Novel three-dimensional coupling non-autonomous chaotic system and circuit

Abstract

The invention discloses a novel three-dimensional coupling non-autonomous chaotic system and circuit, and belongs to the technical field of nonlinear dynamics and analog electronics, the circuit of the coupling chaotic system is developed by using an analog circuit, and the chaotic synchronization phenomenon is known through an electrical experiment under the excitation of a rectangular wave signal. The novel three-dimensional coupling non-autonomous chaotic system is moderate in circuit structure, simple to implement and low in cost, can demonstrate the motion state of a phase track formed by output state variables of the new system only by changing control signal parameters in the circuit, verifies that the new system has a chaotic phenomenon, and is high in practicability. And the chaos synchronization phenomenon in a nonlinear circuit can be researched.

Description

Technical Field

[0001] The present invention belongs to the fields of nonlinear dynamics and analog electronic technology, and particularly relates to a novel three-dimensional coupled non-autonomous chaotic system and circuit. Background Art

[0002] The inherent determinacy, non-equilibrium phase transformation, extremely strong sensitivity to small signals, and strong anti-interference ability to irrelevant noises of chaotic systems make them have a very broad application prospect in the field of weak signal detection. However, generally speaking, the time-domain weak signal detection models based on chaotic oscillators are mainly Duffing, Rossler, Van der Pol, Lorenz, Chua and their extended models, and these traditional nonlinear dynamic systems generally have complex structures. Various existing methods for detecting weak signals require a large number of system parameter regulations, which will not only lead to a sharp increase in workload, but also introduce inevitable human errors, and ultimately result in a large error between the weak signal identification result and the true result. Therefore, constructing a system with a simple structure and finding a new method with higher efficiency and simpler operation are necessary research for applying the weak signal detection technology based on chaotic oscillators to various engineering fields.

[0003] The chaotic synchronization phenomenon of coupled chaotic systems is mainly used to detect non-periodic signals, and the circuit design of coupled chaotic systems for detecting weak periodic signals is extremely rare. Summary of the Invention

[0004] To solve the above technical problems, the present invention provides a novel three-dimensional coupled non-autonomous chaotic system and circuit. The circuit of the coupled chaotic system is developed by using an analog circuit. Under the excitation of a rectangular wave signal, the chaotic synchronization phenomenon is understood through electrical experiments, which has important theoretical significance and potential engineering application value for exploring the weak periodic signal detection technology. It is not only applicable to college non-linear science education, experimental teaching and demonstration, and science popularization, but also can explore the chaotic synchronization phenomenon under the excitation of a rectangular wave signal and be applied to the field of weak periodic signal detection.

[0005] To achieve the above object, the present invention adopts the following technical solution:

[0006] A novel three-dimensional coupled non-autonomous chaotic system, and the dimensionless state equation of the system is:

[0007] ;

[0008] wherein, x1, x2, x3 ∈ R are the state variables of system Q, a is the linear coupling coefficient, reflecting the coupling strength between oscillators, b is the nonlinear coupling coefficient, and a, b > 0, and S(t) is the externally input rectangular wave control signal.

[0009] On the other hand, the present invention provides a novel three-dimensional coupled non-autonomous chaotic circuit, which is applied to the aforementioned novel three-dimensional coupled non-autonomous chaotic system, and includes:

[0010] A three-channel circuit module for outputting three state variable signals X1, X2, and X3;

[0011] A non-linear coupling module for respectively connecting with the three state variable signals X1, X2, and X3 to implement the operation of non-linear coupling terms;

[0012] A linear coupling module for generating a differential signal X2 - X1 to implement the operation of linear coupling terms;

[0013] A power supply module for providing DC power supply for the three-channel circuit module, the non-linear coupling module, and the linear coupling module.

[0014] The beneficial effects of the present invention are as follows:

[0015] The present invention uses electrical experiments for experimental verification. The chaotic system circuit is simple to implement and low in cost. By only changing the control signal parameters in the circuit, the motion state of the phase trajectory composed of the output state variables of the new system and the chaotic synchronization phenomenon can be demonstrated, and it has a wide application prospect and important application value in the field of weak signal detection. Description of the Drawings

[0016] Figure 1 It is a structural diagram of a novel three-dimensional coupled non-autonomous chaotic circuit of the present invention;

[0017] Figure 2 It is the phase trajectory composed of the output signals X1 and X2 based on the circuit of the present invention when the amplitude of the control rectangular wave signal is input as 50 mv;

[0018] Figure 3 It is the chaotic synchronization phenomenon described by the output signals based on the circuit of the present invention when the amplitude of the control rectangular wave signal is input as 50 mv. Specific Embodiments

[0019] The present invention will be further described below with reference to the drawings and embodiments.

[0020] The dimensionless state equation of the novel N-dimensional coupled non-linear system Q is:

[0021] ;

[0022] In the formula, x1, x2,..., x N ∈R are the state variables of the system Q, N represents the number of chaotic oscillators, dx i / dt represents the derivative of the state variable x i with respect to time t. a(xi-1 -x i ) is the linear coupling term between two adjacent oscillators, and a(x N -x1) is the coupling term between the first and the last oscillators. Here, a is a positive linear coupling coefficient, reflecting the coupling strength between oscillators. bx i-1 -x i+1 is the non - linear term of the i - th oscillator, bx N x2 and bx N-1 x1 are the non - linear terms of the first and the N - th oscillators respectively. Here, b is a positive coupling coefficient, that is, the non - linear coupling coefficient.

[0023] Based on the above principle, the present invention proposes a novel three - dimensional coupled non - autonomous chaotic system. Taking the system Q composed of 3 chaotic oscillators as an example, let N = 3, and an external control signal - rectangular wave signal S(t) is added. The system Q becomes a novel three - dimensional coupled non - autonomous chaotic system M, and its dimensionless state equation is:

[0024] ;

[0025] Among them, the control parameter a = 10, b = 5, the rectangular wave signal is a typical square wave signal, the duty cycle is 50%, the frequency is 5Hz, and the amplitude is a variable regulation parameter.

[0026] On the other hand, the present invention also proposes a novel three - dimensional coupled non - autonomous chaotic circuit. A non - linear circuit is constructed according to the novel three - dimensional coupled non - autonomous chaotic system M, as Figure 1 shown. The electronic components included in this circuit are 10 operational amplifiers, 3 analog multipliers, 25 resistors and 3 capacitors, which together constitute three - channel circuits.

[0027] The inverting input terminal of operational amplifier U1 in the first channel circuit is connected to the output terminals of resistors R1, R2, and R3 connected in parallel. The non-inverting input terminal is grounded. Resistor R4 is connected across the inverting input terminal and the output terminal. The output terminal is connected to one end of resistor R5. The inverting input terminal of operational amplifier U2 is connected to the other end of resistor R5. The non-inverting input terminal is grounded. Capacitor C1 is connected across the inverting input terminal and the output terminal (outputting the state variable signal X1). The output terminal (state variable signal X1) is connected to one end of resistor R6. The inverting input terminal of operational amplifier U3 is connected to the other end of resistor R6. The non-inverting input terminal is grounded. Resistor R7 is connected across the inverting input terminal and the output terminal. The output terminal is connected to the input terminal of resistor R2. The first and second non-inverting input terminals of analog multiplier A1 are respectively connected to the output state variable signals X2 and X3. The output terminal is connected to the input terminal of resistor R3. The input terminal of resistor R1 is connected to the output signal X3. Preferably, in the first channel circuit, resistor R1 = R2 = 0.5 kΩ, R3 = 1 kΩ, R4 = 5 kΩ, R5 = 10 kΩ, R6 = R7 = 1 kΩ, and capacitor C1 = 100 μF;

[0028] The inverting input terminal of operational amplifier U5 in the second channel circuit is connected to the output terminals of resistors R8, R9, R10, and R11 connected in parallel. The non-inverting input terminal is grounded. Resistor R12 is connected across the inverting input terminal and the output terminal. The output terminal is connected to one end of resistor R13. The inverting input terminal of operational amplifier U6 is connected to the other end of resistor R13. The non-inverting input terminal is grounded. Capacitor C2 is connected across the inverting input terminal and the output terminal (outputting the state variable signal X2). The output terminal (outputting the state variable signal X2) is connected to one end of resistor R14. The inverting input terminal of operational amplifier U7 is connected to the other end of resistor R14. The non-inverting input terminal is grounded. Resistor R15 is connected across the inverting input terminal and the output terminal. The output terminal is respectively connected to the input terminals of resistor R10 and R24. The first and second non-inverting input terminals of analog multiplier A2 are respectively connected to the output state variable signals X1 and X3. The output terminal is connected to the input terminal of R11. The input terminal of R8 is connected to the output state variable signal X1. The input terminal of R9 is connected to the control signal source S(t). Preferably, in the second channel circuit, the control signal source is a square wave signal, resistor R8 = 0.5 kΩ, R9 = 5 kΩ, R10 = 0.5 kΩ, R11 = 1 kΩ, R12 = 5 kΩ, R13 = 10 kΩ, R14 = R15 = 1 kΩ, and capacitor C2 = 100 μF;

[0029] The inverting input terminal of operational amplifier U8 in the third-channel circuit is connected to the output terminals of resistors R16, R17, and R18 connected in parallel. The non-inverting input terminal is grounded. Resistor R19 is connected across the inverting input terminal and the output terminal. The output terminal is connected to one end of resistor R20. The inverting input terminal of operational amplifier U9 is connected to the other end of resistor R20. The non-inverting input terminal is grounded. Capacitor C3 is connected across the inverting input terminal and the output terminal (outputting the state variable signal X3). The output terminal (outputting the state variable signal X3) is connected to one end of resistor R21. The inverting input terminal of operational amplifier U10 is connected to the other end of resistor R21. The non-inverting input terminal is grounded. Resistor R22 is connected across the inverting input terminal and the output terminal. The output terminal is connected to the input terminal of resistor R17. The first and second non-inverting input terminals of analog multiplier A3 are respectively connected to the output state variable signals X1 and X2. The output terminal is connected to the input terminal of R18. The input terminal of R16 is connected to the output signal X2. Preferably, in the third-channel circuit, resistor R16 = R17 = 0.5 kΩ, R18 = 1 kΩ, R19 = 5 kΩ, R20 = 10 kΩ, R21 = R22 = 1 kΩ, and capacitor C3 = 100 μF.

[0030] In the non-linear circuit of the novel three-dimensional coupled non-autonomous chaotic system M, the inverting input terminal of operational amplifier U4 is connected to the output terminals of resistors R23 and R24 connected in parallel. The non-inverting input terminal is grounded. Resistor R25 is connected across the inverting input terminal and the output terminal (outputting the signal X2 - X1). The input terminal of R23 is connected to the state variable signal X1. The input terminal of R24 is connected to the output terminal of operational amplifier U7. Preferably, resistor R23 = R24 = R25 = 1 kΩ.

[0031] In the non-linear circuit of the novel three-dimensional coupled non-autonomous chaotic system M, the positive power supply terminals and negative power supply terminals of all operational amplifiers and multipliers are respectively connected to the positive pole of DC power supply V1 and the negative pole of DC power supply V2. The negative pole of DC power supply V1 is connected to the positive pole of DC power supply V2 and is grounded simultaneously. DC power supplies V1 and V2 both provide 15 V voltage.

[0032] Figure 2 To control the phase trajectory formed by the output state variable signals X1 and X2 of the circuit based on the present invention when the amplitude of the rectangular wave signal input is 50 mv. Figure 3 To control the chaotic synchronization phenomenon described by the output signal of the circuit based on the present invention when the amplitude of the rectangular wave signal input is 50 mv. It can be seen from this that the three-channel circuits respectively realize the establishment of the three chaotic oscillators in system M, and the real-time changes of the state variables x1, x2, and x3 in system M can be directly observed by this circuit.

[0033] The specific embodiments described above further elaborate on the object, technical solution, and beneficial effects of the present invention. It should be understood that the above are only specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A novel three-dimensional coupled non-autonomous chaotic system, characterized by: The dimensionless state equation of the system is: ， Where x1, x2, x3∈R are the state variables of the system Q, a is the linear coupling coefficient, reflecting the coupling strength between the oscillators, b is the nonlinear coupling coefficient, and a, b>0, S(t) is the external input rectangular wave control signal.

2. A novel three-dimensional coupled non-autonomous chaotic circuit, applied to the novel three-dimensional coupled non-autonomous chaotic system according to claim 1, characterized in that: include: A three-channel circuit module for outputting three state variable signals X1, X2, and X3; A nonlinear coupling module, used to connect to the three state variable signals X1, X2, and X3 respectively to realize the operation of nonlinear coupling terms; The linear coupling module is used to generate a differential signal X2-X1 and realize the operation of the linear coupling term; The power supply module is used to provide DC power supply to the three-channel circuit module, the nonlinear coupling module, and the linear coupling module.

3. A novel three-dimensional coupled non-autonomous chaotic circuit according to claim 2, characterized in that: The three-channel circuit module includes three channel circuits, wherein the first channel circuit generates a state variable signal X1 through operational amplifiers U1-U3, resistors R1-R7 and capacitor C1; the second channel circuit generates a state variable signal X2 through operational amplifiers U5-U7, resistors R8-R15 and capacitor C2, and connects to an external rectangular wave control signal S(t); the third channel circuit generates a state variable signal X3 through operational amplifiers U8-U10, resistors R16-R22 and capacitor C3.

4. A novel three-dimensional coupled non-autonomous chaotic circuit according to claim 3, characterized in that: The inverting input terminal of the operational amplifier U1 in the first channel circuit is connected to the output terminals of the resistors R1, R2, and R3 connected in parallel, the non-inverting input terminal is grounded, the resistor R4 is connected between the inverting input terminal and the output terminal, and the output terminal is connected to one end of the resistor R5; the inverting input terminal of the operational amplifier U2 is connected to the other end of the resistor R5, the non-inverting input terminal is grounded, the capacitor C1 is connected between the inverting input terminal and the output state variable signal X1, and the output state variable signal X1 is connected to one end of the resistor R6; the inverting input terminal of the operational amplifier U3 is connected to the other end of the resistor R6, the non-inverting input terminal is grounded, the resistor R7 is connected between the inverting input terminal and the output terminal, and the output terminal is connected to the input terminal of the resistor R2; the input terminal of the resistor R1 is connected to the output signal X3.

5. The novel three-dimensional coupled non-autonomous chaotic circuit according to claim 3, characterized in that: The inverting input terminal of the operational amplifier U5 in the second channel circuit is connected to the output terminals of the resistors R8, R9, R10, and R11 connected in parallel, the non-inverting input terminal is grounded, the resistor R12 is connected between the inverting input terminal and the output terminal, and the output terminal is connected to one end of the resistor R13; the inverting input terminal of the operational amplifier U6 is connected to the other end of the resistor R13, the non-inverting input terminal is grounded, the capacitor C2 is connected between the inverting input terminal and the output state variable signal X2, and the output state variable signal X2 is connected to one end of the resistor R14; the inverting input terminal of the operational amplifier U7 is connected to the other end of the resistor R14, the non-inverting input terminal is grounded, the resistor R15 is connected between the inverting input terminal and the output terminal, and the output terminal is connected to the input terminals of the resistors R10 and R24 respectively; the input terminal of R8 is connected to the output state variable signal X1; the input terminal of R9 is connected to the control signal source S(t).

6. The novel three-dimensional coupled non-autonomous chaotic circuit according to claim 3, characterized in that: The inverting input terminal of the operational amplifier U8 in the third channel circuit is connected to the output terminals of the resistors R16, R17, and R18 connected in parallel, the non-inverting input terminal is grounded, the resistor R19 is connected between the inverting input terminal and the output terminal, and the output terminal is connected to one end of the resistor R20; the inverting input terminal of the operational amplifier U9 is connected to the other end of the resistor R20, the non-inverting input terminal is grounded, the capacitor C3 is connected between the inverting input terminal and the output state variable signal X3, and the output state variable signal X3 is connected to one end of the resistor R21; the inverting input terminal of the operational amplifier U10 is connected to the other end of the resistor R21, the non-inverting input terminal is grounded, the resistor R22 is connected between the inverting input terminal and the output terminal, and the output terminal is connected to the input terminal of the resistor R17; the input terminal of R16 is connected to the output signal X2.

7. The novel three-dimensional coupled non-autonomous chaotic circuit according to claim 2, characterized in that: The nonlinear coupling module includes analog multipliers A1, A2, and A3.

8. The novel three-dimensional coupled non-autonomous chaotic circuit according to claim 7, characterized in that: The first and second in-phase input terminals of the analog multiplier A1 are connected to the output signals X2 and X3 respectively, and the output terminal is connected to the input terminal of the resistor R3; the first and second in-phase input terminals of the analog multiplier A2 are connected to the output state variable signals X1 and X3 respectively, and the output terminal is connected to the input terminal of R11; the first and second in-phase input terminals of the analog multiplier A3 are connected to the output state variable signals X1 and X2 respectively, and the output terminal is connected to the input terminal of R18.

9. The novel three-dimensional coupled non-autonomous chaotic circuit according to claim 2, characterized in that: The inverting input terminal of the operational amplifier U4 in the linear coupling module is connected to the output terminals of the resistors R23 and R24 connected in parallel, the non-inverting input terminal is grounded, and the resistor R25 is connected between the inverting input terminal and the output signal X2-X1; the input terminal of R23 is connected to the state variable signal X1; the input terminal of R24 is connected to the output terminal of the operational amplifier U7.

10. The novel three-dimensional coupled non-autonomous chaotic circuit according to claim 2, characterized in that: The power supply module includes DC power supplies V1 and V2. The positive power supply terminals and negative power supply terminals of all operational amplifiers and multipliers are respectively connected to the positive pole of the DC power supply V1 and the negative pole of the DC power supply V2; the negative pole of the DC power supply V1 is connected to the positive pole of the DC power supply V2 and are grounded at the same time.

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