CN105227291A - A kind of three-dimensional four-winged chaotic attractor continuous chaotic system and circuit - Google Patents

Abstract

The present invention relates to a kind of chaos system and circuit, in particular to the three-dimensional four-winged chaotic attractor continuous chaotic system of one and circuit, typical chaos system is as Lorenz system, Chen system and Lu system all have three balance points, one is wherein had to be zero balancing point, another two is non-zero balancing point, the attractor that this type systematic produces is double-wing chaotic attractor, by increasing method construct four non-zero balance points of saturation function or continuous function or more balance point on the basis of Typical Chaotic Systems, the chaos system with four wings or multiple wing attractor can be constructed, the present invention is not when increasing saturation function or continuous function, propose a kind of three-dimensional four-winged chaotic attractor continuous chaotic system, add the type of chaos system, a kind of selection is newly provided for chaos system is applied to engineering practice.

Description

A kind of three-dimensional four-winged chaotic attractor continuous chaotic system and circuit

Technical field

The present invention relates to a kind of chaos system and circuit, particularly the three-dimensional four-winged chaotic attractor continuous chaotic system of one and circuit.

Background technology

Typical chaos system is as Lorenz system, Chen system and Lu system all have three balance points, one is wherein had to be zero balancing point, another two is non-zero balancing point, the attractor that this type systematic produces is double-wing chaotic attractor, by increasing method construct four non-zero balance points of saturation function or continuous function or more balance point on the basis of Typical Chaotic Systems, the chaos system with four wings or multiple wing attractor can be constructed, the present invention is not when increasing saturation function or continuous function, propose a kind of three-dimensional four-winged chaotic attractor continuous chaotic system, add the type of chaos system, a kind of selection is newly provided for chaos system is applied to engineering practice.

Summary of the invention

The technical problem to be solved in the present invention is to provide a kind of three-dimensional four-winged chaotic attractor continuous chaotic system and circuit, and the present invention adopts following technological means to realize goal of the invention:

1, a three-dimensional four-winged chaotic attractor continuous chaotic system, is characterized in that:

$\left\{\begin{array}{c}\frac{dx}{dt}=ax+by+cz\\ \frac{dy}{dt}=-x-\left(f\right(z)-d)y\\ \frac{dz}{dt}=-ex+\left(f\right(y)-d)z\end{array}\right.---i$

In i formula, a, b, c, d, e are parameter, and f (z), f (y) are nonlinear function;

Work as a=-0.5, b=2, c=1, d=4, e=0.5, f (z)=z ², f (y)=y ²time system i be:

$\left\{\begin{array}{c}\frac{dx}{dt}=-0.5x+2y+z\\ \frac{dy}{dt}=-x-(z^2-4)y\\ \frac{dz}{dt}=-0.5x+\left(\right|y|-4)z\end{array}\right.---ii$

System ii is three-dimensional four-winged chaotic attractor continuous chaotic system.

2, a kind of three-dimensional four-winged chaotic attractor continuous chaotic system circuit, it is characterized in that: according to the Mathematical Modeling constructing analog circuit of ii, operational amplifier U1, operational amplifier U2 and resistance and electric capacity is utilized to form anti-phase adder and anti-phase fractional order integrator, multiplier U3, multiplier U4 and multiplier U5 is utilized to realize multiplying, described operational amplifier U1, operational amplifier U2 adopt LF347N, and described multiplier U3, multiplier U4 and multiplier U5 adopt AD633JN;

Described operational amplifier U1 concatenation operation amplifier U2, multiplier U4 and multiplier U5, described operational amplifier U2 concatenation operation amplifier U1, multiplier U3 and multiplier U5, described multiplier U3 concatenation operation amplifier U1, described multiplier U4 concatenation operation amplifier U2, described multiplier U5 connects multiplier U3 and multiplier U4;

1st pin of described operational amplifier U1 is connected with the 6th pin by resistance R3, 2nd pin is connected with the 1st pin by resistance R5, 3rd, 5, 10, 12 pin ground connection, 4th pin meets VCC, 11st pin meets VEE, 6th pin is connected with the 7th pin by electric capacity C2, 7th pin connects and exports y, connected with the 13rd pin of operational amplifier U1 by resistance R2, connected with the 2nd pin of operational amplifier U1 by resistance R1, connect the 3rd pin of multiplier U4, connect the 1st pin of multiplier U5, 8th pin connects and exports x, connected with the 2nd pin of operational amplifier U2 by resistance R11, 9th pin is connected with the 8th pin by electric capacity C1, 13rd pin is connected with the 14th pin by resistance R9, 14th pin is connected with 9 pins by resistance R10,

1st pin of described operational amplifier U2 is connected with the 2nd pin of operational amplifier U1 by resistance R4, connected with the 13rd pin of operational amplifier U2 by resistance R18, connected with the 13rd pin of operational amplifier U1 by resistance R19, 1st pin meets output-x, 2nd pin is connected with the 1st pin by resistance R12, 3rd, 5, 10, 12 pin ground connection, 4th pin meets VCC, 11st pin meets VEE, 6th pin is connected with the 7th pin by resistance R14, 7th pin meets output-z, connected with the 13rd pin of operational amplifier U2 by resistance R8, connect the 1st pin of multiplier U3, 8th pin connects and exports z, connect the 3rd pin of multiplier U5, connected with the 13rd pin of operational amplifier U1 by resistance R6, connected with the 6th pin of operational amplifier U2 by resistance R13, 9th pin is connected with the 8th pin by electric capacity C3, 13rd pin completes is crossed resistance R15 and is connected with the 14th pin, 14th pin is connected with the 9th pin by resistance R16,

The equal ground connection of 2nd, 4,6 pin of described multiplier U3, the 5th pin meets VEE, and the 7th pin connects operational amplifier U1 the 2nd pin by resistance R17, and the 8th pin meets VCC;

The equal ground connection of 2nd, 4,6 pin of described multiplier U4, the 5th pin meets VEE, and the 7th pin connects operational amplifier U2 the 13rd pin by resistance R7, and the 8th pin meets VCC;

The equal ground connection of 2nd, 4,6 pin of described multiplier U5, the 5th pin meets VEE, and the 7th pin connects the 3rd pin of multiplier U3, and connect the 1st pin of multiplier U4, the 8th pin meets VCC.

Useful fruit of the present invention is: propose a kind of three-dimensional four-winged chaotic attractor continuous chaotic system, add the type of chaos system, provides a kind of selection newly for chaos system is applied to engineering practice.

Accompanying drawing explanation

Fig. 1 is circuit structure block diagram of the present invention.

Fig. 2 is the actual connection layout of circuit of U1 of the present invention.

Fig. 3 is the actual connection layout of circuit of U2 of the present invention.

Fig. 4 is the actual connection layout of circuit of U3, U4 and U5 of the present invention.

Fig. 5 is the phasor in y-z direction of the present invention.

Embodiment

Below in conjunction with accompanying drawing and preferred embodiment, the present invention is further described in detail, see Fig. 1-Fig. 5.

1, a three-dimensional four-winged chaotic attractor continuous chaotic system, is characterized in that:

$\left\{\begin{array}{c}\frac{dx}{dt}=ax+by+cz\\ \frac{dy}{dt}=-x-\left(f\right(z)-d)y\\ \frac{dz}{dt}=-ex+\left(f\right(y)-d)z\end{array}\right.---i$

In i formula, a, b, c, d, e are parameter, and f (z), f (y) are nonlinear function;

Work as a=-0.5, b=2, c=1, d=4, e=0.5, f (z)=z ², f (y)=y ²time system i be:

$\left\{\begin{array}{c}\frac{dx}{dt}=-0.5x+2y+z\\ \frac{dy}{dt}=-x-(z^2-4)y\\ \frac{dz}{dt}=-0.5x+\left(\right|y|-4)z\end{array}\right.---ii$

System ii is three-dimensional four-winged chaotic attractor continuous chaotic system.

2, a kind of three-dimensional four-winged chaotic attractor continuous chaotic system circuit, it is characterized in that: according to the Mathematical Modeling constructing analog circuit of ii, operational amplifier U1, operational amplifier U2 and resistance and electric capacity is utilized to form anti-phase adder and anti-phase fractional order integrator, multiplier U3, multiplier U4 and multiplier U5 is utilized to realize multiplying, described operational amplifier U1, operational amplifier U2 adopt LF347N, and described multiplier U3, multiplier U4 and multiplier U5 adopt AD633JN;

Described operational amplifier U1 concatenation operation amplifier U2, multiplier U4 and multiplier U5, described operational amplifier U2 concatenation operation amplifier U1, multiplier U3 and multiplier U5, described multiplier U3 concatenation operation amplifier U1, described multiplier U4 concatenation operation amplifier U2, described multiplier U5 connects multiplier U3 and multiplier U4;

1st pin of described operational amplifier U1 is connected with the 6th pin by resistance R3, 2nd pin is connected with the 1st pin by resistance R5, 3rd, 5, 10, 12 pin ground connection, 4th pin meets VCC, 11st pin meets VEE, 6th pin is connected with the 7th pin by electric capacity C2, 7th pin connects and exports y, connected with the 13rd pin of operational amplifier U1 by resistance R2, connected with the 2nd pin of operational amplifier U1 by resistance R1, connect the 3rd pin of multiplier U4, connect the 1st pin of multiplier U5, 8th pin connects and exports x, connected with the 2nd pin of operational amplifier U2 by resistance R11, 9th pin is connected with the 8th pin by electric capacity C1, 13rd pin is connected with the 14th pin by resistance R9, 14th pin is connected with 9 pins by resistance R10,

1st pin of described operational amplifier U2 is connected with the 2nd pin of operational amplifier U1 by resistance R4, connected with the 13rd pin of operational amplifier U2 by resistance R18, connected with the 13rd pin of operational amplifier U1 by resistance R19, 1st pin meets output-x, 2nd pin is connected with the 1st pin by resistance R12, 3rd, 5, 10, 12 pin ground connection, 4th pin meets VCC, 11st pin meets VEE, 6th pin is connected with the 7th pin by resistance R14, 7th pin meets output-z, connected with the 13rd pin of operational amplifier U2 by resistance R8, connect the 1st pin of multiplier U3, 8th pin connects and exports z, connect the 3rd pin of multiplier U5, connected with the 13rd pin of operational amplifier U1 by resistance R6, connected with the 6th pin of operational amplifier U2 by resistance R13, 9th pin is connected with the 8th pin by electric capacity C3, 13rd pin completes is crossed resistance R15 and is connected with the 14th pin, 14th pin is connected with the 9th pin by resistance R16,

The equal ground connection of 2nd, 4,6 pin of described multiplier U3, the 5th pin meets VEE, and the 7th pin connects operational amplifier U1 the 2nd pin by resistance R17, and the 8th pin meets VCC;

The equal ground connection of 2nd, 4,6 pin of described multiplier U4, the 5th pin meets VEE, and the 7th pin connects operational amplifier U2 the 13rd pin by resistance R7, and the 8th pin meets VCC;

The equal ground connection of 2nd, 4,6 pin of described multiplier U5, the 5th pin meets VEE, and the 7th pin connects the 3rd pin of multiplier U3, and connect the 1st pin of multiplier U4, the 8th pin meets VCC.

Resistance R3=R5=R9=R10=R11=R12=R13=R14=R15=R16=10k Ω in circuit, R2=50K Ω, R7=R17=1K Ω, R4=R6=100k Ω, R1=R8=25K Ω, R18=R19=200K Ω, electric capacity C1=C2=C3=10nF in circuit.

Certainly, above-mentioned explanation is not limitation of the present invention, and the present invention is also not limited only to above-mentioned citing, and the change that those skilled in the art make in essential scope of the present invention, remodeling, interpolation or replacement, also belong to protection scope of the present invention.

Claims (2)

1. a three-dimensional four-winged chaotic attractor continuous chaotic system, is characterized in that:

$\left\{\begin{array}{c}\frac{dx}{dt}=ax+by+cz\\ \frac{dy}{dt}=-x-\left(f\right(z)-d)y\\ \frac{dz}{dt}=-ex+\left(f\right(y)-d)z\end{array}\right.---i$

In i formula, a, b, c, d, e are parameter, and f (z), f (y) are nonlinear function;

Work as a=-0.5, b=2, c=1, d=4, e=0.5, f (z)=z ², f (y)=y ²time system i be:

$\left\{\begin{array}{c}\frac{dx}{dt}=-0.5x+2y+z\\ \frac{dy}{dt}=-x-(z^2-4)y\\ \frac{dz}{dt}=-0.5x+\left(\right|y|-4)z\end{array}\right.---ii$

System ii is three-dimensional four-winged chaotic attractor continuous chaotic system.

2. a three-dimensional four-winged chaotic attractor continuous chaotic system circuit, it is characterized in that: according to the Mathematical Modeling constructing analog circuit of ii, operational amplifier U1, operational amplifier U2 and resistance and electric capacity is utilized to form anti-phase adder and anti-phase fractional order integrator, multiplier U3, multiplier U4 and multiplier U5 is utilized to realize multiplying, described operational amplifier U1, operational amplifier U2 adopt LF347N, and described multiplier U3, multiplier U4 and multiplier U5 adopt AD633JN;

Described operational amplifier U1 concatenation operation amplifier U2, multiplier U4 and multiplier U5, described operational amplifier U2 concatenation operation amplifier U1, multiplier U3 and multiplier U5, described multiplier U3 concatenation operation amplifier U1, described multiplier U4 concatenation operation amplifier U2, described multiplier U5 connects multiplier U3 and multiplier U4;

1st pin of described operational amplifier U1 is connected with the 6th pin by resistance R3, 2nd pin is connected with the 1st pin by resistance R5, 3rd, 5, 10, 12 pin ground connection, 4th pin meets VCC, 11st pin meets VEE, 6th pin is connected with the 7th pin by electric capacity C2, 7th pin connects and exports y, connected with the 13rd pin of operational amplifier U1 by resistance R2, connected with the 2nd pin of operational amplifier U1 by resistance R1, connect the 3rd pin of multiplier U4, connect the 1st pin of multiplier U5, 8th pin connects and exports x, connected with the 2nd pin of operational amplifier U2 by resistance R11, 9th pin is connected with the 8th pin by electric capacity C1, 13rd pin is connected with the 14th pin by resistance R9, 14th pin is connected with 9 pins by resistance R10,

1st pin of described operational amplifier U2 is connected with the 2nd pin of operational amplifier U1 by resistance R4, connected with the 13rd pin of operational amplifier U2 by resistance R18, connected with the 13rd pin of operational amplifier U1 by resistance R19, 1st pin meets output-x, 2nd pin is connected with the 1st pin by resistance R12, 3rd, 5, 10, 12 pin ground connection, 4th pin meets VCC, 11st pin meets VEE, 6th pin is connected with the 7th pin by resistance R14, 7th pin meets output-z, connected with the 13rd pin of operational amplifier U2 by resistance R8, connect the 1st pin of multiplier U3, 8th pin connects and exports z, connect the 3rd pin of multiplier U5, connected with the 13rd pin of operational amplifier U1 by resistance R6, connected with the 6th pin of operational amplifier U2 by resistance R13, 9th pin is connected with the 8th pin by electric capacity C3, 13rd pin completes is crossed resistance R15 and is connected with the 14th pin, 14th pin is connected with the 9th pin by resistance R16,

The equal ground connection of 2nd, 4,6 pin of described multiplier U3, the 5th pin meets VEE, and the 7th pin connects operational amplifier U1 the 2nd pin by resistance R17, and the 8th pin meets VCC;

The equal ground connection of 2nd, 4,6 pin of described multiplier U4, the 5th pin meets VEE, and the 7th pin connects operational amplifier U2 the 13rd pin by resistance R7, and the 8th pin meets VCC;

The equal ground connection of 2nd, 4,6 pin of described multiplier U5, the 5th pin meets VEE, and the 7th pin connects the 3rd pin of multiplier U3, and connect the 1st pin of multiplier U4, the 8th pin meets VCC.