]

[0021] From the formula a(y-kx 3 )=0, so y=kx 3 , and then substitute xy-bz=0 to get Substitute formula y and formula z into cx-xz-e(1-x 2 )y=0, then Solve and substitute the solution into formula y and formula z to obtain the five equilibrium points of the system, namely: E1(x1,y1,z1), E2(x2,y2,z2), E3(x3,y3,z3), E4(x4,y4,z4), E5(x5,y5,z5); among which, x1=y1=z1=0, so:

[0022]

[0023]

[0024]

[0025]

[0026] To analyze the stability and type of these five equilibrium points, consider the Jacobian matrix corresponding to the equilibrium point:

[0027]

[0028] according to The characteristic equation of the system is:

[0029] λ 3 +(3akx 2 +e+b)λ 2 +(3aekx 2 +3akbx 2 +eb+x 2 -ac+az+2aexy)λ

[0030] +3abekx 2 +3akx 4 -axy-abz+2abexy=0

[0031] Consider the values ​​of parameters a, b, c, e, and k, and substitute Obtain the characteristic roots of each equilibrium point and determine the state of each equilibrium point to judge the state of the system under the current parameter conditions.

[0032] Optionally, the step of analyzing the bifurcation characteristics and attractors includes:

[0033] When the chaotic characteristics of the system are analyzed with parameter c as the variable parameter, the parameters a, b, e, and k are given fixed values, c is taken as the variable parameter, and the initial value of the system is taken as [1,1,1]; when the parameter c changes, the state variable x changes accordingly, and the corresponding bifurcation diagram is obtained; since the system has five equilibrium points, when the parameter changes from small to large, the system exhibits corresponding dynamic characteristics.

[0034] Optionally, the step of analyzing the transient chaotic phenomenon includes:

[0035] When the parameters are set to a=10, b=3, c=28.665, e=0.1, k=0.1 and the initial condition is [1,1,1], transient chaos occurs in the system, and the phase diagram of the state variables under the current parameter conditions and the corresponding time domain waveform diagram and Lyapunov index diagram are obtained;

[0036] When the parameters are set to a=10, b=3, c=34.9, e=0.1, k=0.1, and the initial conditions are set to [1,1,1], the system exhibits intermittent chaos, and the state variables of the system are sometimes periodic and sometimes chaotic in the process of changing with time.

[0037] Optionally, the specific process of the weak signal detection includes:

[0038] The open-loop control of the chaotic system is realized by using the harmonic control signal with a frequency much larger than the characteristic frequency of the system as the non-resonant parameter excitation. The characteristic frequency of the system is given by The approximate calculation is performed; where N is the number of peak values ​​of the system state variables in the time period T. When the parameters are a=10, b=3, c=28, e=0.1, k=0.1, ω0≈7.8Hz;

[0039] As the parameter c changes, the system will experience transient chaotic phenomena. A weak periodic signal is added to the system, and the autonomous chaotic system is controlled by a harmonic control signal with a frequency much larger than the system characteristic frequency, so that the system enters another chaotic state. The system is used as a detection system to detect weak signals, and the weak signal detection system is constructed in the following form:

[0040]

[0041] In the formula, 1+f cos(ωt) is the harmonic control signal, and the system is adjusted to a transient chaotic state u(t)=δs(t)+n(t) as the input signal by adjusting the f value; s(t) is the signal to be detected, n(t) is Gaussian white noise; x is the output signal, and β is the power of the adjusted input signal; in this system, parameter β is used for weak signal detection, and β=0 is set. When there is no input signal, the other parameters are a=10, b=3, c=28, e=0.1, k=0.1, ω=80, and the f value is adjusted. When f=1.1962, the system is in a steady chaotic state;

[0042] Set β = 0.0001. If the system only inputs Gaussian white noise with power p = 0.1W, the phase trajectory and output signal of the detection system are observed from the phase diagram and time domain waveform. The system is still in a transient chaotic state, which confirms that the noise does not change the chaotic state of the detection system.

[0043] Set β = 0.0001, input a Gaussian white noise with a power of 0.1 into the system, and input a test signal containing harmonic components with the same frequency as ω. Adjust the amplitude of the test signal δ. When δ gradually increases from 0 to 0.0001, the system will enter another stable chaotic state from a transient chaotic state. The simulation data is analyzed by the formula Calculate the minimum signal-to-noise ratio of the detection system, where Ws and W n are the effective powers of signal and noise respectively;

[0044] Adjust the control parameter f of the detection system to put the system in a transient chaotic state, and input the harmonic signal to be tested. When the signal contains a frequency ω component, the system breaks away from the original transient chaos and enters another stable chaotic state. Therefore, the phase change caused by the transition from transient chaos to stable chaotic state is used as the judgment basis to determine whether there is a harmonic signal of a specific frequency in the signal to be tested.

[0045] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0046] The invention discloses a weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms. The method comprises the following steps: replacing the cubic term of a state variable x and the cubic cross terms of x and y in the first equation and the second equation of the Lorenz system respectively, so as to construct a multi-attractor Lorenz-like system containing cubic terms; analyzing the chaotic characteristics and dynamic behavior of the Lorenz-like system, and obtaining chaotic attractors of different shapes by changing the control parameters of the system; utilizing the sensitivity of the Lorenz-like system to parameters under the critical state of transient chaos and steady-state chaos, taking the phase change from transient chaos to steady-state chaos as the basis for judgment, constructing a weak signal detection system, and performing weak signal detection.

[0047] The present invention analyzes the dynamics of a new Lorenz-like chaotic system and finds that as the parameters change, the system may be accompanied by the appearance of transient chaos in the process of the chaotic state of the new Lorenz-like system changing. Based on the data characteristics of the transient chaos and steady-state chaos of the chaotic system, the phase change generated by the transition from transient chaos to steady-state chaos and the time domain waveform of the variable are used as the basis for segment judgment. Therefore, the present invention can effectively detect weak periodic signals by constructing a multi-attractor Lorenz-like system containing cubic terms. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0049] Figure 1 It is a schematic diagram of the process of the weak signal detection method of the present invention;

[0050] Figure 2Schematic diagram of chaotic attractor and equilibrium point in this embodiment; wherein, part (a) is a schematic diagram of xy plane; part (b) is a schematic diagram of xz plane;

[0051] Figure 3 : is the bifurcation diagram and the maximum Lyapunov index diagram in this embodiment; wherein, part (a) is the bifurcation diagram; part (b) is the maximum Lyapunov index diagram;

[0052] Figure 4 Part of the xz plane phase diagram of the present embodiment with the change of parameter c; wherein, a=10, b=3, e=0.1, k=0.1, part (a) is a schematic diagram when c=10; part (b) is a schematic diagram when c=11; part (c) is a schematic diagram when c=14; part (d) is a schematic diagram when c=16; part (e) is a schematic diagram when c=17.1; part (f) is a schematic diagram when c=18.8; part (g) is a schematic diagram when c=24; part (h) is a schematic diagram when c=26; part (i) is a schematic diagram when c=26.09; part (j) is a schematic diagram when c=29; part (k) is a schematic diagram when c=29.23; part (l) is a schematic diagram when c=39;

[0053] Figure 5 Phase diagram of some coexisting attractors in the xz plane in this embodiment; where a=10, b=3, e=0.1, k=0.1, the blue line indicates the initial value is [1,1,1], (a) is a schematic diagram when c=10; (b) is a schematic diagram when c=11; (c) is a schematic diagram when c=14; (d) is a schematic diagram when c=16; (e) is a schematic diagram when c=17.1; (f) is a schematic diagram when c=29;

[0054] Figure 6 Schematic diagram of transient chaotic behavior of the system in this embodiment; wherein, part (a) is an xz phase diagram; part (b) is a time domain waveform diagram;

[0055] Figure 7 Schematic diagram of steady-state chaotic behavior of the system in this embodiment; wherein, part (a) is an xz phase diagram; part (b) is a time domain waveform diagram;

[0056] Figure 8 Schematic diagram of a system output in this embodiment; wherein, part (a) is an xz phase diagram; part (b) is a time domain waveform diagram;

[0057] Fig. 9 Schematic diagram of another system output in this embodiment; wherein, part (a) is an xz phase diagram; and part (b) is a time domain waveform diagram. DETAILED DESCRIPTION

[0058] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0059] The purpose of the present invention is to provide a weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms, which can effectively detect weak signals by constructing a multi-attractor Lorenz-like system containing cubic terms.

[0060] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0061] like Figure 1 As shown, the present invention provides a weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms, comprising:

[0062] Step S100: Based on the Lorenz system, the cubic term of the state variable x and the cubic cross terms of x and y are substituted in the first equation and the second equation of the Lorenz system respectively to construct a multi-attractor-like Lorenz system containing cubic terms.

[0063] Step S200: Analyze the chaotic characteristics and dynamic behaviors of the Lorenz-like system, and obtain chaotic attractors of different shapes by changing the control parameters of the system; the shapes include single wing, double wings and superimposed double wings.

[0064] Step S300: Based on each chaotic attractor, using the sensitivity of the Lorenz-like system to parameters under the critical state of transient chaos and steady-state chaos, taking the phase transition from transient chaos to steady-state chaos as the basis for judgment, constructing a weak signal detection system, and performing weak signal detection.

[0065] Based on the above technical solution, the following is provided: Figure 2-Figure 9 Detailed processing steps are shown.

[0066] Step S100, constructing a Lorenz-like system containing a cubic polynomial.

[0067] Based on the Lorenz system, replace the x term in the first equation of the Lorenz system with x 3 Instead, the y term in the second equation is replaced by the cross term (1-x 2)y and adding two control parameters k and e, a type of quasi-Lorenz system with cubic terms is constructed.

[0068] The theoretical model is as follows:

[0069]

[0070] Where a, b, c, d, and e are parameters.

[0071] The generalized Lorenz system has the following form:

[0072]

[0073] Where X = (x, y, z) T ∈R 3 , λ3∈R, A is a real matrix, its eigenvalues ​​are λ1 and λ2. And λ1>0, λ2<0, λ3<0.

[0074] System (1) can be written in the form of formula (2):

[0075]

[0076] Therefore, the system (1) does not belong to the generalized Lorenz system, that is, the system (1) is not topologically equivalent to the generalized Lorenz system.

[0077] In step S200, the chaotic characteristics and dynamic characteristics of the Lorenz-like system containing cubic terms obtained in step S100 are analyzed, including the symmetry and dissipation of the system, the equilibrium point characteristics of the system, the bifurcation characteristics and attractors of the system, and the transient chaotic phenomena of the system.

[0078] Firstly, the symmetry and dissipation analysis results of the system are obtained.

[0079] Under the coordinate transformation (x, y, z) → (-x, -y, z), the system has rotational symmetry around the z-axis.

[0080] The divergence of the system is defined as:

[0081]

[0082] When x is satisfied 2 <(e+b) / (-3ak+e), yes The system is dissipative.

[0083] The equilibrium point characteristic analysis of the system comprises the following steps:

[0084] According to the equilibrium condition, Bring-in The following formula

[0085]

[0086] From the above formula a(y-kx 3 )=0, so y=kx 3 , substituting xy-bz=0, we get Substitute y and z into cx-xz-e(1-x 2 )y=0, then Solve and substitute y=kx 3 and It is concluded that the system has 5 equilibrium points, namely E1(x1,y1,z1), E2(x2,y2,z2), E3(x3,y3,z3), E4(x4,y4,z4), and E5(x5,y5,z5). Among them, x1=y1=z1=0, so:

[0087]

[0088]

[0089]

[0090]

[0091] In order to study the stability and type of these five equilibrium points, consider the Jacobian matrix corresponding to the equilibrium point:

[0092]

[0093] according to The characteristic equation of the system is:

[0094] λ 3 +(3akx 2 +e+b)λ 2 +(3aekx 2 +3akbx 2 +eb+x 2 -ac+az+2aexy)λ

[0095] +3abekx 2 +3akx 4 -axy-abz+2abexy=0

[0096] Considering the parameters a=10, b=3, c=28, e=0.1, k=0.1, the characteristic root values ​​of the equilibrium points are obtained by substituting them into formula (5) as shown in Table 1.

[0097]

[0098] Under the current parameter conditions, E1 has one positive real root and two negative real roots, so it is a saddle node. The λ values ​​of E2 and E4 both have one positive real root and a pair of conjugate complex roots with negative real parts, so E2 and E4 are both unstable foci. The λ values ​​of E3 and E5 are both composed of two positive real roots and one negative real root, so E3 and E5 are unstable saddle nodes. All five equilibrium points are unstable. Under the current parameter conditions, the system is formed by the superposition of a pair of wing-shaped attractors and two scroll-shaped attractors. Figure 2 Part (a) and Figure 2 The chaotic attractor shown in part (b).

[0099] The bifurcation characteristics and chaotic attractors of the system can be used to describe the dynamic characteristics of the system. Consider using parameter c as a variable parameter to analyze the chaotic characteristics of the system. Let parameter c be the variable parameter and take the initial value of the system as [1,1,1]. When c changes, the state variable x changes accordingly, and we get Figure 3 The bifurcation diagram and maximum Lyapunov exponent diagram are shown.

[0100] When the parameter c changes from small to large, the system exhibits complex dynamic characteristics under the action of the unstable manifolds of the five equilibrium points. From the analysis of the bifurcation diagram and the maximum Lyapunov index diagram, it can be roughly seen that the system (1) appears in different states such as periodic, quasi-periodic, and chaotic as the parameters change. Combining the bifurcation diagram and the maximum Lyapunov index diagram, with a step size of 0.001 and a simulation time t sim =500s, gradually increase the c value, analyze the phase trajectory diagram and time domain waveform diagram of system (1), and obtain the specific dynamic behavior of system (1) as the parameter c changes as shown in Table 2.

[0101] Table 2 Dynamic behavior of the system with the change of parameter c a=10, b=3, e=0.1, k=0.1

[0102]

[0103]

[0104]

[0105] The parameter c changes gradually from 0. In this process, the state of system (1) changes significantly, including periodic state, quasi-periodic state, chaotic state, and even transient chaos in the middle of the system state transition. The attractor undergoes changes in different shapes, such as spiral, symmetrical double-wing, superimposed connected double-wing, asymmetrical double-wing, symmetrical double-wing connected with vortex, two symmetrical double-wing connected with two vortex, left single-wing connected with vortex, right single-wing connected with vortex. And in the process of increasing the parameter c from small to large, an attractor of the same shape may appear multiple times in different value ranges. Figure 3 Part of the xz phase diagram is shown as a function of the parameter c.

[0106] From the symmetry analysis of system (1), it is found that it has rotational symmetry about the z-axis. Figure 3 When the attractor in is an asymmetric shape, under the same parameter conditions, when the initial values ​​are [1,1,1] and [-1,-1,1] respectively, the system will have two coexisting attractors that are symmetric about the z axis. Figure 4 As shown, the blue line represents the phase trajectory of the system (1) in the xz plane when the initial value is [1,1,1], the parameters are a=10, b=3, e=0.1, k=0.1; the red line represents the phase trajectory of the system (1) in the xz plane when the initial value is [-1,-1,1], the parameters are a=10, b=3, e=0.1, k=0.1.

[0107] Then, enter step 300, the transient chaos phenomenon of the system, transient chaos is the phenomenon of different chaotic states appearing on a finite time scale, this phenomenon is common in integer order chaotic systems, with the change of some specific parameters, the chaotic attractor and the boundary of its attraction basin slowly approach each other in the phase space, when the parameter changes to a critical value, the chaotic attractor and the boundary of its attraction basin are close to coincide. At this time, the chaotic attractor will meet a quasi-periodic orbit, and the chaotic attractor will disappear due to a boundary crisis. Under critical value conditions, the system will have instantaneous chaotic behavior in the initial stage on the time axis, and then quickly change to another stable periodic state or another form of chaos.

[0108] When the parameters are set to a=10, b=3, c=28.665, e=0.1, k=0.1., and the initial conditions are [1,1,1]. System (1) also exhibits transient chaos. Figure 8 The phase diagram of the state variables under the current parameter conditions and the corresponding time domain waveform diagram and Lyapunov exponent diagram are displayed.

[0109] When the parameters are set to a=10, b=3, c=34.9, e=0.1, k=0.1, and the initial conditions are set to [1,1,1], the state variables of the system are sometimes periodic and sometimes chaotic in the process of changing with time, and intermittent chaos occurs.

[0110] A weak signal detection system was further constructed based on the above principles to verify the effectiveness of the method.

[0111] Through the analysis of the chaotic characteristics and dynamics of system (1), it is shown that system (1) is an autonomous chaotic system. There is no forced periodic excitation signal with explicit time and frequency. Therefore, the characteristic frequency of the system cannot be calculated analytically, that is, the principle of parameter resonance perturbation cannot be used to suppress the chaotic state of system (1) to detect weak signals. The characteristic frequency of the system is approximately calculated by equation (6), and then the harmonic control signal with a frequency much larger than the characteristic frequency of the system is used as a non-resonant parameter excitation to realize the open-loop control of the chaotic system by chaos suppression:

[0112]

[0113] Where N is the number of peak values ​​of the system state variables in the time period T. Using formula (6), we can calculate that ω0≈7.8Hz when the parameters are a=10, b=3, c=28.665, e=0.1, k=0.1. From the analysis of system (1), it can be seen that the system will experience transient chaos as the parameter c changes. Adding a weak periodic signal to system (1) and using a harmonic control signal with a frequency much larger than the system characteristic frequency to control the autonomous chaotic system can change the chaotic state of the system and make the system enter another chaotic state. Using this principle, the weak signal detection system designed based on system (1) is as follows:

[0114]

[0115] In the formula, 1+fcos(ωt) is the harmonic control signal. Adjusting the f value adjusts the system to the critical state between the transient chaotic state and the steady chaotic state. u(t)=δs(t)+n(t) is the input signal. s(t) is the signal to be detected, n(t) is Gaussian white noise, x(t) is the output signal, and β is used to adjust the power of the input signal. When β=0, that is, there is no input signal, the other parameters are a=10, b=3, c=28, e=0.1, k=0.1, ω=80, and f is adjusted. When f=1.196100, the system is at the critical value of a transient chaotic state. The state variables are as follows: Figure 6 When f=1.196101, the system enters Figure 7 The stable chaotic state shown.

[0116] Set β = 0.0001. If the system only inputs Gaussian white noise with power p = 0.1W, the phase trajectory and output signal of the detection system are as follows: Figure 8 As shown in Figure 2. From the phase diagram and time domain waveform, the system is still in a transient chaotic state. It can be seen that the noise does not change the chaotic state of the detection system.

[0117] Set β = 0.0001, input a Gaussian white noise with a power of 0.1 into the system, and input a signal to be tested containing harmonic components with the same frequency as ω, and adjust the amplitude of the signal to be tested δ, which gradually increases from 0 to 0.0001. Fig. 9 The system shown will enter another stable chaotic state from a transient chaotic state. Analyzing the simulation data, the minimum signal-to-noise ratio of the detection system can be calculated from equation (12) to be -73dB.

[0118]

[0119] Where W s and W n are the effective powers of signal and noise respectively.

[0120] In summary, the control parameter f of the detection system is adjusted to put the system in a transient chaotic state. When the harmonic signal to be tested is input, when the signal contains a component with a frequency of ω, the system will break away from the original transient chaos and enter another stable chaotic state. Therefore, the phase change generated by the transition from transient chaos to stable chaotic state can be used as a judgment basis to determine whether there is a harmonic signal of a specific frequency in the signal to be tested.

[0121] The various embodiments in this specification are described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the various embodiments can be referenced to each other.

[0122] This article uses specific examples to illustrate the principles and implementation methods of the present invention. The above examples are only used to help understand the core idea of ​​the present invention. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting the present invention.

Claims

1. A weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms, characterized in that: include: On the basis of the Lorenz system, the cubic term of the state variable x and the cubic cross terms of x and y are substituted in the first equation and the second equation of the Lorenz system respectively, and a multi-attractor-like Lorenz system containing cubic terms is constructed. The chaotic characteristics and dynamic behaviors of the Lorenz-like system are analyzed, and by changing the control parameters of the system, chaotic attractors of different shapes are obtained; the shapes include single wing, double wings and superimposed double wings; Based on each chaotic attractor and utilizing the sensitivity of the Lorenz-like system to parameters in the critical state between transient chaos and steady-state chaos, the phase transition from transient chaos to steady-state chaos is used as the judgment basis to construct a weak signal detection system and perform weak signal detection.

2. The weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms according to claim 1 is characterized in that: The construction process of the multi-attractor Lorenz-like system containing cubic terms specifically includes: Replace the x term of the first equation in the Lorenz system with x 3 and replace the y term of the second equation with the cross term (1-x 2 )y, and adding two control parameters k and e, the multi-attractor Lorenz-like system with cubic terms is obtained: In the formula, x, y, and z represent system state variables. They represent the first-order derivatives of the state variables, a, b, c represent constant system parameters, and e and k represent the introduced system control parameters.

3. The weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms according to claim 2 is characterized in that: The analysis of chaotic characteristics and dynamic behaviors includes symmetry and dissipation, equilibrium point characteristics, bifurcation characteristics, attractors and transient chaotic phenomena.

4. The weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms according to claim 3 is characterized in that: The analysis steps of symmetry and dissipation include: Perform symmetry analysis on the Lorenz-like system under the coordinate transformation (x, y, z) → (-x, -y, z) to determine the rotational symmetry of the system around the z-axis; The divergence of the Lorenz-like system is defined as: When x is satisfied 2 <(e+b) / (-3ak+e), and , the system is in a dissipative state.

5. The weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms according to claim 3 is characterized in that: The step of analyzing the equilibrium point characteristics comprises: According to the equilibrium condition, Bring-in get: From the formula a(y-kx 3 )=0, so y=kx 3 , and then substitute xy-bz=0 to get Substitute formula y and formula z into cx-xz-e(1-x 2 )y=0, then Solve and substitute the solution into formula y and formula z to obtain the five equilibrium points of the system, namely: E1(x1,y1,z1), E2(x2,y2,z2), E3(x3,y3,z3), E4(x4,y4,z4), E5(x5,y5,z5); among which, x1=y1=z1=0, so: To analyze the stability and type of these five equilibrium points, consider the Jacobian matrix corresponding to the equilibrium point: according to The characteristic equation of the system is: λ 3 +(3only 2 +e+b)λ 2 +(3aex 2 +3akbx 2 +eb+x 2 -ac+az+2aexy)λ+3abekx 2 +3one 4 -axy-abz+2abexy=0 Consider the values ​​of parameters a, b, c, e, and k, and substitute Obtain the characteristic roots of each equilibrium point and determine the state of each equilibrium point to judge the state of the system under the current parameter conditions.

6. The weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms according to claim 3 is characterized in that: The analysis steps of the bifurcation characteristics and attractors include: When the chaotic characteristics of the system are analyzed with parameter c as the variable parameter, the parameters a, b, e, and k are given fixed values, c is taken as the variable parameter, and the initial value of the system is taken as [1,1,1]; when the parameter c changes, the state variable x changes accordingly, and the corresponding bifurcation diagram is obtained; since the system has five equilibrium points, when the parameter changes from small to large, the system exhibits corresponding dynamic characteristics.

7. The weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms according to claim 3 is characterized in that: The analysis steps of the transient chaotic phenomenon include: When the parameters are set to a=10, b=3, c=28.665, e=0.1, k=0.1 and the initial condition is [1,1,1], transient chaos occurs in the system, and the phase diagram of the state variables under the current parameter conditions and the corresponding time domain waveform diagram and Lyapunov index diagram are obtained; When the parameters are set to a=10, b=3, c=34.9, e=0.1, k=0.1, and the initial conditions are set to [1,1,1], the system exhibits intermittent chaos, and the state variables of the system are sometimes periodic and sometimes chaotic in the process of changing with time.

8. The weak signal detection method based on a multi-attractor Lorenz-like system containing cubic terms according to claim 1 is characterized in that: The specific process of the weak signal detection includes: The open-loop control of the chaotic system is realized by using the harmonic control signal with a frequency much larger than the characteristic frequency of the system as the non-resonant parameter excitation. The characteristic frequency of the system is given by Perform an approximate calculation; where N is the number of peak values ​​of the system state variables in the T time period. When the parameters are a=10, b=3, c=28, e=0.1, k=0.1, ω0≈7.8Hz; As the parameter c changes, the system will experience transient chaotic phenomena. A weak periodic signal is added to the system, and the autonomous chaotic system is controlled by a harmonic control signal with a frequency much larger than the system characteristic frequency, so that the system enters another chaotic state. The system is used as a detection system to detect weak signals, and the weak signal detection system is constructed in the following form: In the formula, 1+fcos(ωt) is the harmonic control signal, and the system is adjusted to a transient chaotic state u(t)=δs(t)+n(t) as the input signal by adjusting the f value; s(t) is the signal to be detected, n(t) is Gaussian white noise; x is the output signal, and β is the power of the adjusted input signal; in this system, parameter β is used for weak signal detection, and β=0 is set. When there is no input signal, the other parameters are a=10, b=3, c=28, e=0.1, k=0.1, ω=80, and the f value is adjusted. When f=1.1962, the system is in a steady chaotic state; Set β = 0.0001. If the system only inputs Gaussian white noise with power p = 0.1W, the phase trajectory and output signal of the detection system are observed from the phase diagram and time domain waveform. The system is still in a transient chaotic state, which confirms that the noise does not change the chaotic state of the detection system. Set β = 0.0001, input a Gaussian white noise with a power of 0.1 into the system, and input a test signal containing harmonic components with the same frequency as ω. Adjust the amplitude of the test signal δ. When δ gradually increases from 0 to 0.0001, the system will enter another stable chaotic state from a transient chaotic state. The simulation data is analyzed by the formula Calculate the minimum signal-to-noise ratio of the detection system, where W s and W n are the effective powers of signal and noise respectively; Adjust the control parameter f of the detection system to put the system in a transient chaotic state, and input the harmonic signal to be tested. When the signal contains a frequency ω component, the system breaks away from the original transient chaos and enters another stable chaotic state. Therefore, the phase change caused by the transition from transient chaos to stable chaotic state is used as the judgment basis to determine whether there is a harmonic signal of a specific frequency in the signal to be tested.

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