CN103188072A - Improved four-dimensional chaotic system and device - Google Patents
Improved four-dimensional chaotic system and device Download PDFInfo
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- CN103188072A CN103188072A CN2013101148904A CN201310114890A CN103188072A CN 103188072 A CN103188072 A CN 103188072A CN 2013101148904 A CN2013101148904 A CN 2013101148904A CN 201310114890 A CN201310114890 A CN 201310114890A CN 103188072 A CN103188072 A CN 103188072A
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Abstract
The invention relates to an improved four-dimensional chaotic system and device. The four-dimensional chaotic system is proposed by improving a three-dimensional chaotic system, and a chaotic attractor is in a four-wing form. In the presence of a positive Lyapunov exponent, the chaotic system has complex dynamic behaviors and some basic dynamic characteristics of the system are numerically simulated and analyzed. The system comprises an inverse proportion circuit, a first integrating circuit, a second integrating circuit, a third integrating circuit, a fourth integrating circuit, a first operational amplifier, a second operational amplifier, a third operational amplifier and a fourth operational amplifier; and the output ends of the integrating circuits sequentially output four state variables x(x1), y(x2), z(x3) and (x4) of the chaotic system. The chaotic attractor of the improved four-dimensional chaotic system is in a four-wing form; the circuits are easy to realize; and the system 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 an improved four dimensional chaos system and device thereof, 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 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 chaos system has caused people's attention.
Based on a three-dimensional chaos system, the present invention proposes an improved four dimensional chaos system, 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 four dimensional chaos system is carried out circuit and realize.
Summary of the invention
Technical problem to be solved by this invention provides an improved four dimensional chaos system and device thereof.
In order to solve the problems of the technologies described above, the invention provides an improved four dimensional chaos system and device thereof, it comprises: reverse ratio circuit, first integral circuit, second integral circuit, third integral circuit and the 4th integrating circuit circuit; First amplifier, second amplifier, the 3rd amplifier and four high guaily unit.The output of integrating circuit export successively one of four states variable x (x1), y (x2) as chaos system, z (x3),
(x4), improved four dimensional chaos system, its chaos attractor show four wing formulas.
The corresponding partial differential equation of three-dimensional chaos system of above-mentioned institute foundation are:
Wherein
Be positive constant.
System (1) is improved, and the four dimensional chaos system after can improving is:
Wherein,
,
Be positive constant.And further this four dimensional chaos system is carried out circuit and realize.
Effect of the present invention and effect
(1) the present invention has realized improved four dimensional chaos system, wherein a parameter according to a three-dimensional chaos system
(2) hardware circuit of employing chaos system of the present invention, verified that this four dimensional chaos system output signal has bigger dynamic range, in addition, reduce the capacitance in the chaos 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 an improved four dimensional chaos system and device thereof, 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 be three-dimensional chaos system two dimension and three-dimensional phasor (a) (x, y) (b) (x, z) (c) (z, y) (d) (x, y, z).
Fig. 2 for the two dimension of the four dimensional chaos system after improving and three-dimensional phasor (a) (x, y) (b) (x, z) (c) (z, y) (d) (x, y, z).
Fig. 3 is chaos system
Different initial value responses..
Fig. 4 is chaos system Poincar é mapping, and the cross section is y0=0 (c) z0=0 (d) w0=0 of (a) x0=0(b).
Fig. 5 is that chaos system is with parameter
Change bifurcation graphs.
Fig. 6 is that chaos system is with parameter
Change the Lyapunov exponential spectrum.
Fig. 7 is the four dimensional chaos system circuit theory diagrams.
Embodiment
By a three-dimensional chaos system and its apparatus, its Mathematical Modeling is described as
Wherein
Be positive constant,
,
,
Be state variable.The two dimension of its track and three-dimensional phasor are as shown in Figure 1.
System (1) is improved, and the four dimensional chaos system after can improving is:
Wherein,
,
Be positive constant.As parameter a=7; B=25; C=1; D=11; E=1; During f=0.5, its two dimension and three-dimensional phasor are as shown in Figure 2.
1 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:
Work as a=7; B=25; C=1; D=11; E=1; During f=0.5, there are four balance points in system.Locate at balance point (0,0,0,0), system (2) is carried out linearisation, its Jacobian matrix is:
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, the state status of other balance point as can be known.
Because
=
(6)
As seen, when
The time, each volume element of system's path is retracted to zero.
1.2 initial value sensitivity, Poincar é mapping.
As parameter a=7; B=25; C=1; D=11; E=1; During f=0.5, 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 3, as can be seen from Figure 3, can find at 4.5s, 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. 4 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) four Lyapunonov index of system and be respectively:
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. 5, Fig. 6 be respectively system along with
The bifurcation graphs and the Lyapunov exponential spectrum that change.
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 LM741, is to carry out plus and minus calculation, and analog multiplier adopts AD633 to realize, is the nonlinear terms of finishing in the system.The circuit theory diagrams of chaos system proposed by the invention as shown in Figure 7.
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. improved four dimensional chaos system, its feature comprises: oppositely ratio circuit, first integral circuit, second integral circuit, third integral circuit and the 4th integrating circuit circuit; First amplifier, second amplifier, the 3rd amplifier and four high guaily unit; The output of integrating circuit export successively one of four states variable x (x1), y (x2) as chaos system, z (x3),
(x4), improved four dimensional chaos system, its chaos attractor show four wing formulas.
2. according to four dimensional chaos system and device thereof after the described improvement of claim 1, it is characterized in that the corresponding partial differential equation of described three-dimensional chaos system are:
Wherein
Be positive constant; System (1) is improved, and the four dimensional chaos system after can improving is:
Wherein,
,
Be positive constant.
3. improved four dimensional chaos system according to claim 1 and device is characterized in that: the capacitance of described first electric capacity (C1), second electric capacity (C2), the 3rd electric capacity (C3), the 4th electric capacity (C4) equates,
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 (7)
| 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 |
| CN103441837A (en) * | 2013-09-07 | 2013-12-11 | 王少夫 | Four-dimensional chaotic system with constant lyapunov exponent |
| CN105071924A (en) * | 2015-09-01 | 2015-11-18 | 王宏国 | Four-wing attractor chaotic system with three equilibrium points containing absolute values, and circuit |
| CN105227291A (en) * | 2015-09-01 | 2016-01-06 | 王宏国 | A kind of three-dimensional four-winged chaotic attractor continuous chaotic system and circuit |
| CN110896347A (en) * | 2019-12-13 | 2020-03-20 | 哈尔滨工程大学 | Multi-stability chaotic system with discrete bifurcation graph |
| CN111211885A (en) * | 2019-12-19 | 2020-05-29 | 哈尔滨工程大学 | Multi-stability chaotic system with impulse function form Lyapunov exponent |
| 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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Cited By (9)
| 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 |
| CN103441837A (en) * | 2013-09-07 | 2013-12-11 | 王少夫 | Four-dimensional chaotic system with constant lyapunov exponent |
| CN105071924A (en) * | 2015-09-01 | 2015-11-18 | 王宏国 | Four-wing attractor chaotic system with three equilibrium points containing absolute values, and circuit |
| CN105227291A (en) * | 2015-09-01 | 2016-01-06 | 王宏国 | A kind of three-dimensional four-winged chaotic attractor continuous chaotic system and circuit |
| CN110896347A (en) * | 2019-12-13 | 2020-03-20 | 哈尔滨工程大学 | Multi-stability chaotic system with discrete bifurcation graph |
| CN110896347B (en) * | 2019-12-13 | 2024-02-09 | 哈尔滨工程大学 | Multi-stability chaotic system with discrete bifurcation diagram |
| CN111211885A (en) * | 2019-12-19 | 2020-05-29 | 哈尔滨工程大学 | Multi-stability chaotic system with impulse function form Lyapunov exponent |
| 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 |