CN105406959A - Improved Chua's system of three-scroll attractor capable of generating one self-excited scroll and two hidden scrolls simultaneously - Google Patents

Abstract

The invention discloses an improved Chua's system of a three-scroll attractor capable of generating one self-excited scroll and two hidden scrolls simultaneously. The system comprises three parts, namely, a fundamental circuit a including three integral channels, a piecewise linear function h(x) realization circuit b and a control function h mu (x,z) realization circuit c. Identical ports of the circuit a and the circuit b are sequentially connected so as to generate a two-scroll hidden Chua's chaotic attractor; and identical ports of the circuit a and the circuit b are sequentially connected with identical ports of the circuit c so as to generate a novel three-scroll Chua's chaotic attractor. What's different is that the three-scroll attractor generated by the system includes one self-excited scroll attractor and a two-hidden scroll attractor. The system is easy to carry out numerical simulation and circuit simulation, is easy to perform experimental observation, and greatly promotes development of the multi-scroll chaotic system.

Description

A kind of modified model Cai Shi system that simultaneously can produce 3 scrollwork attractors of 1 self-excitation scrollwork and 2 hiding scrollworks

Technical field

The present invention relates to a kind of modified model Cai Shi system that simultaneously can produce 3 scrollwork attractors of 1 self-excitation scrollwork and 2 hiding scrollworks, this system is based on hiding two scrollwork Cai Shi system, the piece wire characteristic of piecewise linear function is changed by introducing a control function, and then produce the more complicated non-linear phenomena such as self-excitation scrollwork and the simultaneous multi-scroll attractor of hiding scrollwork, achieve a kind of generation simultaneously containing self-oscillation and the chaos signal source hiding vibration.This chaos system structure is simple, is one of simple chaos system that can produce complicated chaotic signal at present, and very easily in physics realization.

Background technology

Famous Chua's chaotic circuit is the autonomous electronic circuit that first catches by experiment, numerical analysis and theoretical proof can produce chaos phenomenon, has become the model of research chaos circuit or chaos system.The Mathematical Modeling of cai's circuit is represented by a differential equation group usually, for ease of analyzing, can carry out dimensionless process, and then obtain a simple Cai Shi system of Algebraic Structure to equation group.In in the past more than 30 year, a large amount of documents is studied the implementation of cai's circuit, theory analysis, numerical analysis and experiment measuring etc.

Classical cai's circuit can produce typical double scroll chaos, and in order to produce multi-scroll attractor, by constructing various nonlinear function, the Saddle focus equilibrium point expanding index 2 in former pair of scrollwork system can produce multiple scrolls.The Saddle focus equilibrium point of extended counter 2 can obtain different multiple scrolls in different directions, as on a direction, both direction or three directions, the Saddle focus equilibrium point of extended counter 2 can obtain one direction distribution multi-scroll chaotic system, two directional spreding plane net trellis multi-scroll chaotic systems and the stereo grid shaped multi-scroll chaotic system of three directional spreding etc. respectively.

On the basis of two scrollwork Cai Shi system, introduce the nonlinear characteristic that control function changes nonlinear resistance, three scroll chaotic attractors can be produced.Based on this, the present invention first proposes the simple two scrollwork Cai Shi system of a kind of structure, this system can produce the two scrollwork of chaos and hide attractor, the nonlinear characteristic of nonlinear resistance is changed by introducing control function, and then the 3 scroll chaotic attractors produced containing 1 self-excitation attractor and 2 hiding attractors, construct a kind of new chaotic signal generator.

Summary of the invention

Technical problem to be solved by this invention is a kind of modified model Cai Shi system that simultaneously can produce 3 scrollwork attractors of 1 self-excitation scrollwork and 2 hiding scrollworks.

For solving the problems of the technologies described above, the invention provides a kind of modified model Cai Shi system that simultaneously can produce 3 scrollwork attractors of 1 self-excitation scrollwork and 2 hiding scrollworks, its structure is as follows:

Described main circuit comprises: containing the basic circuit of three integrating channel as Suo Shi Fig. 1 (a), piecewise linear function h (x) realizing circuit as Suo Shi Fig. 1 (b), control function h _μ(x, z) realizing circuit is as shown in Fig. 1 (c).Three channels configuration are consistent, are made up of anti-phase adder Module, integrator module and inverter modules cascade; The node of three all identical marks of passage is connected, as: the output node " x " of passage 1 is connected with the input " x " of passage 2.In Fig. 1 (a) circuit, the output node " x " of passage 1 is connected with the input " x " of Fig. 1 (b) circuit; The output “ – h of Fig. 1 (b) circuit " with one of them input “ – h of multiplier M in Fig. 1 (c) circuit " be connected.In Fig. 1 (a) circuit, the output node " z " of passage 3 is connected with the input " z " of Fig. 1 (c) circuit, the output node “ – z of passage 3 in Fig. 1 (a) circuit " with the input “ – z of Fig. 1 (c) circuit " be connected; In Fig. 1 (c) circuit, the in-phase input end of operational amplifier mark " μ " connects 0.4V direct voltage, and the 3rd road Shu enters Duan Jie – 1V direct voltage; The output “ – h of Fig. 1 (c) circuit _μ" with the input node “ – h of passage 1 in Fig. 1 (a) circuit _μ" connect.Operational amplifier U ₁, U ₂, U ₃, U ₄, U ₅, U ₆, U ₇, U ₈, U ₉, U _h1, U _h2and U _h5in-phase input end connect " "; Divider resistance R in Fig. 1 (b) _h4all connect for " 1k Ω " divider resistance with two resistance values in Fig. 1 (c) " ".

Basic circuit containing three integrating channel comprises: passage 1, passage 2 and passage 3; Passage 1 has two input “ – h _μ" and " y ", by exporting " x " after anti-phase adder sum-product intergrator, more finally export “ – x through one-level inverter "; Passage 2 has three input " x ", “ – y " and " z ", by exporting " y " after anti-phase ratio adder sum-product intergrator, more finally export “ – y through one-level inverter "; Passage 3 only has an input “ – y ", by exporting " z " after inverter sum-product intergrator, more finally export “ – z through one-level inverter "; Operational amplifier U ₁, U ₂, U ₃, U ₄, U ₅, U ₆, U ₇, U ₈and U ₉in-phase input end connect " ".

In passage 1, two input “ – h _μ" and " y " each series connection one " 10k Ω " resistance after be connected to operational amplifier U ₁inverting input, U ₁inverting input and output between the resistance of in parallel one " 58k Ω ", U ₁output and resistance R ₀left end connect, resistance R ₀right-hand member concatenation operation amplifier U ₂inverting input, U ₂inverting input and output between an electric capacity C in parallel ₀, now U ₂output export " x ", U ₂output and operational amplifier U ₃inverting input between series connection one " 10k Ω " resistance, operational amplifier U ₃inverting input and output between the resistance of in parallel one " 10k Ω ", now U ₃output export “ – x "; Operational amplifier U ₁, U ₂and U ₃in-phase input end all connect " ".

In passage 2, three input " x ", “ – y " and " z " respectively series connection one be connected to operational amplifier U after the resistance of " 10k Ω " ₄inverting input, U ₄inverting input and output between the resistance of in parallel one " 10k Ω ", U ₁output and resistance R ₀left end connect, resistance R ₀right-hand member concatenation operation amplifier U ₅inverting input, U ₅inverting input and output between an electric capacity C in parallel ₀, now U ₅output export " y ", U ₅output and operational amplifier U ₆inverting input between series connection one " 10k Ω " resistance, operational amplifier U ₆inverting input and output between the resistance of in parallel one " 10k Ω ", now U ₆output export “ – y "; Operational amplifier U ₄, U ₅and U ₆in-phase input end all connect " ".

In passage 3, input “ – y " connect one " 10k Ω " resistance after be connected to operational amplifier U ₇inverting input, U ₇inverting input and output between the resistance of in parallel one " 58k Ω ", U ₇output and resistance R ₀left end connect, resistance R ₀right-hand member concatenation operation amplifier U ₈inverting input, U ₈inverting input and output between an electric capacity C in parallel ₀, now U ₈output export " z ", U ₈output and operational amplifier U ₉inverting input between series connection one " 10k Ω " resistance, operational amplifier U ₉inverting input and output between the resistance of in parallel one " 10k Ω ", now U ₉output export “ – z "; Operational amplifier U ₇, U ₈and U ₉in-phase input end all connect " ".

Piecewise linear function h (x) realizing circuit comprises: ABS function realizing circuit and the reverse adder circuit composition of one-level in dotted line frame shown in Fig. 1 (b).Concrete connected mode is: the lower road series resistance R of input " x " _h1after meet operational amplifier U _h1inverting input, U _h1inverting input and output between parallel resistance R _h2, U _h1output connecting resistance R _h3left end, R _h3right-hand member through divider resistance R _h4dividing potential drop Hou Shu Chu – 0.56 (| x+1| – | x – 1|), and as an input of reverse adder, divider resistance R _h4another termination " "; Operational amplifier U is connected to after the resistance of series connection one " the 200k Ω " that set out on a journey of the input " x " of Fig. 1 (b) circuit _h2inverting input, same resistance R _h3right-hand member connect one " 204k Ω " resistance after be connected to operational amplifier U _h2inverting input, U _h2inverting input and output between the resistance of in parallel one " 204k Ω ", operational amplifier U _h1and U _h2in-phase input end all connect " ", now U _h2output export “ – h ".

Control function h _μ(x, z) realizing circuit comprises: sign function circuit and a multiplier circuit.Concrete connected mode is: main circuit has five inputs and is respectively " z ", “ – z ", " μ ", “ – 1V " with “ – h ", an and output “ – h _μ".Operational amplifier U _h3and U _h4in-phase input end all connect " μ ", μ=0.4V." z " holds concatenation operation amplifier U _h3inverting input, U _h3output to connect the resistance of one " 12.5k Ω ", right-hand member Shu Chu – sgn (z – μ) Hou the divider resistance dividing potential drop of " 1k Ω " of this resistance, as an input of anti-phase adder, resistance be the divider resistance of " 1k Ω " another termination " ".In like manner ， “ – z " hold concatenation operation amplifier U _h4inverting input, U _h4output to connect the resistance of one " 12.5k Ω ", the right-hand member of this resistance exports sgn (z – μ) after the divider resistance dividing potential drop of " 1k Ω ", as an input of anti-phase adder, resistance be the divider resistance of " 1k Ω " another termination " ".Three input “ – sgn (z – μ) of anti-phase adder ", " sgn (z – μ) " with “ – 1V " resistance of connecting respectively is concatenation operation amplifier U after the resistance of " 200k Ω " _h5inverting input, U _h5inverting input and output between a resistance in parallel be the resistance of " 200k Ω ", U _h5in-phase input end connect " "; U _h5output connect the input of multiplier M, another input of M is “ – h "; Now, multiplier exports “ – h _μ".

As shown in Figure 1, wherein x, y and z are three state variables of corresponding system to described modified model Cai Shi system main circuit, and have following relation:

Beneficial effect of the present invention is as follows: propose a kind of modified model Cai Shi system that simultaneously can produce 3 scrollwork attractors of 1 self-excitation scrollwork and 2 hiding scrollworks.First two scrollwork Cai Shi systems that can produce hiding double scroll chaos are constructed, the piece wire characteristic of piecewise linear function is changed by introducing a control function, and then produce simultaneously more complicated containing the multi scroll chaotic attactors etc. of self-excitation scrollwork and hiding scrollwork non-linear phenomena, achieve a kind of generation simultaneously containing self-oscillation and the chaos signal source hiding vibration.This Cai Shi chaos system structure is simple, is easy to theory analysis and circuit realiration, and the chaotic signal produced is in there being complicated dynamics.

Accompanying drawing explanation

In order to make content of the present invention be more likely to be clearly understood, below according to specific embodiments also by reference to the accompanying drawings, the present invention is further detailed explanation, wherein:

Fig. 1 is modified model three scrollwork Cai Shi system realizing circuit;

Fig. 2 is the piece wire characteristic curve of nonlinear function;

Fig. 3 is for hiding two scrollwork Cai Shi chaos attractor x-y plane numerical simulation phase rail figure and experimental result;

Fig. 4 is for hiding two scrollwork Cai Shi chaos attractor x-z plane numerical value simulation rail figure and experimental result;

Fig. 5 is novel three scrollwork Cai Shi chaos attractor x-y plane numerical simulation phase rail figure and experimental results;

Fig. 6 is novel three scrollwork Cai Shi chaos attractor x-z plane numerical value simulation rail figure and experimental results;

Fig. 7 is novel three scrollwork Cai Shi chaos attractor y-z plane numerical value simulation rail figure and experimental results;

Embodiment

Mathematical modeling: a kind of modified model Cai Shi circuit system that simultaneously can produce 3 scrollwork attractors of 1 self-excitation scrollwork and 2 hiding scrollworks of the present embodiment builds as shown in Figure 1.First, the present invention constructs one can produce hiding Double Scroll Cai Shi system, and its circuit structure diagram is as shown in Fig. 1 a, 1b.In classical Cai Shi system, nonlinear characteristic is the key producing chaos phenomenon, can be described by three sections of linear functions f (x), wherein a, b are respectively the slope of inner segment and outer section of straight line, its Nonlinear Characteristic Curve as shown in Figure 2 a, wherein a<-1, b<0.The present invention improve three sections of linear functions f (x) Nonlinear Characteristic Curve as shown in Figure 2 b, wherein a<-1, b>0.

Classical Cai Shi system can be described by following dimensionless state equation:

Wherein, α, β and γ are constant, and usual γ is less, can be approximately 0; F (x) is used for three sections of linear characters of description formula (1) system:

f(x)＝bx+0.5(a-b)(|x+1|-|x-1|)(2)

After improving three sections of linear functions f (x), the Cai Shi system that formula (1) describes has three balance points determined:

The Jacobian matrix at balance point place is:

Wherein, zero balancing point P ₀corresponding c=a, two non-zero balance point P _±corresponding c=b.The characteristic equation at the corresponding balance point place of matrix is:

P(λ)＝λ ³+a ₁λ ²+a ₂λ+a ₃＝0(5)

Wherein a ₁=α (c+1)+1, a ₂=α c+ β, a ₃=α β (c+1).Louth-Hull dimension thatch the condition of above-mentioned cubic polynomial is as follows:

a ₁＝α(c+1)+1＞0

a ₃＝αβ(c+1)＞0(6)

a ₁a ₂-a ₃＝αc[α(c+1)+1]+β＞0

Can obtain, P ₀it is a unstable saddle point; For P _±, during c=b, because b>0, formula (6) perseverance is set up, and shows P _±always stable, the attractor that the Cai Shi system therefore improved produces is all hiding.Accordingly, the characteristic value at formula (3) three balance point places can be calculated as:

P ₀:λ ₁＝1.4398,λ _2,3＝-0.9299±j1.2779(7a)

P _±:λ _1,2＝-0.0669±j2.3246,λ ₃＝-6.7821(7b)

P ₀have a positive real root and two real parts to be that the conjugate complex of negative is followed, this shows P ₀it is a unstable saddle point; P _±have a negative real root and two real parts to be that the conjugate complex of negative is followed, this shows P _±that two stable saddles are burnt.Two scrollwork Cai Shi chaos attractor is hiding as can be seen here.

The nonlinear function that definition improves is h (x):

$\begin{array}{c}h\left(x\right)=x+f\left(x\right)\\ =(b+1)x+0.5(a-b)\left(\right|x+1|-|x-1\left|\right)\end{array}---\left(8\right)$

The equivalent form of value of the Cai Shi system then described in formula (1) can be expressed as:

Wherein, h (x) is still the piecewise linear function of three sections.

From formula 7 (a), the zero balancing point P of formula (9) system ₀a unstable saddle point, in order to make P ₀change a unstable saddle into burnt, i.e. the Saddle focus equilibrium point of index 2, introduce the piecewise linearity control function that contains x and z Two Variables, can be described below:

Wherein, controling parameters μ meets 0< μ <1.Therefore, control function h is used _μ(x, z) replaces h (x), can obtain following modified model Cai Shi system:

Wherein α >0, β >0.

Equally, three balance points that the modified model Cai Shi system that formula (11) system describes has formula (3) to provide.Consider control function h _μ(x, z) meets 0< μ <1, zero balancing point P ₀be positioned at the central area of x-z plane, two non-zero balance point P _±be positioned at symmetrical region outside x-z planar central, thus P ₀the stability of point will change, and P _±the stability of point will remain unchanged.

Zero balancing point P ₀corresponding Jacobian matrix is:

Zero balancing point P ₀the characteristic equation at place is:

P(λ)＝λ ³+b ₁λ ²+b ₂λ+b ₃＝0(13)

Wherein, b ₁=1-α (a+1), b ₂=β-α (a+2), b ₃=-α β (a+1).The coefficient of formula (13) cubic polynomial is all non-vanishing, and its Louth-Hull dimension thatch condition is as follows:

b ₁＝1-α(a+1)＞0

b ₃＝-αβ(a+1)＞0(14)

b ₁b ₂-b ₃＝α(a+2)(αa+α-1)+β＞0

Wherein, a and b in h (x) meets a<-1, b>0, then a+1<0.Therefore, the first two condition b ₁>0 and b ₃>0 sets up, but, as β <-α (a+2) (α a+ α-1), the 3rd condition b ₁b ₂-b ₃>0 is false, such as, when modified model Cai Shi system parameters is elected as:

α＝5.8，β＝5.8，a＝-1.1，b＝0.02，μ＝0.4(15)

3rd condition of formula (14) does not meet, so P ₀be that a unstable saddle is burnt.

Two balance point P _±the Jacobian matrix at place can be calculated as:

For P _±, formula (17) is identical with formula (4), therefore, and P _±always stable.

According to the system parameters that formula (15) is given, three balance point place characteristic of correspondence values are:

P ₀:λ _1,2＝0.2496±j1.2472,λ ₃＝-2.0792(17a)

P _±:λ _1,2＝-0.0635±j2.2472,λ ₃＝-6.7891(17b)

Can obtain, P ₀character pair root has a negative real root and two real parts to be positive Conjugate complex roots, so P ₀that a unstable saddle is burnt; P _±character pair root has a negative real root and two real parts to be negative Conjugate complex roots, so P _±that two stable saddles are burnt.Specifically, because this attractor has burnt and two stable non-zero saddle Jiao of unstable zero saddle, its center is a self-excitation attractor scrollwork, and outside is two symmetrical hiding attractor scrollworks.

Numerical simulation: modified model Cai Shi system according to Fig. 1, utilizes MATLAB simulation Software Platform, can carry out Numerical Simulation Analysis to the system described by formula (9), formula (11).Select Runge-Kutta (ODE45) algorithm to solve system equation, the phase rail figure of this chaos circuit state variable can be obtained.

When α=5.8, β=6.2, when a=-1.1, b=0.02 and initial condition are (0.001,0,0), what formula (9) system Different Plane was corresponding hides two scrollwork Cai Shi chaos attractor numerical simulation respectively as shown in Fig. 3 a, Fig. 4 a.Wherein "○" represents zero balancing point, and " " and " * " represents two non-zero balance points.

When α=5.8, β=5.8, a=-1.1, b=0.02, μ=0.4, and initial condition is (0.001,0,0) time, formula (15) system at the chaos attractor numerical simulation phase rail figure of Different Plane respectively as shown in Fig. 5 (a), Fig. 6 (a) He Fig. 7 (a).This result further demonstrate that modified model Cai Shi system can produce the correctness of three novel scrollwork Cai Shi chaos attractor phenomenon analysis.Reach the original intention of inventing a kind of new chaotic signal generator.

Experimental verification: it is the four-quadrant multiplier of AD633JN that the design adopts model to be the operational amplifier of OP07CP and model, and provides ± 15V operating voltage.Resistance adopts accurate adjustable resistance, and electric capacity is leaded multilayer ceramic capacitor.Theory analysis and numerical simulation show, the self-excitation attractor that this circuit produces or hiding attractor insensitive to initial condition, when experimental circuit powers up, node " x " place trigger, be easy to realize required for state variable initial value.

TektronixDPO3034 digital storage oscilloscope is adopted to catch measured waveform, carried out experimental verification to the chaos attractor phase rail figure shown in Fig. 3 (a), Fig. 4 (a), Fig. 5 (a), Fig. 6 (a) He Fig. 7 (a) respectively, experimental result is respectively as shown in Fig. 3 (b), Fig. 4 (b), Fig. 5 (b), Fig. 6 (b) He Fig. 7 (b).

Comparing result can illustrate: a kind of modified model Cai Shi system that simultaneously can produce 3 scrollwork attractors of 1 self-excitation scrollwork and 2 hiding scrollworks that the present invention realizes, and integrated circuit is by nonlinear function h (x) realizing circuit and control function h _μ(x, z) realizing circuit, produces and hides Double Scroll and comprise the 3 scroll chaotic attractors that 1 self-excitation scrollwork and 2 hides scrollwork.The non-linear phenomena observed in experimental circuit and development law thereof and simulation result fit like a glove, can the correctness of proof theory analysis and numerical simulation.Therefore, the modified model Cai Shi system with 3 scrollwork attractors of self-excitation scrollwork and hiding scrollwork constructed by the present invention has theoretical foundation and the realizability physically of science, can play positive impetus to the engineer applied of chaos circuit.

Above-described embodiment is only for example of the present invention is clearly described, and is not the restriction to embodiments of the present invention.For those of ordinary skill in the field, can also make other changes in different forms on the basis of the above description.Here exhaustive without the need to also giving all execution modes.

Claims (5)

1. can produce a modified model Cai Shi system for attractor self-excitation scrollwork and hiding scrollwork simultaneously, it is characterized in that: comprise basic circuit a, piecewise linear function h (x) the realizing circuit b containing three integrating channel and control function h _μ(x, z) realizing circuit c; After wherein a circuit is connected successively with each same port of b circuit, two scrollwork can be produced and hide Cai Shi chaos attractor; A circuit is connected successively with b circuit and each same port of c circuit, can produce three novel scrollwork Cai Shi chaos attractors.

2. a kind of modified model Cai Shi system that simultaneously can produce attractor self-excitation scrollwork and hiding scrollwork according to claim 1, is characterized in that: the described basic circuit a containing three integrating channel comprises passage 1, passage 2 and passage 3; Passage 1 has two input “ – h _μ" and " y ", by exporting " x " after anti-phase adder sum-product intergrator, more finally export “ – x through one-level inverter "; Passage 2 has three input " x ", “ – y " and " z ", by exporting " y " after anti-phase adder sum-product intergrator, more finally export “ – y through one-level inverter "; Passage 3 only has an input “ – y ", by exporting " z " after inverter sum-product intergrator, more finally export “ – z through one-level inverter "; Operational amplifier U ₁, U ₂, U ₃, U ₄, U ₅, U ₆, U ₇, U ₈and U ₉in-phase input end connect " ".

3. a kind of modified model Cai Shi system that simultaneously can produce attractor self-excitation scrollwork and hiding scrollwork according to claim 1, is characterized in that: described piecewise linear function h (x) realizing circuit comprises ABS function realizing circuit in dotted line frame shown in Figure of abstract b and the anti-phase adder circuit of one-level.The lower road series resistance R of input " x " _h1after meet operational amplifier U _h1inverting input, U _h1inverting input and output between parallel resistance R _h2, U _h1output connecting resistance R _h3left end, R _h3right-hand member through divider resistance R _h4dividing potential drop Hou Shu Chu – 0.56 (| x+1| – | x – 1|), and as an input of anti-phase adder, divider resistance R _h4another termination " "; Operational amplifier U is connected to after the resistance of series connection one " the 200k Ω " that set out on a journey of the input " x " of b circuit _h2inverting input, same resistance R _h3right-hand member connect one " 204k Ω " resistance after be connected to operational amplifier U _h2inverting input, U _h2inverting input and output between the resistance of in parallel one " 204k Ω ", operational amplifier U _h1and U _h2in-phase input end connect " ", now U _h2output export “ – h ".

4. a kind of modified model Cai Shi system that simultaneously can produce attractor self-excitation scrollwork and hiding scrollwork according to claim 1, is characterized in that: described control function h _μ(x, z) realizing circuit comprises sign function circuit and a multiplier circuit.Main circuit has five inputs and is respectively " z ", “ – z ", " μ ", “ – 1V " with “ – h ", an and output “ – h _μ".Operational amplifier U _h3and U _h4in-phase input end all connect " μ ", μ=0.4V; " z " holds concatenation operation amplifier U _h3inverting input, U _h3output to connect the resistance of one " 12.5k Ω ", right-hand member Shu Chu – sgn (z – μ) Hou the divider resistance dividing potential drop of " 1k Ω " of this resistance, and as an input of anti-phase adder, resistance be the divider resistance of " 1k Ω " another termination " ".In like manner ， “ – z " hold concatenation operation amplifier U _h4inverting input, U _h4output to connect the resistance of one " 12.5k Ω ", the right-hand member of this resistance exports sgn (z – μ) after the divider resistance dividing potential drop of " 1k Ω ", and as an input of anti-phase adder, resistance be the divider resistance of " 1_k Ω " another termination " ".Three input “ – sgn (z – μ) of anti-phase adder ", " sgn (z – μ) " with “ – 1_V " resistance of connecting respectively is concatenation operation amplifier U after the resistance of " 200k Ω " _h5inverting input, U _h5inverting input and output between a resistance in parallel be the resistance of " 200k Ω ", U _h5in-phase input end connect " "; U _h5output connect the input of multiplier M, another input of M is “ – h "; Now, multiplier exports “ – h _μ".

5. a kind of modified model Cai Shi system that simultaneously can produce attractor self-excitation scrollwork and hiding scrollwork according to claim 1 or 2 or 3 or 4, is characterized in that containing three state variable x, y and z, and having following relation:

$\stackrel{\&CenterDot;}{x}=dx/dt,\stackrel{\&CenterDot;}{y}=dy/dt,\stackrel{\&CenterDot;}{z}=dz/dt$