Hamiltonian conservative chaotic system of two-way grid vortex

Abstract

A Hamiltonian conservative chaos system of a two-way grid scroll is characterized in that: a main channel circuit and an auxiliary channel circuit are arranged, the output end of the auxiliary channel circuit is connected with the input end of the main channel circuit, and the output end of the main channel circuit is connected with the input end of the auxiliary channel circuit; the input end of the auxiliary channel circuit acquires an initial voltage signal (x0, y0, z0, w0) and outputs an auxiliary voltage signal to the main channel circuit, the main channel circuit outputs a first chaotic signal, and the first chaotic signal enters the auxiliary channel circuit as a second input signal of the auxiliary channel circuit. Effect: complex chaotic signals can be generated, the chaotic signals have the characteristics of high ergodicity and strong pseudo-randomness, the phase diagram of the chaotic signals does not have an attractor, the chaotic signals are more suitable for applications in the field of information security, and the chaotic signals provide more choices for information security technologies based on chaos; and the circuit structure is simple and compact, the number of used components is small, the production cost is low, and the circuit is easy to implement.

Description

Technical Field

[0001] This invention relates to the fields of chaotic signal generation and information security technology, and in particular to a Hamiltonian conservative chaotic system with bidirectional grid vortexes. Background Technology

[0002] Chaos is a complex phenomenon of motion that is widespread in nature. Chaotic signals possess characteristics such as randomness and unpredictability, and are considered a natural "code." Since Matthews first proposed using chaos for cryptographic research in 1989, research on chaotic systems has received widespread attention. Chaotic systems can be divided into dissipative chaotic systems and conservative chaotic systems. If the divergence of a system is less than zero, the system is a dissipative chaotic system; if the divergence of a system is zero, the system is a conservative chaotic system. Furthermore, if the Hamiltonian of a conservative chaotic system is also conservative, it is called a Hamiltonian-conservative chaotic system.

[0003] Classical chaotic systems such as the Chua, Chen, and Lorenz systems are all dissipative systems, and their divergence can be calculated to be less than zero. The chaotic motion generated by dissipative systems manifests in phase space as irregular perturbations within a finite subset of space, eventually forming attractors. Therefore, once these attractors are identified by codebreakers, the security of information transmission will be severely threatened.

[0004] For conservative systems, since the phase volume remains constant, there are no attractors, thus avoiding the risk of attractor deciphering. The trajectory of a conservative system is closely related to the initial values ​​of its state variables, and conservative chaotic signals exhibit better quasi-randomness and ergodicity, making them more advantageous for engineering applications. Currently, the design and application of conservative chaotic systems are still immature and difficult to meet the requirements of practical applications. Therefore, further research is needed on the application of conservative chaotic systems and chaotic signals.

[0005] The drawbacks of existing technologies are that the design and application of conservative chaotic systems are not yet mature and it is difficult to meet the requirements of practical applications. Summary of the Invention

[0006] The present invention provides a Hamiltonian conservative chaotic system with bidirectional grid vortex, which can generate complex chaotic signals and has the characteristics of high ergodicity and strong pseudo-randomness. Its phase diagram does not have attractors, making it more suitable for applications in fields such as information security, and providing more options for information security technology based on chaos.

[0007] To achieve the above objectives, the present invention provides a Hamiltonian conservative chaotic system with bidirectional grid vortexes, the key of which is: a main channel circuit and an auxiliary channel circuit are provided, wherein the output terminal of the auxiliary channel circuit is connected to the input terminal of the main channel circuit, and the output terminal of the main channel circuit is connected to the input terminal of the auxiliary channel circuit;

[0008] The auxiliary channel circuit acquires an initial voltage signal (x0, y0, z0, w0) at its input terminal and outputs an auxiliary voltage signal to the main channel circuit. The main channel circuit outputs a first chaotic signal, which is used as the second input signal of the auxiliary channel circuit. The Nth chaotic signal output by the main channel circuit is used as the N+1th input signal of the auxiliary channel circuit, forming a cyclically output chaotic system.

[0009] Through the above design, the Hamiltonian conservative chaotic system with bidirectional grid vortex can be used to design a random number generator. By using random numbers, i.e. chaotic signals, to scramble and spread images, voice, etc., the purpose of encryption can be achieved. It can be widely used in the field of information security.

[0010] Preferably, the main channel circuit includes a first main channel circuit, a second main channel circuit, a third main channel circuit, and a fourth main channel circuit, and the auxiliary channel circuit includes a first auxiliary channel circuit, a second auxiliary channel circuit, a third auxiliary channel circuit, a fourth auxiliary channel circuit, a fifth auxiliary channel circuit, a sixth auxiliary channel circuit, a seventh auxiliary channel circuit, an eighth auxiliary channel circuit, and a ninth auxiliary channel circuit.

[0011] The first main channel circuit outputs an x ​​signal, the second main channel circuit outputs a y signal and a -y signal, the third main channel circuit outputs a z signal and a -z signal, and the fourth main channel circuit outputs a w signal and a -w signal.

[0012] The first auxiliary channel circuit outputs y 3 The signal, the second auxiliary channel circuit outputs -yz 3 The signal, the third auxiliary channel circuit outputs -w 3 Signal, output of the 4th auxiliary channel circuit -xw 3 The fifth auxiliary channel circuit outputs an xw signal, the sixth auxiliary channel circuit outputs a -xy signal, and the seventh auxiliary channel circuit outputs an xy signal. 3 The 8th auxiliary channel circuit outputs a yz signal, and the 9th auxiliary channel circuit outputs an xz signal.

[0013] The main channel circuit is used to output chaotic signals, and the auxiliary channel circuit is used to output auxiliary signals. The circuit structure is simple, uses fewer components, and has a low production cost.

[0014] Preferably, the first main channel circuit includes a resistor R1, a resistor R2, a capacitor C1, and an operational amplifier U1.

[0015] The front end of resistor R1 obtains -yz 3The signal is connected to the inverting input terminal of the operational amplifier U1 at the rear end; the front end of the resistor R2 acquires the yz signal, and the rear end is connected to the inverting input terminal of the operational amplifier U1; the non-inverting input terminal of the operational amplifier U1 is grounded, the output terminal of the operational amplifier U1 outputs the x signal, and the output terminal of the operational amplifier U1 is connected to the inverting input terminal via a series capacitor C1.

[0016] The resistors R1 and R2 are connected in parallel to the inverting input of the operational amplifier U1, and then connected to the output of the operational amplifier U1 through the capacitor C1. The non-inverting input of the operational amplifier U1 is grounded, forming an inverting adder circuit to realize the signal x.

[0017] Preferably, the second main channel circuit includes resistors R3, R4, R5, and R6, capacitor C2, operational amplifier U2, and operational amplifier U3.

[0018] The front end of resistor R3 is obtained as -w 3 The signal is connected to the inverting input terminal of the operational amplifier U2 at the rear end; the front end of the resistor R4 receives the w signal, and the rear end is connected to the inverting input terminal of the operational amplifier U2; the non-inverting input terminal of the operational amplifier U2 is grounded, the output terminal of the operational amplifier U2 outputs the y signal, and the output terminal of the operational amplifier U2 is connected to the inverting input terminal via a series capacitor C2.

[0019] The output terminal of the operational amplifier U2 is connected in series with resistor R5 to the inverting input terminal of the operational amplifier U3. The non-inverting input terminal of the operational amplifier U3 is grounded. The output terminal of the operational amplifier U3 outputs a -y signal. The output terminal of the operational amplifier U3 is connected in series with resistor R6 to the inverting input terminal.

[0020] Resistors R3 and R4 are connected in parallel to the inverting input of operational amplifier U2, and then connected to the output of operational amplifier U2 through capacitor C2. The non-inverting input of operational amplifier U2 is grounded, forming an inverting adder circuit to realize signal y. Resistor R5 is connected to the inverting input of operational amplifier U3, and then connected to the output of operational amplifier U3 through resistor R6. The non-inverting input of operational amplifier U3 is grounded, converting signal y into signal -y, forming an inverting circuit.

[0021] Preferably, the third main channel circuit includes resistors R7, R8, R9, R10, R11, and R12, capacitor C3, operational amplifier U4, and operational amplifier U5.

[0022] The front end of resistor R7 receives the -xy signal, and the rear end is connected to the inverting input of operational amplifier U4; the front end of resistor R8 receives the xy signal. 3The signal is connected to the inverting input of the operational amplifier U4 at the rear end; the front end of the resistor R9 obtains -xw. 3 The signal is connected to the inverting input terminal of the operational amplifier U4 at the rear end; the front end of the resistor R10 acquires the xw signal, and the rear end is connected to the inverting input terminal of the operational amplifier U4; the non-inverting input terminal of the operational amplifier U4 is grounded; the output terminal of the operational amplifier U4 outputs the z signal, and the output terminal of the operational amplifier U4 is connected to the inverting input terminal via a series capacitor C3;

[0023] The output terminal of the operational amplifier U4 is connected in series with resistor R11 to the inverting input terminal of the operational amplifier U5. The non-inverting input terminal of the operational amplifier U5 is grounded. The output terminal of the operational amplifier U5 outputs a -z signal. The output terminal of the operational amplifier U5 is connected in series with resistor R12 to the inverting input terminal.

[0024] Resistors R7, R8, R9, and R10 are connected in parallel to the inverting input of operational amplifier U4, and then connected to the output of operational amplifier U4 through capacitor C3. The non-inverting input of operational amplifier U4 is grounded, forming an inverting adder circuit to realize signal z. Resistor R11 is connected to the inverting input of operational amplifier U5, and then connected to the output of operational amplifier U5 through resistor R12. The non-inverting input of operational amplifier U5 is grounded, converting signal z into signal -z, forming an inverting circuit.

[0025] Preferably, the fourth main channel circuit includes resistors R13, R14, R15, R16, R17, capacitor C4, operational amplifier U6, and operational amplifier U7.

[0026] The front end of resistor R13 obtains y 3 The signal is connected to the inverting input terminal of the operational amplifier U6 at the rear end; the front end of the resistor R14 acquires the xz signal, and the rear end is connected to the inverting input terminal of the operational amplifier U6; the front end of the resistor R15 acquires the -y signal, and the rear end is connected to the inverting input terminal of the operational amplifier U6; the non-inverting input terminal of the operational amplifier U6 is grounded; the output terminal of the operational amplifier U6 outputs the w signal; and the output terminal of the operational amplifier U6 is connected to the inverting input terminal via series capacitor C4.

[0027] The output terminal of the operational amplifier U6 is connected in series with resistor R16 to the inverting input terminal of the operational amplifier U7. The non-inverting input terminal of the operational amplifier U7 is grounded. The output terminal of the operational amplifier U7 outputs a -w signal. The output terminal of the operational amplifier U7 is connected in series with resistor R17 to the inverting input terminal.

[0028] Resistors R13, R14, and R15 are connected in parallel to the inverting input of operational amplifier U6, and then connected to the output of operational amplifier U6 through capacitor C4. The non-inverting input of operational amplifier U6 is grounded, forming an inverting adder circuit to realize signal w. Resistor R16 is connected to the inverting input of operational amplifier U7, and then connected to the output of operational amplifier U7 through resistor R17. The non-inverting input of operational amplifier U7 is grounded, converting signal w into signal -w, forming an inverting circuit.

[0029] Preferably, the first auxiliary channel circuit includes multipliers A1 and A2. The two inputs of multiplier A1 acquire the y signal, the output of multiplier A1 is connected to one input of multiplier A2, the other input of multiplier A2 acquires the y signal, and the output of multiplier A2 is the output of the first auxiliary channel circuit, outputting y. 3 Signal;

[0030] The second auxiliary channel circuit is equipped with a multiplier A3, one input of which acquires y. 3 The signal is obtained from the other input terminal, which is the -z signal. The output terminal of the multiplier A3 is the output terminal of the second auxiliary channel circuit, which outputs -zy. 3 Signal;

[0031] The third auxiliary channel circuit includes multipliers A4 and A5. Multiplier A4 acquires the 'w' signal at its two inputs. The output of multiplier A4 is connected to one input of multiplier A5. The other input of multiplier A5 acquires the '-w' signal. The output of multiplier A5 is the output of the third auxiliary channel circuit, outputting '-w'. 3 Signal;

[0032] The fourth auxiliary channel circuit is equipped with a multiplier A6, one input of which acquires -w. 3 The signal is obtained from the other input terminal, and the output terminal of the multiplier A6 is the output terminal of the fourth auxiliary channel circuit, outputting -xw. 3 Signal;

[0033] The fifth auxiliary channel circuit is equipped with a multiplier A7. One input terminal of the multiplier A7 acquires the x signal, and the other input terminal acquires the w signal. The output terminal of the multiplier A7 is the output terminal of the fifth auxiliary channel circuit, which outputs the xw signal.

[0034] The sixth auxiliary channel circuit includes a multiplier A8, resistors R18 and R19, and an operational amplifier U8. One input of the multiplier A8 acquires the y signal, and the other input acquires the x signal. The output of the multiplier A8 is connected in series with resistor R18 to the inverting input of the operational amplifier U8, and the non-inverting input of the operational amplifier U8 is grounded. The output of the operational amplifier U8 is the output of the sixth auxiliary channel circuit, outputting the -xy signal. The output of the operational amplifier U8 is connected in series with resistor R19 to the inverting input.

[0035] The seventh auxiliary channel circuit includes multipliers A9 and A10. One input of multiplier A9 is connected to the output of multiplier A8, and the other input acquires the y signal. The output of multiplier A9 is connected to one input of multiplier A10, and the other input of multiplier A10 acquires the y signal. The output of multiplier A10 is the output of the seventh auxiliary channel circuit, outputting xy. 3 Signal;

[0036] The 8th auxiliary channel circuit is equipped with a multiplier A11. One input terminal of the multiplier A11 acquires the y signal, the other input terminal acquires the z signal, and the output terminal of the multiplier A11 is the output terminal of the 8th auxiliary channel circuit, outputting the yz signal.

[0037] The 9th auxiliary channel circuit is equipped with a multiplier A12. One input terminal of the multiplier A12 acquires the x signal, and the other input terminal acquires the z signal. The output terminal of the multiplier A11 is the output terminal of the 9th auxiliary channel circuit, which outputs the xz signal.

[0038] Through the above design, using multiple multipliers A1-A12, resistors R18 and R19, and operational amplifier U8, auxiliary signal outputs for nine auxiliary channel circuits were achieved.

[0039] Preferably, the output model expression of the main channel circuit is:

[0040]

[0041] Where x, y, z, w are the state variables of the system, that is, the chaotic signal output by the bidirectional grid vortex Hamiltonian conservative chaotic system. The phase diagram of the system in the phase space yw is a 2×2 grid vortex conservative chaotic flow.

[0042] Typically, the divergence of a system is used to determine whether it is a conservative or dissipative system. The divergence of the bidirectional grid vortex Hamiltonian conservative chaotic system can be expressed as:

[0043] This indicates that the phase volume of the system is conservative, and also indicates that the system has no attractors, because the phase volume continuously contracts and eventually becomes zero before an attractor is formed. However, the phase volume of the bidirectional grid vortex Hamiltonian conservative chaotic system remains unchanged, so it will not form an attractor.

[0044] According to Helmholtz's theorem, the matrix form of the bidirectional grid vortex Hamiltonian conservative chaotic system can be expanded into a gradient field F. c (X) and vortex field F d (X):

[0045]

[0046] That is, F d (X) = 0; Hamiltonian energy H and F c The relationship between (X) is: Right now

[0047]

[0048] Find Hamiltonian energy By calculating the derivative of the Hamiltonian with respect to time This indicates that the system is a Hamiltonian energy conservative system.

[0049] The initial values ​​of the state variables of the bidirectional grid vortex Hamiltonian conservative chaotic system are (x0, y0, z0, w0) = (1, 1, 1, 1). Using the Wolf algorithm, the Lyapunov exponents of the system are calculated as follows: LE1 = 2.338, LE2 = LE3≈0, LE4 = -2.338. According to the chaos criterion, a system is considered chaotic when at least one Lyapunov exponent is greater than zero. Therefore, the bidirectional grid vortex Hamiltonian conservative chaotic system is a Hamiltonian conservative chaotic system.

[0050] Given initial values, the bidirectional grid vortex Hamiltonian conservative chaotic system exhibits a double vortex shape in both the y and w directions.

[0051] Through the above design, the bidirectional grid vortex Hamiltonian conservative chaotic system, compared to dissipative systems, possesses characteristics such as white noise-like properties, high ergodicity, and strong pseudo-randomness. It can also generate complex chaotic signals without forming attractors, effectively avoiding the risks associated with attractor reconstruction. Therefore, the bidirectional grid vortex Hamiltonian conservative chaotic system is more suitable for applications in fields such as information security.

[0052] Preferably, the chaotic signal is an analog voltage signal.

[0053] The beneficial effects of this invention are: it can generate complex chaotic signals with high ergodicity and strong pseudo-randomness; its phase diagram does not contain attractors, making it more suitable for applications in fields such as information security, and providing more options for information security technology based on chaos; moreover, the circuit structure is simple and compact, uses fewer components, has lower production costs, and is easy to implement. Attached Figure Description

[0054] Figure 1 This is a schematic diagram of the overall structure of the present invention;

[0055] Figure 2 The schematic diagrams are of the four main channel circuits in the embodiment;

[0056] Figure 3 This is a schematic diagram of the circuit for the nine auxiliary channels in the embodiment;

[0057] Figure 4 This is a two-dimensional phase diagram of state variables y and w in the embodiment;

[0058] Figure 5 This is a two-dimensional phase diagram of state variable y and state variable x in the embodiment;

[0059] Figure 6 This is a two-dimensional phase diagram of state variables z and w in the embodiment. Detailed Implementation

[0060] The present invention will be further described in detail below with reference to the accompanying drawings and specific examples. The following embodiments or drawings are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0061] like Figure 1 As shown: A Hamiltonian conservative chaotic system with bidirectional grid vortexes, the key feature of which is: a main channel circuit and an auxiliary channel circuit are provided, the output terminal of the auxiliary channel circuit is connected to the input terminal of the main channel circuit, and the output terminal of the main channel circuit is connected to the input terminal of the auxiliary channel circuit;

[0062] The auxiliary channel circuit acquires an initial voltage signal (x0, y0, z0, w0) at its input terminal and outputs an auxiliary voltage signal to the main channel circuit. The main channel circuit outputs a first chaotic signal, which is used as the second input signal of the auxiliary channel circuit. The Nth chaotic signal output by the main channel circuit is used as the N+1th input signal of the auxiliary channel circuit, forming a cyclically output chaotic system.

[0063] The main channel circuit is provided with a first main channel circuit, a second main channel circuit, a third main channel circuit, and a fourth main channel circuit, and the auxiliary channel circuit is provided with a first auxiliary channel circuit, a second auxiliary channel circuit, a third auxiliary channel circuit, a fourth auxiliary channel circuit, a fifth auxiliary channel circuit, a sixth auxiliary channel circuit, a seventh auxiliary channel circuit, an eighth auxiliary channel circuit, and a ninth auxiliary channel circuit.

[0064] The first main channel circuit outputs an x ​​signal, the second main channel circuit outputs a y signal and a -y signal, the third main channel circuit outputs a z signal and a -z signal, and the fourth main channel circuit outputs a w signal and a -w signal.

[0065] The first auxiliary channel circuit outputs y 3 The signal, the second auxiliary channel circuit outputs -yz 3 The signal, the third auxiliary channel circuit outputs -w 3 Signal, output of the 4th auxiliary channel circuit -xw 3 The fifth auxiliary channel circuit outputs an xw signal, the sixth auxiliary channel circuit outputs a -xy signal, and the seventh auxiliary channel circuit outputs an xy signal. 3 The 8th auxiliary channel circuit outputs a yz signal, and the 9th auxiliary channel circuit outputs an xz signal.

[0066] like Figure 2 As shown: The first main channel circuit is equipped with resistor R1, resistor R2, capacitor C1 and operational amplifier U1;

[0067] The front end of resistor R1 obtains -yz 3 The signal is connected to the inverting input terminal of the operational amplifier U1 at the rear end; the front end of the resistor R2 acquires the yz signal, and the rear end is connected to the inverting input terminal of the operational amplifier U1; the non-inverting input terminal of the operational amplifier U1 is grounded, the output terminal of the operational amplifier U1 outputs the x signal, and the output terminal of the operational amplifier U1 is connected to the inverting input terminal via a series capacitor C1.

[0068] The second main channel circuit is equipped with resistors R3, R4, R5, and R6, capacitor C2, operational amplifier U2, and operational amplifier U3;

[0069] The front end of resistor R3 is obtained as -w 3 The signal is connected to the inverting input terminal of the operational amplifier U2 at the rear end; the front end of the resistor R4 receives the w signal, and the rear end is connected to the inverting input terminal of the operational amplifier U2; the non-inverting input terminal of the operational amplifier U2 is grounded, the output terminal of the operational amplifier U2 outputs the y signal, and the output terminal of the operational amplifier U2 is connected to the inverting input terminal via a series capacitor C2.

[0070] The output terminal of the operational amplifier U2 is connected in series with resistor R5 to the inverting input terminal of the operational amplifier U3. The non-inverting input terminal of the operational amplifier U3 is grounded. The output terminal of the operational amplifier U3 outputs a -y signal. The output terminal of the operational amplifier U3 is connected in series with resistor R6 to the inverting input terminal.

[0071] The third main channel circuit is equipped with resistors R7, R8, R9, R10, R11, R12, capacitor C3, operational amplifier U4, and operational amplifier U5.

[0072] The front end of resistor R7 receives the -xy signal, and the rear end is connected to the inverting input of operational amplifier U4; the front end of resistor R8 receives the xy signal. 3 The signal is connected to the inverting input of the operational amplifier U4 at the rear end; the front end of the resistor R9 obtains -xw. 3 The signal is connected to the inverting input terminal of the operational amplifier U4 at the rear end; the front end of the resistor R10 acquires the xw signal, and the rear end is connected to the inverting input terminal of the operational amplifier U4; the non-inverting input terminal of the operational amplifier U4 is grounded; the output terminal of the operational amplifier U4 outputs the z signal, and the output terminal of the operational amplifier U4 is connected to the inverting input terminal via a series capacitor C3;

[0073] The output terminal of the operational amplifier U4 is connected in series with resistor R11 to the inverting input terminal of the operational amplifier U5. The non-inverting input terminal of the operational amplifier U5 is grounded. The output terminal of the operational amplifier U5 outputs a -z signal. The output terminal of the operational amplifier U5 is connected in series with resistor R12 to the inverting input terminal.

[0074] The fourth main channel circuit is equipped with resistors R13, R14, R15, R16, R17, capacitor C4, operational amplifier U6, and operational amplifier U7.

[0075] The front end of resistor R13 obtains y 3 The signal is connected to the inverting input terminal of the operational amplifier U6 at the rear end; the front end of the resistor R14 acquires the xz signal, and the rear end is connected to the inverting input terminal of the operational amplifier U6; the front end of the resistor R15 acquires the -y signal, and the rear end is connected to the inverting input terminal of the operational amplifier U6; the non-inverting input terminal of the operational amplifier U6 is grounded; the output terminal of the operational amplifier U6 outputs the w signal; and the output terminal of the operational amplifier U6 is connected to the inverting input terminal via series capacitor C4.

[0076] The output terminal of the operational amplifier U6 is connected in series with resistor R16 to the inverting input terminal of the operational amplifier U7. The non-inverting input terminal of the operational amplifier U7 is grounded. The output terminal of the operational amplifier U7 outputs a -w signal. The output terminal of the operational amplifier U7 is connected in series with resistor R17 to the inverting input terminal.

[0077] like Figure 3 As shown: The first auxiliary channel circuit includes multipliers A1 and A2. The two inputs of multiplier A1 acquire the y signal. The output of multiplier A1 is connected to one input of multiplier A2. The other input of multiplier A2 acquires the y signal. The output of multiplier A2 is the output of the first auxiliary channel circuit, outputting y. 3 Signal;

[0078] The second auxiliary channel circuit is equipped with a multiplier A3, one input of which acquires y. 3 The signal is obtained from the other input terminal, which is the -z signal. The output terminal of the multiplier A3 is the output terminal of the second auxiliary channel circuit, which outputs -zy. 3 Signal;

[0079] The third auxiliary channel circuit includes multipliers A4 and A5. Multiplier A4 acquires the 'w' signal at its two inputs. The output of multiplier A4 is connected to one input of multiplier A5. The other input of multiplier A5 acquires the '-w' signal. The output of multiplier A5 is the output of the third auxiliary channel circuit, outputting '-w'. 3 Signal;

[0080] The fourth auxiliary channel circuit is equipped with a multiplier A6, one input of which acquires -w. 3 The signal is obtained from the other input terminal, and the output terminal of the multiplier A6 is the output terminal of the fourth auxiliary channel circuit, outputting -xw. 3 Signal;

[0081] The fifth auxiliary channel circuit is equipped with a multiplier A7. One input terminal of the multiplier A7 acquires the x signal, and the other input terminal acquires the w signal. The output terminal of the multiplier A7 is the output terminal of the fifth auxiliary channel circuit, which outputs the xw signal.

[0082] The sixth auxiliary channel circuit includes a multiplier A8, resistors R18 and R19, and an operational amplifier U8. One input of the multiplier A8 acquires the y signal, and the other input acquires the x signal. The output of the multiplier A8 is connected in series with resistor R18 to the inverting input of the operational amplifier U8, and the non-inverting input of the operational amplifier U8 is grounded. The output of the operational amplifier U8 is the output of the sixth auxiliary channel circuit, outputting the -xy signal. The output of the operational amplifier U8 is connected in series with resistor R19 to the inverting input.

[0083] The seventh auxiliary channel circuit includes multipliers A9 and A10. One input of multiplier A9 is connected to the output of multiplier A8, and the other input acquires the y signal. The output of multiplier A9 is connected to one input of multiplier A10, and the other input of multiplier A10 acquires the y signal. The output of multiplier A10 is the output of the seventh auxiliary channel circuit, outputting xy. 3 Signal;

[0084] The 8th auxiliary channel circuit is equipped with a multiplier A11. One input terminal of the multiplier A11 acquires the y signal, the other input terminal acquires the z signal, and the output terminal of the multiplier A11 is the output terminal of the 8th auxiliary channel circuit, outputting the yz signal.

[0085] The 9th auxiliary channel circuit is equipped with a multiplier A12. One input terminal of the multiplier A12 acquires the x signal, and the other input terminal acquires the z signal. The output terminal of the multiplier A11 is the output terminal of the 9th auxiliary channel circuit, which outputs the xz signal.

[0086] The output model expression of the main channel circuit is:

[0087]

[0088] Where x, y, z, and w are the state variables of the system, i.e., the chaotic signal output by the bidirectional grid vortex Hamiltonian conservative chaotic system. The phase diagram of this system in the phase space yw is a 2×2 grid vortex conservative chaotic flow. Figure 4 As shown; the phase diagram of the system in phase space yx is as follows. Figure 5 As shown; the phase diagram of the system in phase space zw is as follows Figure 6 As shown.

[0089] Typically, the divergence of a system is used to determine whether it is a conservative or dissipative system. The divergence of the bidirectional grid vortex Hamiltonian conservative chaotic system can be expressed as:

[0090] This indicates that the phase volume of the system is conservative, and also indicates that the system has no attractors, because the phase volume continuously contracts and eventually becomes zero before an attractor is formed. However, the phase volume of the bidirectional grid vortex Hamiltonian conservative chaotic system remains unchanged, so it will not form an attractor.

[0091] According to Helmholtz's theorem, the matrix form of the bidirectional grid vortex Hamiltonian conservative chaotic system can be expanded into a gradient field F. c (X) and vortex field F d (X):

[0092]

[0093] That is, F d (X) = 0; Hamiltonian energy H and F c The relationship between (X) is: ▽H T F c (X) = 0, that is

[0094] Find Hamiltonian energy By calculating the derivative of the Hamiltonian with respect to time This indicates that the system is a Hamiltonian energy conservative system.

[0095] The initial values ​​of the state variables of the bidirectional grid vortex Hamiltonian conservative chaotic system are (x0, y0, z0, w0) = (1, 1, 1, 1). Using the Wolf algorithm, the Lyapunov exponents of the system are calculated as follows: LE1 = 2.338, LE2 = LE3≈0, LE4 = -2.338. According to the chaos criterion, a system is considered chaotic when at least one Lyapunov exponent is greater than zero. Therefore, the bidirectional grid vortex Hamiltonian conservative chaotic system is a Hamiltonian conservative chaotic system.

[0096] In this embodiment of the invention, multipliers A1-A12 are all multipliers in the Virtual Electronics Lab software, with a gain of 1V / V. The operational amplifiers are all LF347N, and their DC power supply voltage is: V. CC =15V, V EE = -15V.

[0097] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A Hamiltonian conservative chaotic system with bidirectional grid vortices, characterized in that: The circuit is equipped with a main channel circuit and an auxiliary channel circuit. The output terminal of the auxiliary channel circuit is connected to the input terminal of the main channel circuit, and the output terminal of the main channel circuit is connected to the input terminal of the auxiliary channel circuit. The input terminal of the auxiliary channel circuit acquires the initial voltage signal ( The system outputs an auxiliary voltage signal to the main channel circuit, and the main channel circuit outputs a first chaotic signal. The first chaotic signal is used as the second input signal of the auxiliary channel circuit. The Nth chaotic signal output by the main channel circuit is used as the N+1th input signal of the auxiliary channel circuit, forming a cyclically output chaotic system. The main channel circuit is provided with a first main channel circuit, a second main channel circuit, a third main channel circuit, and a fourth main channel circuit, and the auxiliary channel circuit is provided with a first auxiliary channel circuit, a second auxiliary channel circuit, a third auxiliary channel circuit, a fourth auxiliary channel circuit, a fifth auxiliary channel circuit, a sixth auxiliary channel circuit, a seventh auxiliary channel circuit, an eighth auxiliary channel circuit, and a ninth auxiliary channel circuit. The first main channel circuit outputs an x ​​signal, the second main channel circuit outputs a y signal and a -y signal, the third main channel circuit outputs a z signal and a -z signal, and the fourth main channel circuit outputs a w signal and a -w signal. The first auxiliary channel circuit output The signal, the second auxiliary channel circuit outputs -z The signal, the output of the third auxiliary channel circuit - Signal, output of the 4th auxiliary channel circuit -x The fifth auxiliary channel circuit outputs an xw signal, the sixth auxiliary channel circuit outputs a -xy signal, and the seventh auxiliary channel circuit outputs an x ​​signal. The 8th auxiliary channel circuit outputs a yz signal, and the 9th auxiliary channel circuit outputs an xz signal; The output model expression of the main channel circuit is: , Where x, y, z, w are the state variables of the system, that is, the chaotic signal output by the bidirectional grid vortex Hamiltonian conservative chaotic system. The phase diagram of the system in the phase space yw is a 2×2 grid vortex conservative chaotic flow.

2. The Hamiltonian conservative chaotic system with bidirectional grid vortices according to claim 1, characterized in that: The first main channel circuit is equipped with resistor R1, resistor R2, capacitor C1 and operational amplifier U1; The front end of resistor R1 obtains -z The signal is connected to the inverting input terminal of the operational amplifier U1 at the rear end; the front end of the resistor R2 acquires the yz signal, and the rear end is connected to the inverting input terminal of the operational amplifier U1; the non-inverting input terminal of the operational amplifier U1 is grounded, the output terminal of the operational amplifier U1 outputs the x signal, and the output terminal of the operational amplifier U1 is connected to the inverting input terminal via a series capacitor C1.

3. The Hamiltonian conservative chaotic system with bidirectional grid vortices according to claim 1, characterized in that: The second main channel circuit is equipped with resistors R3, R4, R5, and R6, capacitor C2, operational amplifier U2, and operational amplifier U3; The front end of resistor R3 is obtained - The signal is connected to the inverting input terminal of the operational amplifier U2 at the rear end; the front end of the resistor R4 receives the w signal, and the rear end is connected to the inverting input terminal of the operational amplifier U2; the non-inverting input terminal of the operational amplifier U2 is grounded, the output terminal of the operational amplifier U2 outputs the y signal, and the output terminal of the operational amplifier U2 is connected to the inverting input terminal via a series capacitor C2. The output terminal of the operational amplifier U2 is connected in series with resistor R5 to the inverting input terminal of the operational amplifier U3. The non-inverting input terminal of the operational amplifier U3 is grounded. The output terminal of the operational amplifier U3 outputs a -y signal. The output terminal of the operational amplifier U3 is connected in series with resistor R6 to the inverting input terminal.

4. The Hamiltonian conservative chaotic system with bidirectional grid vortexes according to claim 1, characterized in that: The third main channel circuit is equipped with resistors R7, R8, R9, R10, R11, R12, capacitor C3, operational amplifier U4, and operational amplifier U5. The front end of resistor R7 receives the -xy signal, and the rear end is connected to the inverting input of operational amplifier U4; the front end of resistor R8 receives the x signal. The signal is connected to the inverting input of the operational amplifier U4 at the rear end; the front end of the resistor R9 obtains -x The signal is connected to the inverting input terminal of the operational amplifier U4 at the rear end; the front end of the resistor R10 acquires the xw signal, and the rear end is connected to the inverting input terminal of the operational amplifier U4; the non-inverting input terminal of the operational amplifier U4 is grounded; the output terminal of the operational amplifier U4 outputs the z signal, and the output terminal of the operational amplifier U4 is connected to the inverting input terminal via a series capacitor C3; The output terminal of the operational amplifier U4 is connected in series with resistor R11 to the inverting input terminal of the operational amplifier U5. The non-inverting input terminal of the operational amplifier U5 is grounded. The output terminal of the operational amplifier U5 outputs a -z signal. The output terminal of the operational amplifier U5 is connected in series with resistor R12 to the inverting input terminal.

5. The Hamiltonian conservative chaotic system with bidirectional grid vortexes according to claim 1, characterized in that: The fourth main channel circuit is equipped with resistors R13, R14, R15, R16, R17, capacitor C4, operational amplifier U6, and operational amplifier U7. The front end of the resistor R13 is obtained The signal is connected to the inverting input terminal of the operational amplifier U6 at the rear end; the front end of the resistor R14 acquires the xz signal, and the rear end is connected to the inverting input terminal of the operational amplifier U6; the front end of the resistor R15 acquires the -y signal, and the rear end is connected to the inverting input terminal of the operational amplifier U6; the non-inverting input terminal of the operational amplifier U6 is grounded; the output terminal of the operational amplifier U6 outputs the w signal; and the output terminal of the operational amplifier U6 is connected to the inverting input terminal via series capacitor C4. The output terminal of the operational amplifier U6 is connected in series with resistor R16 to the inverting input terminal of the operational amplifier U7. The non-inverting input terminal of the operational amplifier U7 is grounded. The output terminal of the operational amplifier U7 outputs a -w signal. The output terminal of the operational amplifier U7 is connected in series with resistor R17 to the inverting input terminal.

6. The Hamiltonian conservative chaotic system with bidirectional grid vortexes according to claim 1, characterized in that: The first auxiliary channel circuit includes multipliers A1 and A2. Multiplier A1 acquires the y signal at both inputs. The output of multiplier A1 is connected to one input of multiplier A2. The other input of multiplier A2 acquires the y signal. The output of multiplier A2 is the output of the first auxiliary channel circuit. Signal; The second auxiliary channel circuit is equipped with a multiplier A3, and one input terminal of the multiplier A3 acquires... The signal is obtained from the other input terminal, which is the -z signal. The output terminal of the multiplier A3 is the output terminal of the second auxiliary channel circuit, which outputs -z. Signal; The third auxiliary channel circuit includes multipliers A4 and A5. Multiplier A4 acquires the 'w' signal at its two inputs. The output of multiplier A4 is connected to one input of multiplier A5. The other input of multiplier A5 acquires the '-w' signal. The output of multiplier A5 is the output of the third auxiliary channel circuit, outputting a '-w' signal. Signal; The fourth auxiliary channel circuit is equipped with a multiplier A6, one input of which acquires - The signal is obtained from the other input terminal, and the output terminal of the multiplier A6 is the output terminal of the fourth auxiliary channel circuit, outputting -x. Signal; The fifth auxiliary channel circuit is equipped with a multiplier A7. One input terminal of the multiplier A7 acquires the x signal, and the other input terminal acquires the w signal. The output terminal of the multiplier A7 is the output terminal of the fifth auxiliary channel circuit, which outputs the xw signal. The sixth auxiliary channel circuit includes a multiplier A8, resistors R18 and R19, and an operational amplifier U8. One input of the multiplier A8 acquires the y signal, and the other input acquires the x signal. The output of the multiplier A8 is connected in series with resistor R18 to the inverting input of the operational amplifier U8, and the non-inverting input of the operational amplifier U8 is grounded. The output of the operational amplifier U8 is the output of the sixth auxiliary channel circuit, outputting the -xy signal. The output of the operational amplifier U8 is connected in series with resistor R19 to the inverting input. The seventh auxiliary channel circuit includes multipliers A9 and A10. One input of multiplier A9 is connected to the output of multiplier A8, and the other input acquires the y signal. The output of multiplier A9 is connected to one input of multiplier A10, and the other input of multiplier A10 acquires the y signal. The output of multiplier A10 is the output of the seventh auxiliary channel circuit, outputting x. Signal; The 8th auxiliary channel circuit is equipped with a multiplier A11. One input terminal of the multiplier A11 acquires the y signal, the other input terminal acquires the z signal, and the output terminal of the multiplier A11 is the output terminal of the 8th auxiliary channel circuit, which outputs the yz signal. The 9th auxiliary channel circuit is equipped with a multiplier A12. One input terminal of the multiplier A12 acquires the x signal, and the other input terminal acquires the z signal. The output terminal of the multiplier A11 is the output terminal of the 9th auxiliary channel circuit, which outputs the xz signal.

7. The Hamiltonian conservative chaotic system with bidirectional grid vortexes according to claim 1, characterized in that: The chaotic signal is an analog voltage signal.

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