CN103227711A - Three-dimensional chaotic system and device for producing two-winged, three-winged and four-winged attractors - Google Patents

Three-dimensional chaotic system and device for producing two-winged, three-winged and four-winged attractors Download PDF

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CN103227711A
CN103227711A CN2013100836199A CN201310083619A CN103227711A CN 103227711 A CN103227711 A CN 103227711A CN 2013100836199 A CN2013100836199 A CN 2013100836199A CN 201310083619 A CN201310083619 A CN 201310083619A CN 103227711 A CN103227711 A CN 103227711A
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winged
chaotic system
circuit
chaos
dimensional chaotic
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Abstract

The invention relates to a three-dimensional chaotic system and device for producing two-winged, three-winged and four-winged attractors. With the introduction of five parameters, the novel three-dimensional chaotic system is provided and can produce the attractors with two-scroll, three-scroll and four-scroll attractor topological structures by adjusting the parameters, and the chaotic system has a complicated dynamic behavior. The three-dimensional chaotic system is characterized in that output ends of a reverse proportion circuit, a first integral circuit, a second integral circuit and a third integral circuit sequentially output three state variables of the chaotic system; and output ends of a first reverse circuit, a second reverse circuit and a third reverse circuit sequentially output three state variables of the chaotic system. According to the simple three-dimensional chaotic system, the circuits of the system can be simply implemented, and the system has wide application prospect and important application value in the field of radar, secret communication, electronic countermeasures and the like.

Description

Three-dimensional chaotic system and a device that produces two, three and four wing attractors
Technical field
The present invention relates to three-dimensional chaotic system and a device that produces two, three and four wing attractors, belong to electronic communication field.
Background technology
In recent years, along with the continuous exploration of people to the chaos attractor dynamic behavior, and the further investigation of, Generalized Projective Synchronization synchronous to the self adaptation of chaos system and the synchronous technology such as anti-synchronous, chaos has obtained major progress in the application of engineering field, and becomes study hotspot in fields such as chaos encryption, secure communication, chaotic radars.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.
Three-dimensional chaotic system and the device that produces two, three and four wing attractors that the present invention proposes, carried out numerical simulation and theory analysis to some basic motive characteristics of system.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 studied system parameters sensitiveness.
Summary of the invention
Technical problem to be solved by this invention is to provide the three-dimensional chaotic system and the device that produce two, three and four wing attractors.
In order to solve the problems of the technologies described above, the invention provides three-dimensional chaotic system and a device that produces two, three and four wing attractors, it comprises: oppositely the output of ratio circuit, first integral circuit, second integral circuit, third integral circuit is exported three state variables as chaos system successively ,
Figure 726516DEST_PATH_IMAGE002
,
Figure 960051DEST_PATH_IMAGE003
; First oppositely, second oppositely and the output of the 3rd negater circuit export successively three state variables as chaos system
Figure 462708DEST_PATH_IMAGE004
,
Figure 613460DEST_PATH_IMAGE005
,
Figure 275385DEST_PATH_IMAGE006
.
Above-mentioned three-dimensional chaotic system institute corresponding equation is:
Figure 555188DEST_PATH_IMAGE007
Figure 669775DEST_PATH_IMAGE008
(1)
effect of the present invention and effect
(1) three-dimensional chaotic system and the device that provide one to produce two, three and four wing attractors have been provided in the present invention, wherein, and parameter .
(2) adopt the hardware circuit of chaos system of the present invention, verified that this hyperchaotic system output signal has larger dynamic range and has bifurcated characteristic preferably, in addition, reduce the capacitance in the hyperchaotic system circuit, can make the signal spectrum of output move to high frequency direction, show that this hyperchaos 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 that produces two, three and four wing attractors, 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.
The accompanying drawing explanation
For content of the present invention is more likely to be clearly understood, below the specific embodiment by reference to the accompanying drawings of basis, the present invention is further detailed explanation, wherein
Fig. 1 be chaos system two dimension and three-dimensional both wings attractor phasor (
Figure 89889DEST_PATH_IMAGE011
) (a)
Figure 862410DEST_PATH_IMAGE012
; (b)
Figure 322342DEST_PATH_IMAGE013
3; (c)
Figure 427701DEST_PATH_IMAGE014
; (d)
Figure 111623DEST_PATH_IMAGE015
.
Fig. 2 be chaos system two dimension and three-dimensional three wings attractor phasor (
Figure 810589DEST_PATH_IMAGE016
) (a)
Figure 933266DEST_PATH_IMAGE012
; (b)
Figure 535542DEST_PATH_IMAGE013
3; (c)
Figure 390365DEST_PATH_IMAGE014
; (d)
Figure 435682DEST_PATH_IMAGE015
.
Fig. 3 be chaos system two dimension and three-dimensional both wings attractor phasor (
Figure 362049DEST_PATH_IMAGE017
) (a)
Figure 786209DEST_PATH_IMAGE012
; (b)
Figure 513731DEST_PATH_IMAGE013
3; (c) ; (d)
Figure 245244DEST_PATH_IMAGE015
.
Fig. 4 is chaos system
Figure 117385DEST_PATH_IMAGE018
different initial value responses..
Fig. 5 is chaos system Poincar é mapping, and cross section is (a) x0=0, (b) y0=1, (c) z0=5.
Fig. 6 is that chaos system is with parameter
Figure 782852DEST_PATH_IMAGE019
change Lyapunov exponential spectrum and bifurcation graphs (a) Lyapunov exponential spectrum; (b) bifurcation graphs.
Fig. 7 is that chaos system is with parameter
Figure 537182DEST_PATH_IMAGE020
change Lyapunov exponential spectrum and bifurcation graphs (a) Lyapunov exponential spectrum; (b) bifurcation graphs.
Fig. 8 is that chaos system is with parameter
Figure 651025DEST_PATH_IMAGE021
change Lyapunov exponential spectrum and bifurcation graphs (a) Lyapunov exponential spectrum; (b) bifurcation graphs.
Fig. 9 is that chaos system is with parameter change Lyapunov exponential spectrum and bifurcation graphs (a) Lyapunov exponential spectrum; (b) bifurcation graphs.
Figure 10 is that chaos system is with parameter
Figure 10779DEST_PATH_IMAGE023
change Lyapunov exponential spectrum and bifurcation graphs (a) Lyapunov exponential spectrum; (b) bifurcation graphs.
Figure 11 is two of chaos systems double scroll chaos (a) x that deposits, z (b) y, z (c) z, x, y
Figure 12 is the chaos system circuit theory diagrams.
Embodiment
By building three-dimensional chaotic system and a device that produces two, three and four wing attractors, its Mathematical Modeling is described as
Figure 527528DEST_PATH_IMAGE008
(1)
Wherein
Figure 178269DEST_PATH_IMAGE010
,
Figure 641611DEST_PATH_IMAGE024
for state variable, when
Figure 687803DEST_PATH_IMAGE011
, initial condition is [1 10 10] tthe time, Fig. 1
Figure 592305DEST_PATH_IMAGE025
for system (1) both wings chaos attractor phasor.。When
Figure 973739DEST_PATH_IMAGE016
, initial condition is [1 10 10] tthe time, Fig. 2
Figure 455535DEST_PATH_IMAGE025
for system (1) three wings chaos attractor phasor.When
Figure 234312DEST_PATH_IMAGE017
, initial condition is [1 10 10] tthe time, Fig. 3
Figure 586796DEST_PATH_IMAGE025
for system (1) four wing chaos attractor phasor.
1 basic dynamics
1.1 balance point, dissipativeness.
Make the right of system (1) equation equal 0, balance point can be separated following algebraic equation and tries to achieve:
Figure 467027DEST_PATH_IMAGE026
Figure 436120DEST_PATH_IMAGE027
(2)
Figure 263262DEST_PATH_IMAGE028
Clearly, system has the balance balance point
Figure 673514DEST_PATH_IMAGE029
, another five balance points are
Figure 786964DEST_PATH_IMAGE030
,
Figure 243353DEST_PATH_IMAGE031
,
Figure 638300DEST_PATH_IMAGE032
,
Figure 168638DEST_PATH_IMAGE033
.
At balance point
Figure 125093DEST_PATH_IMAGE034
place, carry out linearisation to system (1), and its Jacobian matrix is shown in following (3) formula
(3)
In order to ask balance point
Figure 706564DEST_PATH_IMAGE029
, corresponding characteristic value, order
Figure 592874DEST_PATH_IMAGE036
(4)
Can be balanced a little
Figure 313705DEST_PATH_IMAGE029
corresponding characteristic value
Figure 620053DEST_PATH_IMAGE037
, according to condition, known balance point
Figure 956673DEST_PATH_IMAGE029
it is unsettled saddle point.In like manner, the character of known other balance point.
Due to
Figure 254931DEST_PATH_IMAGE039
=
Figure 376470DEST_PATH_IMAGE040
， (5)
Because
Figure 448069DEST_PATH_IMAGE041
, this chaos system is dissipative system and restrains as shown in following (6) formula with exponential form:
Figure 10769DEST_PATH_IMAGE042
(6)
Visible, when
Figure 73403DEST_PATH_IMAGE043
the time, each volume element of system path is with index percent
Figure 885501DEST_PATH_IMAGE044
be retracted to zero.
1.2 initial value sensitivity, Poincar é mapping.
Work as parameter
Figure 199939DEST_PATH_IMAGE017
the time, the time domain sequences of system x (t) has very strong sensitiveness to initial value, as the initial value as x0 differs 0.000001, other initial value is constant, can obtain its initial value sensitiveness as shown in Figure 4, as can be seen from Figure 4, can find at 18s, it is fully different that its sequence becomes, and absolutely proved the sensitiveness of system to initial value.
Poincar é mapping has reflected the folding and Bifurcation Characteristics of system, and Fig. 5 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 476199DEST_PATH_IMAGE045
A positive Lyapunov index is wherein arranged, and one is zero, and all the other one is negative value, shows that this system has strange attractor, and its motion is chaos, the Lyapunov dimension can be calculated as follows into:
Figure 352145DEST_PATH_IMAGE046
Figure 651539DEST_PATH_IMAGE047
Figure 894302DEST_PATH_IMAGE048
(7)
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
This section is by the analysis to Lyapunov exponential spectrum and bifurcation graphs, the sensitivity characteristic of Study system parameter to chaotic behavior.
Work as parameter
Figure 369276DEST_PATH_IMAGE049
exist respectively ;
Figure 996622DEST_PATH_IMAGE051
;
Figure 43075DEST_PATH_IMAGE052
;
Figure 638136DEST_PATH_IMAGE053
;
Figure 213473DEST_PATH_IMAGE054
in interval, change, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Figure 10 be respectively system along with
Figure 425143DEST_PATH_IMAGE049
the Lyapunov exponential spectrum and the bifurcation graphs that change can find out that system (1) is in chaos state.
In addition, work as parameter
Figure 183277DEST_PATH_IMAGE011
the time, system (1), while getting different initial value, can obtain two two scrollwork attractors that coexist as shown in figure 11.
By above-mentioned trouble figure and Lyapunov index spectrogram labor, can find out, 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.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 resistance, linear capacitance, operational amplifier, analog multiplier to realize.Operational amplifier adopts LM741, for carrying out plus and minus calculation, analog multiplier adopts AD633 to realize, the nonlinear terms in completion system. the allowable voltage of operational amplifier LM741 is ± 18V, the allowable voltage of multiplier AD633 is only ± 10V, the circuit theory diagrams of chaos system proposed by the invention as shown in figure 12, wherein
Figure 288636DEST_PATH_IMAGE055
, other component value as shown in the figure.
Above-described embodiment is only for example of the present invention clearly is described, and be 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.

Claims (3)

1. three-dimensional chaotic system and a device that produces two, three and four wing attractors, its feature comprises: oppositely the output of ratio circuit, first integral circuit, second integral circuit, third integral circuit is exported three state variables as chaos system successively
Figure 706659DEST_PATH_IMAGE001
,
Figure 684019DEST_PATH_IMAGE002
,
Figure 237491DEST_PATH_IMAGE003
; First oppositely, second oppositely and the output of the 3rd negater circuit export successively three state variables as chaos system
Figure 531069DEST_PATH_IMAGE004
,
Figure 915914DEST_PATH_IMAGE005
,
Figure 777691DEST_PATH_IMAGE006
.
2. three-dimensional chaotic system according to claim 1, is characterized in that, described three-dimensional chaotic system institute corresponding equation is:
Figure 848153DEST_PATH_IMAGE007
Figure 414263DEST_PATH_IMAGE008
(1)
Figure 919194DEST_PATH_IMAGE009
Wherein
Figure 545347DEST_PATH_IMAGE010
for state variable, parameter for arithmetic number.
3. three-dimensional chaotic system according to claim 1, 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 646476DEST_PATH_IMAGE012
and, by regulate the capacitance of each electric capacity simultaneously, can adjust described three state variables of chaos system ,
Figure 773274DEST_PATH_IMAGE002
,
Figure 913268DEST_PATH_IMAGE003
frequency of oscillation.
CN2013100836199A 2013-03-17 2013-03-17 Three-dimensional chaotic system and device for producing two-winged, three-winged and four-winged attractors Pending CN103227711A (en)

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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN103414551A (en) * 2013-08-02 2013-11-27 南京师范大学 Three-dimensional four-winged chaotic circuit
CN105099663A (en) * 2015-09-01 2015-11-25 王忠林 Construction method of chaotic system comprising folding double-wing chaotic attractor, and circuit
CN105281887A (en) * 2014-06-20 2016-01-27 马英杰 2-14 scroll chaotic attractor system and circuit

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
谢国波: "几类混沌系统的设计及应用研究", 《中国博士学位论文全文数据库》 *

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN103414551A (en) * 2013-08-02 2013-11-27 南京师范大学 Three-dimensional four-winged chaotic circuit
CN103414551B (en) * 2013-08-02 2016-02-24 南京师范大学 A kind of three-dimensional four wing chaos circuits
CN105281887A (en) * 2014-06-20 2016-01-27 马英杰 2-14 scroll chaotic attractor system and circuit
CN105281887B (en) * 2014-06-20 2018-05-08 马英杰 2~14 scroll chaotics attract subsystem and circuit
CN105099663A (en) * 2015-09-01 2015-11-25 王忠林 Construction method of chaotic system comprising folding double-wing chaotic attractor, and circuit

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Application publication date: 20130731