CN106877997A - A three-dimensional chaotic system that can generate self-excited or hidden attractors - Google Patents

Abstract

The invention discloses a special three-dimensional chaotic system which is both conservative and dissipative. With the change of the control parameters, the system equilibrium point switches between no equilibrium point and two non-zero equilibrium points, forming an automatic Coexistence behavior that stimulates or hides multi-attractors. Using commercial discrete components, the hardware circuit corresponding to the system is designed. The circuit system is easy to numerical value, circuit simulation and experimental observation, and plays a greater role in promoting the application of chaotic systems in the engineering field.

Description

A three-dimensional chaotic system that can generate self-excited or hidden attractors

technical field

The invention realizes a special chaotic signal source capable of generating hidden or self-excited oscillation attractors.

Background technique

As a unique kinematic behavior in nonlinear dynamical systems, chaos has extreme sensitivity to the initial value and inherent randomness, which has attracted great attention from scholars in the fields of electronics, communication, information processing and other science or engineering; When Lorenz was studying atmospheric motion in the 1960s, when processing the collected atmospheric data, he unexpectedly found that the data changed infinitely and non-cyclically with time within a limited space range, that is, a strange attractor appeared. OE proposed A chaotic system, which is simpler than the Lorenz system and has a different topology than the Lorenz system. Since then, the research and excavation of the chaotic system composed of ordinary differential equations in the academic circle has never stopped. Chen Guanrong proposed the Chen system, and then Lu Jinhu proposed the Lü chaotic system. Since then, new types of chaotic systems have been produced continuously, including Liu chaotic system, super chaotic Lü system, Bao chaotic system and so on.

With the in-depth study of the chaotic system, the basic characteristics and dynamic behavior of the chaos are more and more understood, and the application of the chaotic system in the fields of science and engineering is more in-depth. Because the chaos is extremely sensitive to the initial conditions and parameters, it has excellent randomness. ; At present, it is more convenient to use discrete components such as operational amplifiers, resistors, and capacitors to build a chaotic signal generator, which can achieve purposeful control of chaos and enable chaos to be applied in more fields.

In recent years, a newly defined phenomenon—multi-stability, that is, in the case of constant circuit parameters, the initial state is different, and the system trajectory may be stable in different states such as point attractor, chaos, quasi-period, and cycle. This phenomenon is called multi-stability; applying multi-stability or super multi-stability chaotic system to chaotic secure communication can effectively improve the security performance of the system. Therefore, it is of great theoretical and practical significance to study the multi-stability of chaotic systems.

Contents of the invention

The technical problem to be solved by the present invention is to design a three-dimensional chaotic system that can generate self-excited or hidden attractors, and realize its hardware circuit.

In order to solve the above technical problems, the present invention provides a three-dimensional chaotic system capable of generating self-excited or hidden attractors, and a corresponding hardware circuit is designed, the structure of which is as follows:

The main circuit includes: integration channel 1, integration channel 2 and integration channel 3; integration channel 1 has 5 input terminals, respectively 1 "-v _y ", 2 "-v _x ", 1 "v _y ” and 1 “v _z ”, output “v _x ” after passing through the multiplier and integrator, and finally output “–v _x ” through a first-stage inverter; the integration channel 2 has 4 input terminals, which are respectively “– V ₁ ”, “v _x ”, “–v _y ” and “v _z ”, output “v _y ” after passing through the multiplier and integrator, and then output “–v _y ” after passing through a first-stage inverter; integration channel There are 3 input terminals "–v _x ", "v _x " and "v _y ", and output "v _z " after passing through the multiplier and integrator; operational amplifiers U ₁ , U ₂ , U ₃ , U ₄ and U The non-inverting input terminal of ₅ is connected to "ground", and the "-V ₁ " terminal provides "-1V" DC voltage.

In integration channel 1, the input terminal “–v _y ” is connected in series with a “20kΩ” resistor to the inverting input terminal of the operational amplifier U ₁ ; the input terminals “–v _x ” and “ _{vy y} ” are multiplied by the multiplier M ₁ The rear series resistance R _a is connected to the inverting input terminal of the operational amplifier U ₁ ; the input terminals “–v _x ” and “v _z ” are multiplied by the multiplier M ₂ and then a “20kΩ” resistor is connected in series to the operational amplifier U ₁ The inverting input terminal of U ₁ ; the capacitor C ₁ is connected in parallel between the inverting input terminal and the output terminal of U 1, at this time, the output terminal of U ₁ outputs "v _x "; the output terminal of U ₁ and the inverting input terminal of the operational amplifier U ₂ A "10kΩ" resistor is connected in series between the terminals; a "10kΩ" resistor is connected in parallel between the inverting input terminal and the output terminal of U ₂ , and the output terminal of U ₂ outputs "–v _x " at this time; the operational amplifier U ₁ and The noninverting input terminals of U ₂ are all connected to "ground".

In the integration channel 2, the input terminal “–V ₁ ” series resistance R _b is connected to the inverting input terminal of the operational amplifier U ₃ ; the input terminal “v _x ” is multiplied by the multiplier M ₃ for square operation, and the series resistance R _c is connected to At the inverting input terminal of the operational amplifier U ₃ ; the input terminals “–v _y ” and “v _z ” are multiplied by the multiplier M ₄ and connected in series with a “20kΩ” resistor to the inverting input terminal of the operational amplifier U ₃ ; A capacitor C ₂ is connected in parallel between the inverting input terminal and the output terminal of U ₃ , at this time, the output terminal of U ₃ outputs " _vy "; a "vy" is connected in series between the output terminal of U ₃ and the inverting input terminal of the operational amplifier U ₄ 10kΩ” resistor; a “10kΩ” resistor is connected in parallel between the inverting input terminal and output terminal of U ₄ , and the output terminal of U ₄ outputs “–v _y ” at this time; the non-inverting input terminals of operational amplifiers U ₃ and U ₄ All connected to "ground".

In the integration channel three, the input terminal “–v _x ” is connected in series with a “20kΩ” resistor to the inverting input terminal of the operational amplifier U ₅ ; the input terminal “v _x ” is multiplied by the multiplier M ₅ for square operation and then connected in series with a The "20kΩ" resistor is connected to the inverting input terminal of the operational amplifier _U5 ; the input terminal " _vy " is multiplied by the multiplier _M6 for square operation, and then a "20kΩ" resistor is connected in series to the inverting input terminal of the operational amplifier _U5 The input terminal; a capacitor C ₃ is connected in parallel between the inverting input terminal and the output terminal of U ₅ , at this time, the output terminal of U ₅ outputs "v _z "; the non-inverting input terminal of the operational amplifier U ₅ is connected to "ground".

The main circuit corresponding to the three-dimensional chaotic system that can produce self-excited or hidden attractors is shown in Figure 1. The system equation contains three state variables x, y, and z; the corresponding circuit state equation contains three state variables v _x , v _y , and v _z .

The beneficial effects of the present invention are as follows: a three-dimensional chaotic system capable of generating self-excited or hidden attractors is proposed, and its hardware circuit is designed to realize a hidden or self-excited oscillation chaotic signal source. The system is simple in structure, easy for theoretical analysis and circuit integration, and has great engineering application value.

Description of drawings

In order to make the content of the present invention more easily understood clearly, the present invention will be described in further detail below according to specific embodiments in conjunction with the accompanying drawings:

Fig. 1 is a hardware implementation circuit of a three-dimensional chaotic system that can generate self-excited or hidden attractors;

Figure 2 selects the initial condition (0,0,0) in the numerical simulation phase orbit diagram and experimental verification results of the v _x -v _y plane;

Figure 3 selects the initial condition (1,0,1) in the numerical simulation phase orbit diagram and experimental verification results of the v _x -v _y plane;

Figure 4 selects the initial condition (2,0,0) in the numerical simulation phase orbit diagram and experimental verification results of the v _x -v _y plane;

detailed description

Mathematical modeling: The construction of a three-dimensional autonomous oscillation circuit capable of generating self-excited or hidden attractors in this embodiment is shown in Figure 1 . First of all, the present invention is based on a three-dimensional chaotic system, which can be described by the following dimensionless state equation:

Among them, x, y, z are 3 state variables, a, b, c are 3 newly introduced control parameters and they are all normal numbers.

Let the left side of the equation of system (1) be equal to zero, we have

can be transformed into

x[(ca)x ² +(a-1)x+1-b]＝0 (3)

It can be verified that x=0 is not a solution of equation (2), that is, system (1) does not have a non-zero equilibrium point. When c=2 is fixed, then discuss the solution of formula (2) according to the following five situations, and determine the equilibrium point of system (1).

Case 1: a=2, b=1. Equation (2) has no solution, that is, system (1) has no equilibrium point.

Case 2: a=2, 1<b<2. Equation (2) has 2 solutions, so it can be analyzed that the 2 non-zero equilibrium points of system (1) are

Case three: a=2, b≤1 or b≥2. Equation (2) has no solution, and system (1) has no equilibrium point.

Situation 4: 1<a<1.5, b=1. Equation (2) has two solutions, and the two non-zero equilibrium points of system (1) can be obtained analytically as

Case five: a≤1 or a≥1.5, b=1. Equation (2) has no solution, and system (1) has no equilibrium point.

In summary, it can be found that the system (1) switches between no equilibrium point and two non-zero equilibrium points when the parameters change, that is, the system can generate self-excited or hidden attractors.

Numerical simulation: Using the MATLAB simulation software platform, the numerical simulation analysis of the system described by formula (4) can be carried out. The Runge-Kutta (ODE23) algorithm is selected to solve the system equations, and the phase orbit diagram of the state variables of the chaotic system can be obtained. The typical parameters a=2, b=1, c=2 are selected, and the MATLAB numerical simulation phase orbit diagrams corresponding to different initial values are shown in Fig. 2(a), Fig. 3(a) and Fig. 4(a) respectively.

Experimental verification: This design uses an operational amplifier model AD711KN, and provides ±15V operating voltage. Wherein, v _x , v _y , and v _z respectively represent three capacitive voltage state variables, C=C ₁ =C ₂ =C ₃ =0.1 μF, R _a =10 kΩ, R _b =20 kΩ, and R _c =10 kΩ. The resistor is a precision adjustable resistor, and the capacitor is a monolithic capacitor. Theoretical analysis and numerical simulation show that the chaotic attractor generated by this circuit is more sensitive to the initial state, and it is easy to realize the initial value of the required state variable by continuously turning on and off the power supply. The Tektronix DPO3034 digital storage oscilloscope was used to capture the measurement waveform, and the phase track diagram of the chaotic attractor in the numerical simulation was verified experimentally. The experimental results are shown in Fig. 2(b), Fig. 3(b) and Fig. 4(b). Show.

The comparison results can show that the nonlinear phenomenon observed in the experimental circuit is completely consistent with the simulation results, which can verify the correctness of the theoretical analysis and numerical simulation. Therefore, a kind of three-dimensional autonomous system hardware circuit that can generate self-excited or hidden attractors constructed by the present invention has scientific theoretical basis and physical realizability, and can play a positive role in promoting the engineering application of chaotic system circuits .

The above-mentioned embodiments are only examples for clearly illustrating the present invention, rather than limiting the implementation of the present invention. For those of ordinary skill in the art, other changes or changes in different forms can be made on the basis of the above description. It is not necessary and impossible to exhaustively list all the implementation manners here.

Claims (3)

1. it is a kind of may result from swashing or hiding attractor three-dimensional chaotic system, it is characterised in that：Circuit includes that three integrations are logical Road, with the change of circuit parameter, the circuit system equalization point is formed without being changed between equalization point and 2 non-zero equalization points Self-excitation or hiding multi attractor coexist behavior.

2. it is according to claim 1 it is a kind of may result from swashing or hiding attractor three-dimensional chaotic system, it is characterised in that： The three-dimensional chaotic system includes integrating channel one, integrating channel two and integrating channel three；Integrating channel one has 5 inputs, Respectively 1 "-v_y", 2 "-v_x", 1 " v_y" and 1 " v_z", by exporting " v after multiplier and integrator_x", then by one Level phase inverter final output "-v_x”；Integrating channel two has 4 inputs, respectively "-V₁”、“v_x”、“–v_y" and " v_z", by multiplying " v is exported after musical instruments used in a Buddhist or Taoist mass and integrator_y", then by one-level phase inverter final output "-v_y”；Integrating channel three have 3 inputs "- v_x”、“v_x" and " v_y", by exporting " v after multiplier and integrator_z”；Operational amplifier U₁、U₂、U₃、U₄And U₅Homophase input Termination " ", "-V₁" end offer " -1V " DC voltage.

3. it is according to claim 1 and 2 it is a kind of may result from swashing or hiding attractor three-dimensional chaotic system, its feature exists In system equation contains three state variables x, y and z；Corresponding circuits state equation contains three state variable v_x、v_yAnd v_z。