A novel three-dimensional hidden multi-scroll chaotic oscillator based on jerk system

Abstract

The application provides a novel three-dimensional hidden multi-vortex chaotic oscillator based on a JERK system, and the continuous multi-vortex chaotic oscillation signal is output through a reverse phase integral summation operation circuit, a reverse phase proportional operation circuit, a sine conversion operation circuit and a multiplier circuit. The resistance of the variable resistor in the three branches is changed to realize the change of the parameters, the flexibility of the hardware circuit control is increased, the difficulty of the circuit debugging is reduced, a circuit basis is provided for the application of the multi-vortex chaotic signal in the technical field of electronic communication and information engineering, and the research of the multi-vortex chaotic oscillator in the application field such as image encryption is facilitated. Two different types of hidden attractors can be obtained through the control of the parameters, and the advantage lies in that it is more difficult to numerically locate and calculate the hidden attractor system than the self-excited attractor because there is no systematic method to select the initial condition, so that the cracking difficulty of the encryption system based on the multi-vortex chaotic oscillator is improved.

Description

Technical Field

[0001] This invention belongs to the field of electronic, communication and information engineering technology, and relates to the field of oscillator technology, and in particular to a novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system. Background Technology

[0002] Chaos is a pseudo-random phenomenon generated by deterministic nonlinear systems, characterized by initial condition sensitivity, unpredictability, and broadband properties. Because simple nonlinear dynamical systems can exhibit complex dynamic behaviors, chaos has found wide application in information engineering fields such as information encryption, secure communication, and chaotic radar.

[0003] Chaotic signal generators are a prerequisite for chaotic applications, and their circuit implementation and the characteristics of the generated chaotic signals are the focus of research and the foundation for practical applications. Among them, multi-vortex chaotic oscillators generate multi-vortex attractors with more complex attraction domains and more random orbital motions, which greatly improves the security of encryption algorithms designed based on these characteristics. Multi-vortex chaotic oscillators have important applications in chaotic communication, information security, and parameter estimation, and have become a hot topic in chaos theory research.

[0004] Existing multi-vortex chaotic signal generators are complex in structure, sensitive to parameters, have poor robustness, are difficult to control, have a single type of chaotic attractor, and are limited in the number of vortices they generate. When the number of vortices generated by the system is large, the number of nonlinear circuit modules will also increase accordingly, and the increase in circuit components will lead to high power consumption. At the same time, the complexity of the output low-dimensional chaotic signal is not high, so the encryption system obtained by the chaotic signal generator is also easy to crack.

[0005] In summary, the design of a novel multi-vortex chaotic oscillator to avoid the problems of difficult integration and increased power consumption caused by the stacking of nonlinear circuit modules, while further increasing the complexity of the system, has attracted further attention from experts and scholars. Summary of the Invention

[0006] The technical problem solved by this invention is to address the issues of poor integration, increased power consumption, and system complexity caused by the stacking of nonlinear circuit modules in existing technologies. This invention proposes a novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system. Building upon the JERK chaotic system, a multi-vortex chaotic system with a hidden attractor is constructed using a step function. In this system, the number of vortices is directly proportional to the time T, thus avoiding the problems of poor integration and increased power consumption caused by the stacking of nonlinear circuit modules, while further increasing the system's complexity.

[0007] The technical solution of the present invention is: a novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system, comprising a first product branch, a second product branch and a third product branch;

[0008] The first integration branch includes an operational amplifier U1, a resistor R1, a capacitor C1, a gain amplifier G, and a sine conversion unit F; wherein the operational amplifier U1, resistor R1, and capacitor C1 form an inverting integration circuit, and the gain amplifier G and the sine conversion unit F form a sine conversion circuit; the signal -y is input to the input terminal of the inverting integration circuit, and the output terminal of the inverting integration circuit outputs the signal x; the signal x is input to the sine conversion circuit, and the final output signal is singx;

[0009] The second integration branch includes operational amplifier U2, operational amplifier U3, resistor R2, resistor R3, resistor R4, and capacitor C2; wherein operational amplifier U2, resistor R2, and capacitor C2 form an inverting integrating operational circuit, and operational amplifier U3, resistor R3, and resistor R4 form an inverting proportional operational circuit; signal -z is input to the input terminal of the inverting integrating operational circuit, and the output terminal of the inverting integrating operational circuit outputs signal y; signal y is input to the input terminal of the inverting proportional operational circuit, and the output terminal of the inverting proportional operational circuit finally outputs signal -y;

[0010] The third integration branch includes operational amplifier U4, operational amplifier U5, resistors R5, R6, R7, R8, R9, and R10, capacitor C3, multiplier A, and DC power supply V; wherein operational amplifier U4, resistors R5, R6, R7, R8, and capacitor C3 form an inverting integrator circuit, and operational amplifier U5, resistors R9, and R10 form an inverting proportional circuit; signals -x and -y are input to the input terminal of multiplier A, the output terminal of multiplier A, along with signals y and z and DC power supply V, is connected to the input terminal of the inverting integrator circuit, and the output terminal of the inverting integrator circuit outputs signal z; signal z is input to the input terminal of the inverting proportional circuit, and the output terminal of the inverting proportional circuit outputs signal -z.

[0011] Furthermore, in the first integration branch, the signal -y is connected to the inverting input terminal of the operational amplifier U1 via resistor R1, and the non-inverting input terminal of the amplifier U1 is grounded; the inverting input terminal of the amplifier U1 is connected to the output terminal of the amplifier U1 via capacitor C1, the output terminal of the amplifier U1 outputs signal x, the output terminal of the amplifier U1 is connected to the input terminal of the sine conversion unit F via gain amplifier G, and the output terminal of the sine conversion unit F outputs signal singx.

[0012] Furthermore, in the second integration branch, the signal -z is connected to the inverting input terminal of operational amplifier U2 via resistor R2, and the non-inverting input terminal of amplifier U2 is grounded; the inverting input terminal of amplifier U2 is connected to the output terminal of amplifier U2 via capacitor C2, and the output terminal of amplifier U2 outputs signal y; the output terminal of amplifier U2 is connected to the inverting input terminal of operational amplifier U3 via resistor R3, and the non-inverting input terminal of amplifier U3 is grounded; the inverting input terminal of amplifier U3 is connected to the output terminal of amplifier U3 via resistor R4, and the output terminal of amplifier U3 outputs signal -y.

[0013] Furthermore, in the third integration branch, signal y is connected to the inverting input terminal of operational amplifier U4 via resistor R5, signal z is connected to the inverting input terminal of amplifier U4 via resistor R6, and the positive terminal of DC power supply V is grounded; the negative terminal of DC power supply V is connected to the inverting input terminal of amplifier U4 via resistor R7, signals singx and y are respectively connected to the input terminals of multiplier A, and the output terminal of multiplier A is connected to the inverting input terminal of amplifier U4 via resistor R8; the non-inverting input terminal of amplifier U4 is grounded; the inverting input terminal of amplifier U4 is connected to the output terminal of amplifier U4 via capacitor C3, and the output terminal of amplifier U4 outputs signal z; the output terminal of amplifier U4 is connected to the inverting input terminal of operational amplifier U5 via resistor R9, and the non-inverting input terminal of amplifier U5 is grounded; the inverting input terminal of amplifier U5 is connected to the output terminal of amplifier U5 via resistor R10, and the output terminal of amplifier U5 outputs signal -z.

[0014] Furthermore, the capacitance values ​​of capacitors C1, C2, and C3 are all 10μF.

[0015] Furthermore, resistors R1, R5, R6, R7, and R8 are all variable resistors, and resistors R2, R3, R4, R9, and R10 all have a resistance of 100kΩ.

[0016] Furthermore, the gain coefficient g of the gain amplifier G is π.

[0017] Furthermore, the voltage value of the DC power supply V is 1V.

[0018] Invention Effects

[0019] The technical effects of this invention are as follows: This invention proposes a novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system. Through an inverted integral summation circuit, an inverted proportional circuit, a sine conversion circuit, and a multiplier circuit, it outputs a continuous multi-vortex chaotic oscillation signal. By changing the resistance value of the variable resistor in the three branches to achieve parameter changes, the flexibility of hardware circuit control is increased, the difficulty of circuit debugging is reduced, and a circuit foundation is provided for the application of multi-vortex chaotic signals in the fields of electronics, communication and information engineering. This facilitates the research of multi-vortex chaotic oscillators in application fields such as image encryption. Furthermore, by controlling the parameters, two different types of hidden attractors can be obtained. By setting d≠0 in formula (1), the hidden attractor when the system has no equilibrium point can be obtained; by setting d=0 in formula (1), the hidden attractor when the system has a linear equilibrium point can be obtained. The advantage of the hidden attractor over the self-excited attractor is that, since there is no systematic method to select the initial conditions, the numerical positioning and calculation of the hidden attractor system is more difficult than that of the self-excited attractor, thereby increasing the difficulty of cracking the encryption system based on this multi-vortex chaotic oscillator. Attached Figure Description

[0020] Figure 1 The circuit diagram of the present invention is shown below, wherein (a) is the circuit diagram of the first product branch, (b) is the circuit diagram of the second product branch, and (c) is the circuit diagram of the third product branch.

[0021] Figure 2 The waveform of the state signal x in the time domain is shown.

[0022] Figure 3 The waveform of the state signal y in the time domain is shown.

[0023] Figure 4 This is the attractor phase diagram for the xy plane. Detailed Implementation

[0024] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," and "counterclockwise," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0025] See Figures 1-4 The technical solution of the present invention is: a novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system, including a first product branch, a second product branch and a third product branch;

[0026] The first integration branch includes one input terminal for signal -y, which is connected to the input terminal of operational amplifier U1 through resistor R1. The output terminal of operational amplifier U1 passes through a sine conversion circuit composed of gain amplifier G and sine conversion unit F, and finally outputs signal -x. The second integration branch includes one input terminal for signal -z, which is connected to the input terminal of operational amplifier U2 through resistor R2. The output terminal of operational amplifier U2 is connected to the input terminal of operational amplifier U3, and finally outputs signal -y. The third integration branch includes five input terminals for signal y, signal z, DC power supply V, signal -x, and signal -y. Signal y is input to the input terminal of operational amplifier U4 through resistor R5, signal z is input to the input terminal of operational amplifier U4 through resistor R6, DC power supply V is connected to the input terminal of operational amplifier U4 through resistor R7, signal -x and signal -y are input to the input terminals of multiplier A, and after multiplier A, they are connected to the input terminal of operational amplifier U4 through resistor R8. The output terminal of operational amplifier U4 is connected to the input terminal of operational amplifier U5, and finally outputs signal -z.

[0027] The first integration branch includes operational amplifier U1, resistor R1, capacitor C1, gain amplifier G, and sine converter F. Signal -y is connected to the input of the inverting integrator circuit built from operational amplifier U1, resistor R1, and capacitor C1. The output of the inverting integrator circuit is signal x. Signal x is input to the sine converter circuit composed of gain amplifier G and sine converter F, and finally outputs the signal singx.

[0028] The second integration branch includes operational amplifier U2, operational amplifier U3, resistors R2, R3, and R4, and capacitor C2. Signal -z is connected to the input of the inverting integrator circuit built with operational amplifier U2, resistor R2, and capacitor C2; the output of the inverting integrator circuit is signal y. Signal y is input to the input of the inverting proportional circuit built with operational amplifier U3, resistors R3 and R4; the output of the inverting proportional circuit is signal -y.

[0029] The third integration branch includes operational amplifier U4, operational amplifier U5, resistors R5, R6, R7, R8, R9, and R10, capacitor C3, multiplier A, and DC power supply V. Signals -x and -y serve as inputs to multiplier A. The output of multiplier A is connected to signals y, z, and DC power supply V. The inverting integrator circuit, constructed using operational amplifier U4, resistors R5, R6, R7, R8, and capacitor C3, outputs signal z. Signal z is also input to the inverting proportional circuit, constructed using operational amplifier U5, resistors R9 and R10, which outputs signal -z.

[0030] The three product branches will be explained further below.

[0031] Its circuit diagram is as follows Figure 1 As shown, the first integration branch includes an operational amplifier U1, a resistor R1, a capacitor C1, a gain amplifier G, and a sine wave converter F. The signal -y is connected to the inverting input terminal of the operational amplifier U1 via the resistor R1. The non-inverting input terminal of the amplifier U1 is grounded. The inverting input terminal of the amplifier U1 is connected to the output terminal of the amplifier U1 via the capacitor C1. The output terminal of the amplifier U1 outputs the signal x. The output terminal of the amplifier U1 is connected to the input terminal of the sine wave converter F via the gain amplifier G. The output terminal of the sine wave converter F outputs the signal singx.

[0032] The second integration branch includes operational amplifier U2, operational amplifier U3, resistors R2, R3, and R4, and capacitor C2. The signal -z is connected to the inverting input terminal of operational amplifier U2 via resistor R2. The non-inverting input terminal of operational amplifier U2 is grounded. The inverting input terminal of operational amplifier U2 is connected to the output terminal of operational amplifier U2 via capacitor C2, and the output terminal of operational amplifier U2 outputs signal y. The output terminal of operational amplifier U2 is connected to the inverting input terminal of operational amplifier U3 via resistor R3. The non-inverting input terminal of operational amplifier U3 is grounded. The inverting input terminal of operational amplifier U3 is connected to the output terminal of operational amplifier U3 via resistor R4, and the output terminal of operational amplifier U3 outputs signal -y.

[0033] The third multiplication branch includes operational amplifier U4, operational amplifier U5, resistors R5, R6, R7, R8, R9, and R10, capacitor C3, multiplier A, and DC power supply V. Signal y is connected to the inverting input of operational amplifier U4 via resistor R5, signal z is connected to the inverting input of amplifier U4 via resistor R6, the positive terminal of DC power supply V is grounded, and the negative terminal of DC power supply V is connected to the inverting input of amplifier U4 via resistor R7. Signals x and y are respectively connected to the input terminals of multiplier A. The output terminal of amplifier A is connected to the inverting input terminal of amplifier U4 via resistor R8. The non-inverting input terminal of amplifier U4 is grounded. The inverting input terminal of amplifier U4 is connected to the output terminal of amplifier U4 via capacitor C3. The output terminal of amplifier U4 outputs signal z. The output terminal of amplifier U4 is connected to the inverting input terminal of operational amplifier U5 via resistor R9. The non-inverting input terminal of amplifier U5 is grounded. The inverting input terminal of amplifier U5 is connected to the output terminal of amplifier U5 via resistor R10. The output terminal of amplifier U5 outputs signal -z.

[0034] Preferably, the capacitance values ​​of capacitors C1, C2, and C3 are all 10μF.

[0035] Preferably, resistors R1, R5, R6, R7, and R8 are all variable resistors, and resistors R2, R3, R4, R9, and R10 all have a resistance of 100kΩ.

[0036] Preferably, the gain coefficient g of the gain amplifier G is π.

[0037] Preferably, the voltage value of the DC power supply V is 1V.

[0038] The dimensionless mathematical model of a novel three-dimensional hidden multi-vortex chaotic oscillator system based on the JERK system is as follows:

[0039]

[0040] Formula (1) can be implemented by three integral operation circuits. The circuit equations are consistent with the dynamic equations. The coefficients of each feedback term in the system are achieved by the combined setting of resistors and capacitors. The circuit equations corresponding to Formula (1) are:

[0041]

[0042] Where R1C1=1 / c, R2C2=1, R5C3=1 / a, R6C3=1 / a, R8C3=1 / b, R7C3=V / d.

[0043] In formula (1), x, y, z are system state variables, and parameters a, b, c, d are constants. When parameters a = 0.2, b = 2, c = 0.1, d = 0.1, and IC = (0, -1, -1), the system has a typical attractor. Let the capacitance values ​​of capacitors C1, C2, and C3 all be 10μF, the DC power supply V output a voltage of 0.1V, the resistance values ​​of resistors R1 and R7 all be 1000kΩ, the resistance values ​​of resistors R2, R3, R4, R9, and R10 all be 100kΩ, the resistance values ​​of resistors R5 and R6 all be 500kΩ, and the resistance value of resistor R8 be 50kΩ.

[0044] The time-domain waveform of the system output state signal x is shown below. Figure 2 As shown, the overall structure presents a stepped shape with a period of time T0, achieving the positioning of a single vortex center on the x-axis. The time-domain waveform of the state signal y is shown in the figure. Figure 3 As shown, the chaotic time-domain waveforms in the system all exhibit synchronization with the center of the vortex; the chaotic attractor output by the system is as follows: Figure 4 As shown, Figure 4The attractor phase diagram in the xy plane is presented, clearly demonstrating the multi-vortex characteristics of this system. (Summary) Figure 2 , Figure 3 , Figure 4 It can be seen that the number of vortices in this system is proportional to time. The state signal x completes the location of a new vortex center after each step time. That is, the longer the time, the more vortices there are.

[0045] The specific embodiments are only for illustrating the technical ideas of the present invention and should not be used to limit the scope of protection of the present invention. Any modifications made to the technical solutions based on the technical ideas proposed in the present invention shall fall within the scope of protection of the present invention.

Claims

1. A novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system, characterized in that, This includes the first product branch, the second product branch, and the third product branch; The first integration branch includes an operational amplifier U1, a resistor R1, a capacitor C1, a gain amplifier G, and a sine conversion unit F; wherein the operational amplifier U1, resistor R1, and capacitor C1 form an inverting integration circuit, and the gain amplifier G and the sine conversion unit F form a sine conversion circuit; the signal -y is input to the input terminal of the inverting integration circuit, and the output terminal of the inverting integration circuit outputs the signal x; the signal x is input to the sine conversion circuit, and the final output signal is singx; The second integration branch includes operational amplifier U2, operational amplifier U3, resistor R2, resistor R3, resistor R4, and capacitor C2; wherein operational amplifier U2, resistor R2, and capacitor C2 form an inverting integrating operational circuit, and operational amplifier U3, resistor R3, and resistor R4 form an inverting proportional operational circuit; signal -z is input to the input terminal of the inverting integrating operational circuit, and the output terminal of the inverting integrating operational circuit outputs signal y; signal y is input to the input terminal of the inverting proportional operational circuit, and the output terminal of the inverting proportional operational circuit finally outputs signal -y; The third integration branch includes operational amplifier U4, operational amplifier U5, resistors R5, R6, R7, R8, R9, and R10, capacitor C3, multiplier A, and DC power supply V; wherein operational amplifier U4, resistors R5, R6, R7, R8, and capacitor C3 form an inverting integrator circuit, and operational amplifier U5, resistors R9, and R10 form an inverting proportional circuit; signals -x and -y are input to the input terminal of multiplier A, the output terminal of multiplier A, along with signals y and z and DC power supply V, is connected to the input terminal of the inverting integrator circuit, and the output terminal of the inverting integrator circuit outputs signal z; signal z is input to the input terminal of the inverting proportional circuit, and the output terminal of the inverting proportional circuit outputs signal -z.

2. The novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system as described in claim 1, characterized in that, In the first integration branch, the signal -y is connected to the inverting input terminal of the operational amplifier U1 via resistor R1, and the non-inverting input terminal of the amplifier U1 is grounded; the inverting input terminal of the amplifier U1 is connected to the output terminal of the amplifier U1 via capacitor C1, and the output terminal of the amplifier U1 outputs the signal x; the output terminal of the amplifier U1 is connected to the input terminal of the sine conversion unit F via gain amplifier G, and the output terminal of the sine conversion unit F outputs the signal singx.

3. The novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system as described in claim 1, characterized in that, In the second integration branch, the signal -z is connected to the inverting input terminal of operational amplifier U2 via resistor R2, and the non-inverting input terminal of amplifier U2 is grounded; the inverting input terminal of amplifier U2 is connected to the output terminal of amplifier U2 via capacitor C2, and the output terminal of amplifier U2 outputs signal y; the output terminal of amplifier U2 is connected to the inverting input terminal of operational amplifier U3 via resistor R3, and the non-inverting input terminal of amplifier U3 is grounded; the inverting input terminal of amplifier U3 is connected to the output terminal of amplifier U3 via resistor R4, and the output terminal of amplifier U3 outputs signal -y.

4. A novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system as described in claim 1, characterized in that, In the third multiplication branch, signal y is connected to the inverting input of operational amplifier U4 via resistor R5, signal z is connected to the inverting input of amplifier U4 via resistor R6, and the positive terminal of DC power supply V is grounded; the negative terminal of DC power supply V is connected to the inverting input of amplifier U4 via resistor R7, signals singx and y are respectively connected to the input of multiplier A, and the output of multiplier A is connected to the inverting input of amplifier U4 via resistor R8; the non-inverting input of amplifier U4 is grounded; the inverting input of amplifier U4 is connected to the output of amplifier U4 via capacitor C3, and the output of amplifier U4 outputs signal z; the output of amplifier U4 is connected to the inverting input of operational amplifier U5 via resistor R9, and the non-inverting input of amplifier U5 is grounded; the inverting input of amplifier U5 is connected to the output of amplifier U5 via resistor R10, and the output of amplifier U5 outputs signal -z.

5. A novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system as described in claim 1, characterized in that, The capacitance values ​​of capacitors C1, C2, and C3 are all 10μF.

6. A novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system as described in claim 1, characterized in that, The resistors R1, R5, R6, R7, and R8 are all variable resistors, and the resistors R2, R3, R4, R9, and R10 all have a resistance of 100kΩ.

7. A novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system as described in claim 1, characterized in that, The gain coefficient g of the gain amplifier G is π.

8. A novel three-dimensional hidden multi-vortex chaotic oscillator based on the JERK system as described in claim 1, characterized in that, The voltage value of the DC power supply V is 1V.

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