CN105099657A

CN105099657A - Left-leaning three-dimensional single-scroll chaotic system and circuit

Info

Publication number: CN105099657A
Application number: CN201510551155.9A
Authority: CN
Inventors: 王晓红
Original assignee: Individual
Current assignee: Individual
Priority date: 2015-09-01
Filing date: 2015-09-01
Publication date: 2015-11-25

Abstract

This invention relates to a chaotic system and circuit, and specifically relates to a left-leaning three-dimensional single-scroll chaotic system and circuit. Typical chaotic systems, such as Lorenz system, Chen system and Lu system, are all have three balance points, wherein one of three balance points is a zero balance point, and the other two are non-zero balance points; this type of systems generally contain two nonlinear terms; and attractors generated by this type of systems are double-wing attractors. The invention provides a three-dimensional chaotic system with only one nonlinear term; the chaotic system is capable of generating single-scroll chaotic attractors; thus, the types of the chaotic systems are increased, and a new choice for applying the chaotic system to the engineering practice is provided.

Description

A kind of left-leaning three-dimensional single scroll chaotic system and circuit

Technical field

The present invention relates to a kind of chaos system and circuit, particularly the left-leaning three-dimensional single scroll chaotic system of one and circuit.

Background technology

Typical chaos system such 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, general containing two nonlinear terms in this type systematic, the attractor that this type systematic produces is double-wing chaotic attractor, the present invention proposes the three-dimensional chaotic system that only has nonlinear terms, this chaos system can produce the chaos attractor of single scrollwork, add the type of chaos system, provide a kind of selection newly 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 left-leaning three-dimensional single scrollwork continuous chaotic system and circuit, and the present invention adopts following technological means to realize goal of the invention:

1, a left-leaning three-dimensional single scroll chaotic system, is characterized in that:

\{\begin{matrix} \frac{d x}{d t} = a x + b y + c z \\ \frac{d y}{d t} = - x - (z - d) y \\ \frac{d z}{d t} = - e x + (y - d) z \end{matrix} - - - i

In i formula, a, b, c, d, e are parameter;

Work as a=-0.5, when b=5, c=-2.5, d=1, e=0.3, system i is:

\{\begin{matrix} \frac{d x}{d t} = - 0.5 x + 5 y - 2.5 z \\ \frac{d y}{d t} = - x - (z - 1) y \\ \frac{d z}{d t} = - 0.3 x + (y - 1) z \end{matrix} - - - i i

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

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

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

The 1st of described operational amplifier U1, 2 pins are unsettled, 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 R12, 7th pin meets output-x, connected with the 13rd pin of operational amplifier U1 by resistance R19, connected with the 13rd pin of operational amplifier U2 by resistance R4, connected with the 13rd pin of operational amplifier U3 by resistance R18, 8th pin connects and exports x, connected with the 6th pin of operational amplifier U1 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,

The 1st of described operational amplifier U2, 2 pins are unsettled, 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 R31, 7th pin meets output-y, connect the 3rd pin of multiplier U4, 8th pin connects and exports y, connect the 1st pin of multiplier U5, connected with the 13rd pin of operational amplifier U1 by resistance R2, connected with the 13rd pin of operational amplifier U2 by resistance R1, connected with the 6th pin of operational amplifier U2 by resistance R30, 9th pin is connected with the 8th pin by electric capacity C2, 13rd pin completes is crossed resistance R5 and is connected with the 14th pin, 14th pin is connected with the 9th pin by resistance R3,

The 1st of described operational amplifier U3, 2 pins are unsettled, 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 U1 by resistance R6, connected with the 13rd pin of operational amplifier U3 by resistance R8, 8th pin connects and exports z, connect the 1st pin of multiplier U4, connect the 3rd pin of multiplier U5, connected with the 6th pin of operational amplifier U3 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 U4, the 5th pin meets VEE, and the 7th pin is connected with operational amplifier U2 the 13rd pin by resistance R17, 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 is connected with the 13rd pin of operational amplifier U3 by resistance R7, and the 8th pin meets VCC.

Useful fruit of the present invention is: propose a kind of left-leaning three-dimensional single scroll 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 and U4 of the present invention.

Fig. 4 is the actual connection layout of circuit of U3 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.

\{\begin{matrix} \frac{d x}{d t} = a x + b y + c z \\ \frac{d y}{d t} = - x - (z - d) y \\ \frac{d z}{d t} = - e x + (y - d) z \end{matrix} - - - i

In i formula, a, b, c, d, e are parameter;

Work as a=-0.5, when b=5, c=-2.5, d=1, e=0.3, system i is:

\{\begin{matrix} \frac{d x}{d t} = - 0.5 x + 5 y - 2.5 z \\ \frac{d y}{d t} = - x - (z - 1) y \\ \frac{d z}{d t} = - 0.3 x + (y - 1) z \end{matrix} - - - i i

Resistance R3=R5=R9=R10=R11=R12=R13=R14=R15=R16=R30=R31=10k Ω in circuit, R7=R17=10K Ω, R1=R4=R8=100K Ω, R2=20K Ω, R6=40k Ω, R19=200K Ω, R18=333.3K 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

1. a left-leaning three-dimensional single scroll chaotic system, is characterized in that:

\{\begin{matrix} \frac{d x}{d t} = a x + b y + c z \\ \frac{d y}{d t} = - x - (z - d) y \\ \frac{d z}{d t} = - e x + (y - d) z \end{matrix} - - - i

In i formula, a, b, c, d, e are parameter;

Work as a=-0.5, when b=5, c=-2.5, d=1, e=0.3, system i is:

\{\begin{matrix} \frac{d x}{d t} = - 0.5 x + 5 y - 2.5 z \\ \frac{d y}{d t} = - x - (z - 1) y \\ \frac{d z}{d t} = - 0.3 x + (y - 1) z \end{matrix} - - - i i

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