CN103236920A - Improved four-dimensional unified multi-wing chaotic system - Google Patents

Improved four-dimensional unified multi-wing chaotic system Download PDF

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CN103236920A
CN103236920A CN2013101428375A CN201310142837A CN103236920A CN 103236920 A CN103236920 A CN 103236920A CN 2013101428375 A CN2013101428375 A CN 2013101428375A CN 201310142837 A CN201310142837 A CN 201310142837A CN 103236920 A CN103236920 A CN 103236920A
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multiple wing
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Abstract

The invention relates to an improved four-dimensional unified multi-wing chaotic system and discloses a new method for generating multi-wing butterfly chaotic attractors. On the basis of a three-dimensional unified chaotic system, by adding a state variable and a piecewise symbolic linear function, the improved four-dimensional unified multi-wing chaotic system is constructed. By setting parameters and different symbolic function values, the butterfly chaotic attractors of the four-dimensional unified multi-wing chaotic system can be obtained. The feasibility and the effectiveness of the method are verified by experimental results. The improved four-dimensional unified multi-wing chaotic system has a wide application prospect and an important application value in the fields of radar, secret communication, electronic countermeasure and the like.

Description

A kind of improved four-dimensional unified multiple wing chaos system
Technical field
Invention relates to a kind of improved four-dimensional unified multiple wing chaos system, belongs to electronic communication field.
Background technology
Since Lorenz in 1963 proposed first chaotic model, people had produced great interest to the chaos phenomenon in the non linear system.2002,
Figure 774749DEST_PATH_IMAGE001
Find unified chaotic system.2003, Liu etc. constructed four wing chaos systems, had caused that but people are to constructing the interest of four wings and multiple wing chaos system.But for the structure of multiple wing chaos system, study also less, and for the structure such chaos system still challenging.
This paper is on the basis of unified chaotic system, by increasing a state variable and a break sign linear function, propose a kind of improved four-dimensional unified multiple wing chaos system, this system can produce the butterfly-like chaos attractor of multiple wing by certain parameter is set.Experiment show feasibility and the validity of this method.To have a wide range of applications in fields such as radar, secure communication, electronic countermeasuress and important use value.
Summary of the invention
Technical problem to be solved by this invention provides a kind of improved four-dimensional unified multiple wing chaos system, and this system can produce the butterfly-like chaos attractor of multiple wing by certain parameter is set.
In order to solve the problems of the technologies described above, on the basis of unified chaotic system, the invention provides a kind of improved four-dimensional unified chaotic system, by different parameters is set, this system can produce the butterfly-like chaos attractor of multiple wing.The quantic of the chaos system of this method construct is simple, and this system has bigger using value in engineering, especially the application in secure communication.
The corresponding partial differential equation of described three dimensional unification chaos system are:
Figure 539443DEST_PATH_IMAGE002
(1)
Wherein, work as parameter
Figure 180115DEST_PATH_IMAGE003
The time, chaos is the Lorenz chaos system; When
Figure 298244DEST_PATH_IMAGE004
The time, chaos is
Figure 960169DEST_PATH_IMAGE001
Chaos system; When
Figure 646497DEST_PATH_IMAGE005
The time, chaos is the Chen chaos system; Wherein
Figure 229925DEST_PATH_IMAGE006
,
Figure 151744DEST_PATH_IMAGE007
,
Figure 933756DEST_PATH_IMAGE008
Be state variable.
By increasing a state variable and a break sign linear function, equation (1) is carried out conversion, unified chaotic system is carried out
Figure 817351DEST_PATH_IMAGE009
Conversion, and increase a state variable and a break sign linear function, put in order then, can obtain partial differential is equation:
Figure 888076DEST_PATH_IMAGE010
(2)
Wherein, function
Figure 738220DEST_PATH_IMAGE011
For:
Figure 250104DEST_PATH_IMAGE012
(3)
Wherein,
Figure 278234DEST_PATH_IMAGE013
, work as parameter When getting different numerical value respectively, can obtain the dissimilar four-dimension and unify the butterfly-like chaos attractor of multiple wing.
Effect of the present invention and effect
(1) the present invention has realized providing a kind of improved four-dimensional unified multiple wing chaos system, works as parameter
Figure 631035DEST_PATH_IMAGE014
When getting different numerical value respectively, can obtain the dissimilar butterfly-like chaos attractors of unified multiple wing.
(2) adopt improved four-dimensional unified multiple wing chaos system of the present invention, its output signal has bigger dynamic range, and this chaos signal source has the wideband section characteristic of different frequency range scope, indicates that it is at radar, secure communication, fields such as the electronic countermeasures value that has a wide range of applications.
Description of drawings
Content of the present invention is easier clearly to be understood in order to make, below the specific embodiment and by reference to the accompanying drawings of basis, the present invention is further detailed explanation.
Fig. 1 for improved four-dimensional unified multiple wing chaos system produce the butterfly-like chaos attractor two dimension of 2 wing chaos phasor ( ,
Figure 318161DEST_PATH_IMAGE015
).
Fig. 2 for improved four-dimensional unified multiple wing chaos system produce the butterfly-like chaos attractor two dimension of 4 wing chaos phasor (
Figure 910947DEST_PATH_IMAGE003
, ).
Fig. 3 for improved four-dimensional unified multiple wing chaos system produce the butterfly-like chaos attractor two dimension of 4 wing chaos phasor (
Figure 527053DEST_PATH_IMAGE004
,
Figure 411833DEST_PATH_IMAGE016
).
Fig. 4 for improved four-dimensional unified multiple wing chaos system produce the butterfly-like chaos attractor two dimension of 4 wing chaos phasor (
Figure 226336DEST_PATH_IMAGE005
,
Figure 956395DEST_PATH_IMAGE016
).
Fig. 5 for improved four-dimensional unified multiple wing chaos system produce the butterfly-like chaos attractor two dimension of 6 wing chaos phasor ( ,
Figure 293671DEST_PATH_IMAGE017
).
Fig. 6 for improved four-dimensional unified multiple wing chaos system produce the butterfly-like chaos attractor two dimension of 6 wing chaos phasor ( ,
Figure 129220DEST_PATH_IMAGE017
).
Fig. 7 for improved four-dimensional unified multiple wing chaos system produce the butterfly-like chaos attractor two dimension of 6 wing chaos phasor (
Figure 324709DEST_PATH_IMAGE005
,
Figure 36444DEST_PATH_IMAGE017
).
Embodiment
The corresponding partial differential equation of described three dimensional unification chaos system are:
Figure 74807DEST_PATH_IMAGE002
(1)
Wherein, work as parameter
Figure 160050DEST_PATH_IMAGE003
The time, chaos is the Lorenz chaos system; When
Figure 537942DEST_PATH_IMAGE004
The time, chaos is
Figure 561524DEST_PATH_IMAGE001
Chaos system; When
Figure 821604DEST_PATH_IMAGE005
The time, chaos is the Chen chaos system; Wherein
Figure 716398DEST_PATH_IMAGE006
,
Figure 230687DEST_PATH_IMAGE007
,
Figure 267913DEST_PATH_IMAGE008
Be state variable.
By increasing a state variable and a break sign linear function, equation (1) is carried out conversion, unified chaotic system is carried out
Figure 156234DEST_PATH_IMAGE009
Conversion, and increase a state variable and a break sign linear function, put in order then, can obtain partial differential is equation:
(2)
Wherein, function
Figure 732502DEST_PATH_IMAGE011
For:
Figure 956941DEST_PATH_IMAGE012
(3)
Wherein,
Figure 863717DEST_PATH_IMAGE013
, work as parameter
Figure 97383DEST_PATH_IMAGE014
When getting different numerical value respectively, can obtain the dissimilar four-dimension and unify the butterfly-like chaos attractor of multiple wing.
For system (2), work as parameter
Figure 510566DEST_PATH_IMAGE003
,
Figure 624015DEST_PATH_IMAGE015
The time, the butterfly-like chaos attractor of 2 wings that system produces is as shown in Figure 1; Work as parameter ,
Figure 789865DEST_PATH_IMAGE016
The time, the butterfly-like chaos attractor of 4 wings that system produces is as shown in Figure 2; Other parameter constant, when The time, the butterfly-like chaos attractor of its 4 wing is as shown in Figure 3; Work as parameter
Figure 290041DEST_PATH_IMAGE005
The time, the butterfly-like chaos attractor of its 4 wing is as shown in Figure 4; Work as parameter ,
Figure 215719DEST_PATH_IMAGE017
The time, the butterfly-like chaos attractor of 6 wings that system produces is as shown in Figure 5; Other parameter constant, when
Figure 69406DEST_PATH_IMAGE004
The time, the butterfly-like chaos attractor of its 6 wing is as shown in Figure 6; Work as parameter
Figure 524658DEST_PATH_IMAGE005
The time, the butterfly-like chaos attractor of its 6 wing is as shown in Figure 7; From Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6 and Fig. 7 as can be seen, the four-dimension after the above-mentioned improvement is unified the multiple wing chaos system in parameter
Figure 221219DEST_PATH_IMAGE014
When getting different numerical value respectively, can obtain the dissimilar butterfly-like chaos attractors of unified multiple wing.
Above-described embodiment only is 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. an improved four-dimension is unified the multiple wing chaos system, its feature comprises: on the basis of unified chaotic system, the present invention proposes a kind of improved four-dimensional unified chaotic system, by different parameters and break sign function are set, this four dimensional chaos system can produce the butterfly-like chaos attractor of multiple wing, the quantic that the four-dimension of this method construct is unified the multiple wing chaos system is simple, and this system has bigger using value in engineering, especially the application in secure communication.
2. a kind of improved three dimensional unification multiple wing chaos system according to claim 1 is characterized in that the corresponding partial differential equation of described three dimensional unification chaos system are:
Figure 800449DEST_PATH_IMAGE001
(1)
Wherein, work as parameter
Figure 110338DEST_PATH_IMAGE002
The time, chaos is the Lorenz chaos system; When
Figure 760763DEST_PATH_IMAGE003
The time, chaos is
Figure 216015DEST_PATH_IMAGE004
Chaos system; When
Figure 663308DEST_PATH_IMAGE005
The time, chaos is the Chen chaos system; Wherein
Figure 26156DEST_PATH_IMAGE006
,
Figure 606785DEST_PATH_IMAGE007
,
Figure 905043DEST_PATH_IMAGE008
Be state variable.
3. three dimensional unification multiple wing chaos system after the improvement according to claim 1 is characterized in that: equation (1) is carried out conversion, unified chaotic system is carried out Conversion, and increase a state variable and a break sign linear function, put in order then, can obtain partial differential is equation:
(2)
Wherein, function
Figure 631187DEST_PATH_IMAGE011
For:
Figure 100346DEST_PATH_IMAGE012
(3)
Wherein,
Figure 505920DEST_PATH_IMAGE013
, work as parameter
Figure 229812DEST_PATH_IMAGE014
When getting different numerical value respectively, can obtain the dissimilar four-dimension and unify the butterfly-like chaos attractor of multiple wing.
CN2013101428375A 2013-04-24 2013-04-24 Improved four-dimensional unified multi-wing chaotic system Pending CN103236920A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN103957098A (en) * 2014-04-14 2014-07-30 重庆邮电大学 Chaotic circuit for producing multiple butterfly-shaped attractors and implementation method
CN106160999A (en) * 2016-09-28 2016-11-23 重庆邮电大学 A kind of grid many butterflies wing chaos attractor production method

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN103957098A (en) * 2014-04-14 2014-07-30 重庆邮电大学 Chaotic circuit for producing multiple butterfly-shaped attractors and implementation method
CN103957098B (en) * 2014-04-14 2017-05-10 重庆邮电大学 Chaotic circuit for producing multiple butterfly-shaped attractors and implementation method
CN106160999A (en) * 2016-09-28 2016-11-23 重庆邮电大学 A kind of grid many butterflies wing chaos attractor production method

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Application publication date: 20130807