CN103188071A - Three-dimensional chaotic system and device thereof - Google Patents
Three-dimensional chaotic system and device thereof Download PDFInfo
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- CN103188071A CN103188071A CN2013101148798A CN201310114879A CN103188071A CN 103188071 A CN103188071 A CN 103188071A CN 2013101148798 A CN2013101148798 A CN 2013101148798A CN 201310114879 A CN201310114879 A CN 201310114879A CN 103188071 A CN103188071 A CN 103188071A
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Abstract
The invention relates to a three-dimensional chaotic system containing square terms and a device thereof. The new three-dimensional chaotic system is proposed by introducing two parameters, parameters can be adjusted so that the system can produce attractors having different topological structures, and the chaotic system has complex dynamic behaviors. The system comprises an integrating circuit and an inverse proportion circuit; and the output end of a first operational amplifier sequentially outputs three state variables x, y and z of the chaotic system. The simple three-dimensional chaotic system containing the square terms is simple in circuit realization, and has wide application prospect and important application value in the fields of radar, secret communication, electronic countermeasures and the like.
Description
Technical field
The present invention relates to a three-dimensional chaos system and its apparatus that contains quadratic term, belong to electronic communication 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 had obtained using widely 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.
This paper has at first proposed a three-dimensional chaos system that contains quadratic term, 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 provides a three-dimensional chaos system and its apparatus that contains quadratic term.
In order to solve the problems of the technologies described above, the invention provides a three-dimensional chaos system that contains quadratic term, it comprises: integrating circuit, reverse ratio circuit; The output of first amplifier is exported three state variables as this chaos system successively
,
,
The above-mentioned corresponding partial differential equation of three-dimensional chaos system that contain quadratic term are:
(1)
Wherein
And
,
,
,
Be state variable.
Effect of the present invention and effect
(1) the three-dimensional chaos system that provides one to contain quadratic term, wherein parameter have been provided in the present invention
And
(2) hardware circuit of employing chaos system of the present invention, verified that this three-dimensional chaos system output signal has bigger 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 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.
(3) the present invention proposes a three-dimensional chaos system and its apparatus that contains quadratic term, realized the bigger dynamic range that has of chaotic signal output.Theory analysis, results of study such as numerical simulation and Experiment of Electrical Circuits have also been verified the validity of this system.
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 is chaos system two dimension and three-dimensional phasor (a)
(b)
(c)
(d)
Fig. 2 is chaos system
Different initial value responses..
Fig. 3 is chaos system Poincar é mapping, and the cross section is (a) x0=0, (b) y0=1, (c) z0=1.
Fig. 4 is that chaos system is with parameter
Change bifurcation graphs.
Fig. 5 is that chaos system is with parameter
Change the Lyapunov exponential spectrum.
Fig. 6 is the chaos system circuit theory diagrams.
Embodiment
By making up a three-dimensional chaos system and its apparatus that contains quadratic term, its Mathematical Modeling is described as
Wherein
And
,
,
,
Be state variable.When
, when initial condition is [1 1 1] T, Fig. 1
Two dimension and three-dimensional phasor for system (1) track.As can be seen from Figure 1, chaos system has complicated dynamic behavior.
The basic dynamic characteristic
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:
When
The time, system have three balance points (
).At the balance point place, system (1) is carried out linearisation, its Jacobian matrix is shown in following (3) formula
In order to ask balance point
, corresponding characteristic value, order
Can obtain balance point
Corresponding characteristic value
,
According to
Condition, balance point as can be known
It is unsettled saddle point.In like manner, balance point as can be known
Be unsettled saddle point.
Because
Because
, then this chaos system be dissipative system and with exponential form convergence shown in following (6) formula:
As seen, when
The time, each volume element of system's path is retracted to zero with index percent-0.55.
1.2 initial value sensitivity, Poincar é mapping.
Work as parameter
The time, the time domain sequences of system x (t) has very strong sensitiveness to initial value, differ d0=0.000001 as the initial value as x0, 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 72s, it is different fully that its sequence becomes, and proved absolutely the sensitiveness of system to initial value.
Poincar é mapping has reflected the folding of system and fork characteristic, 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 the 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:
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:
Therefore, the Lyapunov dimension is the mark dimension as can be seen, shows that this system has the characteristic of chaos.
1.4. system parameters sensitivity analysis
By the analysis to Lyapunov exponential spectrum and bifurcation graphs, the research system parameters is to the sensitivity characteristic of chaotic behavior.
Allow
Change in the scope, other parameter immobilizes, Fig. 4, Fig. 5 be respectively system along with
The bifurcation graphs and the Lyapunov exponential spectrum that change, as can be seen: remove the small part point, when
, system (1) is in chaos state.
By above-mentioned trouble figure and Lyapunov index spectrogram labor as can be seen, system parameters has very large sensitiveness, and the influence of different parameters also has nothing in common with each other, and along with the variation of parameter, system experiences different courses.So this system has a wide range of applications in 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 TL081, be to carry out plus and minus calculation, analog multiplier adopts AD633 to realize, be the nonlinear terms of finishing in the system. the allowable voltage of operational amplifier TL081 is ± 15V, the allowable voltage of multiplier AD633 only is ± 10V, the circuit theory diagrams of chaos system proposed by the invention as shown in Figure 6, wherein, adjustable resistance
,
,
,
,
,
,
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. three-dimensional chaos system and its apparatus, its feature comprises: integrating circuit, reverse ratio circuit; The output of first amplifier is exported three state variables as chaos system successively
,
,
2. three-dimensional chaos system and its apparatus according to claim 1 is characterized in that, the corresponding partial differential equation of described three-dimensional chaos system are:
Wherein
And
,
,
,
Be state variable.
3. three-dimensional chaos system according to claim 1 and device is characterized in that: the described first electric capacity (C
1), the second electric capacity (C
2), the 3rd electric capacity (C
3) capacitance equate,
And by regulating the capacitance of each electric capacity simultaneously, can adjust described three state variables of chaos system
,
,
Frequency of oscillation.
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105897397A (en) * | 2016-06-06 | 2016-08-24 | 南京信息工程大学 | Chaotic circuit capable of realizing amplitude-frequency control by time constant |
| CN108365946A (en) * | 2018-01-31 | 2018-08-03 | 国网河南省电力公司潢川县供电公司 | A kind of energy internet communication security system and method based on chaos system array |
| CN116054786A (en) * | 2023-03-28 | 2023-05-02 | 南京信息工程大学 | Simplified Jerk-like amplitude-adjustable frequency-modulated chaotic oscillator and regulation and control method thereof |
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| CN102930762A (en) * | 2012-11-19 | 2013-02-13 | 湖南大学 | Three-dimensional chaotic circuit |
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| CN102930762A (en) * | 2012-11-19 | 2013-02-13 | 湖南大学 | Three-dimensional chaotic circuit |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105897397A (en) * | 2016-06-06 | 2016-08-24 | 南京信息工程大学 | Chaotic circuit capable of realizing amplitude-frequency control by time constant |
| CN105897397B (en) * | 2016-06-06 | 2019-01-08 | 南京信息工程大学 | The chaos circuit of pot life constant realization amplitude-frequency control |
| CN108365946A (en) * | 2018-01-31 | 2018-08-03 | 国网河南省电力公司潢川县供电公司 | A kind of energy internet communication security system and method based on chaos system array |
| CN116054786A (en) * | 2023-03-28 | 2023-05-02 | 南京信息工程大学 | Simplified Jerk-like amplitude-adjustable frequency-modulated chaotic oscillator and regulation and control method thereof |
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Application publication date: 20130703 |