CN103152159A - Three-dimensional chaotic system with only one balance point and device thereof - Google Patents

Three-dimensional chaotic system with only one balance point and device thereof Download PDF

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CN103152159A
CN103152159A CN2013100836220A CN201310083622A CN103152159A CN 103152159 A CN103152159 A CN 103152159A CN 2013100836220 A CN2013100836220 A CN 2013100836220A CN 201310083622 A CN201310083622 A CN 201310083622A CN 103152159 A CN103152159 A CN 103152159A
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dimensional chaotic
chaotic system
balance point
circuit
chaos
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Abstract

The invention relates to a three-dimensional chaotic system with only one balance point and a device thereof, and provides a new three-dimensional chaotic system by introducing three parameters. The three-dimensional chaotic system can generate attractors with different topological structures by adjusting the parameters a, b and c, and has complex dynamic actions. The comprises an operational amplifying circuit, an integrating circuit, a reverse proportional circuit and the like. The simple three-dimensional chaotic system with only one balance point has obvious bifurcation characteristic, is simple in circuit implementation, and has wide application prospect and important application value in the fields of radars, secret communication, electronic countermeasure and the like.

Description

The three-dimensional chaotic system and the device thereof that only have a balance point
Technical field
The present invention relates to three-dimensional chaotic system and a device thereof that only has a balance point, belong to electronic communication and nonlinear Control field.
Background technology
Since Lorenz in 1963 found chaos phenomenon, chaos phenomenon had all been obtained significant development in the research of every field.In recent years, chaos was widely used in secure communication.And make coded signal for the chaos system signal that only has a positive Lyapunov index, its secret signal ratio is easier to be decrypted; And it is more complicated to have the contrafunctional Time Chaotic Dynamical Systems character of hyperbolic, and its signal has application prospect extremely widely as the chaos encryption signal.In recent years, the method for various structure chaos and hyperchaotic system has caused people's attention.
At first this paper proposed a three-dimensional chaotic system of only having a balance point, and some basic motive characteristics of system have been carried out numerical simulation and theory analysis.As behaviors such as initial value sensitiveness, balance point, dissipativeness, Poincar é mappings.By the analysis to Lyapunov exponential spectrum and bifurcation graphs, and further this chaos system is carried out circuit and realize.
Summary of the invention
Technical problem to be solved by this invention is to provide the three-dimensional chaotic system and the device thereof that only have a balance point.
In order to solve the problems of the technologies described above, the invention provides a three-dimensional chaotic system of only having a balance point, it comprises: integrating circuit, reverse ratio circuit and discharge circuit, the output of negater circuit is exported three state variables as this chaos system successively ,
Figure 723992DEST_PATH_IMAGE002
,
Figure 592853DEST_PATH_IMAGE003
Above-mentioned only have the corresponding partial differential equation of three-dimensional chaotic system of a balance point to be:
Figure 254778DEST_PATH_IMAGE004
(1)
Wherein
Figure 800160DEST_PATH_IMAGE005
,
Figure 304960DEST_PATH_IMAGE006
Be state variable.
Effect of the present invention and effect
(1) three-dimensional chaotic system that provides one to only have a balance point, wherein parameter have been provided in the present invention
Figure 351413DEST_PATH_IMAGE005
, Be state variable.
(2) adopt the hardware circuit of chaos system of the present invention, verified that this three-dimensional chaotic system output signal has larger dynamic range, in addition, reduce the capacitance in the hyperchaotic system circuit, the signal spectrum of output is moved to high frequency direction, show that this chaos signal source has the wide-band characteristic of different frequency range scope, indicates that it is at radar, secure communication, the fields such as the electronic countermeasures value that has a wide range of applications.
(3) the present invention proposes three-dimensional chaotic system and a device thereof that only has a balance point, realized the larger dynamic range that has of chaotic signal output.Theory analysis, the results of study such as numerical simulation and Experiment of Electrical Circuits have also been verified the validity of this system.
Description of drawings
For content of the present invention is more likely to be clearly understood, below the specific embodiment and by reference to the accompanying drawings of basis, the present invention is further detailed explanation, wherein
Fig. 1 is chaos system two dimension and three-dimensional phase diagram (a)
Figure 584128DEST_PATH_IMAGE007
(b) (c)
Figure 806129DEST_PATH_IMAGE009
(d)
Figure 177068DEST_PATH_IMAGE010
Fig. 2 is chaos system
Figure 579099DEST_PATH_IMAGE011
Different initial value responses..
Fig. 3 is chaos system Poincar é mapping, and the cross section is (a) x0=0, (b) y0=0, (c) z0=0.
Fig. 4 is that chaos system is with parameter Change Liapunov (LE) exponential spectrum and bifurcation graphs (a) Liapunov (LE) exponential spectrum (b) trouble figure.
Fig. 5 is that chaos system is with parameter
Figure 931900DEST_PATH_IMAGE013
Change Liapunov (LE) exponential spectrum and bifurcation graphs (a) Liapunov (LE) exponential spectrum (b) trouble figure.
Fig. 6 is that chaos system is with parameter
Figure 626187DEST_PATH_IMAGE014
Change Liapunov (LE) exponential spectrum and bifurcation graphs (a) Liapunov (LE) exponential spectrum (b) trouble figure.
Fig. 7 is the chaos system circuit theory diagrams.
Embodiment
By building three-dimensional chaotic system and a device thereof that only has a balance point, its Mathematical Modeling is described as
Figure 700584DEST_PATH_IMAGE004
(1)
Wherein
Figure 152425DEST_PATH_IMAGE005
, Be state variable.Work as parameter
Figure 814537DEST_PATH_IMAGE015
, initial condition is [0.1 0.1 0.1] TThe time, Fig. 1
Figure 309103DEST_PATH_IMAGE016
Two dimension and three-dimensional phase diagram for system (1) track.As can be seen from Figure 1, chaos system has the dynamic behavior of single scrollwork.
1 basic dynamics
1.1 balance point, dissipativeness.
Make the right of system (1) equation equal 0, namely balance point can be separated following algebraic equation and tries to achieve:
Figure 107295DEST_PATH_IMAGE017
(2)
When
Figure 726101DEST_PATH_IMAGE015
The time, system only have a balance point (
Figure 67084DEST_PATH_IMAGE018
Figure 591606DEST_PATH_IMAGE019
).At the balance point place, system (1) is carried out linearisation, its Jacobian matrix is
Figure 267307DEST_PATH_IMAGE020
(3)
In order to ask balance point
Figure 535477DEST_PATH_IMAGE021
, corresponding characteristic value, order
Figure 730967DEST_PATH_IMAGE022
(4)
Can be balanced a little
Figure 318068DEST_PATH_IMAGE021
Corresponding characteristic value
Figure 356431DEST_PATH_IMAGE023
Figure 303658DEST_PATH_IMAGE024
According to Condition, balance point as can be known
Figure 734826DEST_PATH_IMAGE021
It is unsettled saddle point.
Due to
Figure 870272DEST_PATH_IMAGE026
= ， (5)
Because
Figure 392270DEST_PATH_IMAGE028
, this chaos system is dissipative system and restrains as shown in following (6) formula with exponential form:
Figure 836020DEST_PATH_IMAGE029
(6)
As seen, when The time, each volume element of system's path is retracted to zero with index percent-1.
1.2 initial value sensitivity, Poincar é mapping.
Work as parameter
Figure 121694DEST_PATH_IMAGE015
The time, the time domain sequences of system x (t) has very strong sensitiveness to initial value, differs d as the initial value as x0 0=0.000001, other initial value is constant, can get its initial value sensitiveness as shown in Figure 2, as can be seen from Figure 2, can find at 73s, and it is fully different that its sequence becomes, and proved absolutely system's sensitivity to initial.
Poincar é mapping has reflected the folding and Bifurcation Characteristics of system, and Fig. 3 is the Poincar é mapping of system (1) when different cross section.
1.3 Liapunov exponent and dimension thereof
Liapunov exponent is a key character of chaos system.Chaos attractor between adjacent orbit demonstrates the trend of separating by index percent.At present, there are many kinds of methods can calculate largest Lyapunov exponent, use the single argument decomposition method, can obtain (1) three Lyapunonov index of system and be respectively:
Figure 146282DEST_PATH_IMAGE031
Figure 57923DEST_PATH_IMAGE033
(7)
A positive Lyapunov index is wherein arranged, and one is zero, and all the other one is negative value, shows that there is strange attractor in this system, and its motion is chaos, the Lyapunov dimension can be calculated as follows into:
Figure 901376DEST_PATH_IMAGE034
Figure 780471DEST_PATH_IMAGE035
Figure 346450DEST_PATH_IMAGE036
(8)
Therefore, can find out that the Lyapunov dimension is dimension, show that this system has the characteristic of chaos.
1.4. system parameters sensitivity analysis
Work as parameter
Figure 271681DEST_PATH_IMAGE037
Respectively in the interval
Figure 168093DEST_PATH_IMAGE038
,
Figure 760748DEST_PATH_IMAGE039
And When changing in scope, and during another two parameter constants, its Lyapunov exponential spectrum and trouble figure are respectively as Fig. 4, Fig. 5 and shown in Figure 6.
Can find out by above-mentioned trouble figure and Lyapunov index spectrogram labor, system parameters has very large sensitiveness, and the impact of different parameters is also different, and along with the variation of parameter, system experiences different courses and has obvious Bifurcation Characteristics.So this system has a wide range of applications in the fields such as secure communication.
This chaos system circuit design is comparatively simple, adopts linear adjustable resistance, linear capacitance, operational amplifier, analog multiplier to realize.Operational amplifier adopts LM741, is to carry out plus and minus calculation, and analog multiplier adopts AD633 to realize, is the nonlinear terms in completion system. and the allowable voltage of operational amplifier LM741 is ± 18V that the allowable voltage of multiplier AD633 is only ± 10V to make
Figure 549636DEST_PATH_IMAGE041
，
Figure 108793DEST_PATH_IMAGE042
， (9)
Figure 211748DEST_PATH_IMAGE043
.
Can obtain
Figure 932579DEST_PATH_IMAGE044
，
Figure 504506DEST_PATH_IMAGE045
， ，
Figure 326279DEST_PATH_IMAGE047
，
Figure 358957DEST_PATH_IMAGE048
，
Figure 401869DEST_PATH_IMAGE049
The circuit theory diagrams of chaos system proposed by the invention as shown in Figure 7, electric capacity wherein , resistance
Figure 459003DEST_PATH_IMAGE051
,
Figure 662583DEST_PATH_IMAGE052
,
Figure 697185DEST_PATH_IMAGE053
,
Figure 401835DEST_PATH_IMAGE054
, , ,
Figure 273342DEST_PATH_IMAGE057
,
Figure 516105DEST_PATH_IMAGE058
,
Figure 804129DEST_PATH_IMAGE059
,
Figure 349511DEST_PATH_IMAGE060
, , ,
Figure 292562DEST_PATH_IMAGE063
,
Figure 133479DEST_PATH_IMAGE064
,
Figure 624110DEST_PATH_IMAGE065
,
Figure 349621DEST_PATH_IMAGE066
Be 10k Ω,
Figure 720559DEST_PATH_IMAGE067
=
Figure 122590DEST_PATH_IMAGE057
=5k Ω,
Figure 946190DEST_PATH_IMAGE068
K Ω.
Above-described embodiment is only for example of the present invention clearly is described, and be not to be 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.

Claims (3)

1. only have three-dimensional chaotic system and the device thereof of a balance point, its feature comprises: integrating circuit, reverse ratio circuit and discharge circuit etc., the output of negater circuit is exported three state variables as chaos system successively
Figure 701045DEST_PATH_IMAGE001
, ,
2. three-dimensional chaotic system and the device thereof that only has a balance point according to claim 1, is characterized in that, the corresponding partial differential equation of described three-dimensional chaotic system are:
Figure 110925DEST_PATH_IMAGE004
(1)
Wherein
Figure 277465DEST_PATH_IMAGE005
,
Figure 89432DEST_PATH_IMAGE006
Be state variable.
3. three-dimensional chaotic system according to claim 1 and device, is characterized in that: described the first electric capacity (C 1), the second electric capacity (C 2), the 3rd electric capacity (C 3) capacitance equate,
Figure 558590DEST_PATH_IMAGE007
And by regulating simultaneously the capacitance of each electric capacity, can adjust described three state variables of chaos system
Figure 698585DEST_PATH_IMAGE001
,
Figure 26404DEST_PATH_IMAGE002
,
Figure 709190DEST_PATH_IMAGE003
Frequency of oscillation.
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Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN105227159A (en) * 2015-08-26 2016-01-06 韩敬伟 A kind of spherical five quasi-periodic oscillation systems and circuit
CN105245204A (en) * 2015-08-26 2016-01-13 韩敬伟 Five-item quasi-period spherical oscillation system and circuit thereof
CN109039582A (en) * 2016-04-28 2018-12-18 王志 A kind of simple chaos system circuit exporting Lorenz type attractor
CN117424787A (en) * 2023-11-20 2024-01-19 常州大学 Chaotic orbit folding method and system based on complex operation

Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN202475450U (en) * 2012-04-04 2012-10-03 滨州学院 Single-scroll three-dimensional chaotic circuit
CN202475446U (en) * 2012-03-01 2012-10-03 滨州学院 Random digit sequence based on three-dimensional chaos system
CN202503530U (en) * 2012-04-06 2012-10-24 滨州学院 Three-dimensional chaotic system
CN102930762A (en) * 2012-11-19 2013-02-13 湖南大学 Three-dimensional chaotic circuit
CN103199982A (en) * 2013-01-09 2013-07-10 王少夫 Three-dimensional chaotic system with quadratic component

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN202475446U (en) * 2012-03-01 2012-10-03 滨州学院 Random digit sequence based on three-dimensional chaos system
CN202475450U (en) * 2012-04-04 2012-10-03 滨州学院 Single-scroll three-dimensional chaotic circuit
CN202503530U (en) * 2012-04-06 2012-10-24 滨州学院 Three-dimensional chaotic system
CN102930762A (en) * 2012-11-19 2013-02-13 湖南大学 Three-dimensional chaotic circuit
CN103199982A (en) * 2013-01-09 2013-07-10 王少夫 Three-dimensional chaotic system with quadratic component

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN105227159A (en) * 2015-08-26 2016-01-06 韩敬伟 A kind of spherical five quasi-periodic oscillation systems and circuit
CN105245204A (en) * 2015-08-26 2016-01-13 韩敬伟 Five-item quasi-period spherical oscillation system and circuit thereof
CN109039582A (en) * 2016-04-28 2018-12-18 王志 A kind of simple chaos system circuit exporting Lorenz type attractor
CN109039581A (en) * 2016-04-28 2018-12-18 王志 A kind of simple chaos system circuit of output Lorenz type switching attractor
CN117424787A (en) * 2023-11-20 2024-01-19 常州大学 Chaotic orbit folding method and system based on complex operation

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Application publication date: 20130612