CN103441837A - Four-dimensional chaotic system with constant lyapunov exponent - Google Patents
Four-dimensional chaotic system with constant lyapunov exponent Download PDFInfo
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Abstract
The invention relates to a three-dimensional chaotic system, and provides a novel four-dimensional chaotic system via introducing a partial differential equation. With a numerical simulation and a power spectrum analysis, chaotic behaviors of the system are researched. The analysis reveals that an attractor with a more complicated topological structure can be generated by improving the system, and the attractor has a constant lyapunov exponent characteristic. A chaotic signal with the constant lyapunov exponent can be applied to secret communication so that security of information encryption is enhanced. The four-dimensional chaotic system with the constant lyapunov exponent has wide application prospects and important application values in the fields of radar, secret communication, electronic countermeasures, etc.
Description
Technical field
The present invention relates to a four dimensional chaos system with permanent Liapunov exponent, 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.
The present invention, on the basis of a three-dimensional chaotic system, has proposed an improved four dimensional chaos system, by numerical simulation and power spectrumanalysis, has studied the chaotic behavior of this system.The analysis showed that improved system can produce the more attractor of complex topology structure, and there is constant Liapunov exponent characteristic.
Summary of the invention
Technical problem to be solved by this invention is to provide a four dimensional chaos system with permanent Liapunov exponent.
In order to solve the problems of the technologies described above, the invention provides a four dimensional chaos system, it comprises: according to a three-dimensional chaotic system, by introducing partial differential equation, a new four dimensional chaos system has been proposed, by numerical simulation and power spectrumanalysis, studied the chaotic behavior of this system.The analysis showed that improved system can produce the attractor of complex topology structure more and have constant Liapunov exponent, can change system parameters, can make system produce complicated non-linear chaotic dynamics behavior.
Described three-dimensional chaotic system institute corresponding equation is:
Wherein
for state variable, parameter
for arithmetic number.
On the base of above-mentioned three-dimensional chaotic system goes out, the four dimensional chaos system institute corresponding equation after described improvement is:
effect of the present invention and effect
(1) four dimensional chaos system that provides to have permanent Liapunov exponent has been provided in the present invention, wherein, and parameter
.
(2) adopt the attractor structure of four dimensional chaos system complexity of the present invention, realized the larger dynamic range that has of chaotic signal output, and there is constant Liapunov exponent characteristic.Indicate that it is at radar, secure communication, the fields such as the electronic countermeasures value that has a wide range of applications.
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 is three-dimensional chaotic system two dimension and three-dimensional phase diagram (a)
; (b)
3; (c)
.
Fig. 2 is four dimensional chaos system two dimension and three-dimensional phase diagram (a)
; (b)
3; (c)
; (d)
.
Fig. 3 is four dimensional chaos system temporal evolution Lyapunov exponential spectrum.
Embodiment
The non-linear three-dimensional chaos system is:
Wherein
for state variable, parameter
for arithmetic number.When parameter a=3.8; B=2.5; C=7 o'clock, the two dimension of this system and three-dimensional phase diagram are as shown in Fig. 1 (a), (b), (c)
On the basis of system (1), improved, by introducing gamma controller u, thereby the kinetics equation that constructs an improved four dimensional chaos system is:
When parameter a=3.8; B=2.5; C=7;
=5;
=40 o'clock, its two-dimentional phasor was as shown in Fig. 2 (a), (b), (c), (d).
The dynamic behavior of 1 system
1.1 system Dissipative Analysis
For system equation (2), have
(3)
Therefore, it is the dissipativeness system, and with index speed
convergence.Therefore work as
, each the small size unit that comprises the system path is retracted to 0 with index speed, and its asymptotic dynamic behavior can be fixed on an attractor, and the existence of attractor has been described.
1.2 system balancing point and Lee Lyapunov index
When (2) formula left end is zero, unique balance point that can go out to calculate system (2) is
(0,0,0,0), carry out linearisation at the balance point place to system, can obtain corresponding Jacobian matrix to be:
When parameter a=3.8; B=2.5; C=7;
=5;
=40 o'clock, the characteristic value that can obtain matrix was
-3.8,
2.5,
-10.6101,
-26.3899, because
for positive root, according to the Routh-Hurwitz condition, balance point is stable needs four characteristic roots entirely for negative or real part is negative compound radical, and therefore, this balance point is unsettled saddle point.As shown in Figure 3, as can be seen from Figure 3, this four-dimensional system has constant Liapunov exponent to the time dependent curve of its Lyapunov exponential spectrum.
The present invention, by a three-dimensional chaotic system is improved, utilizes the analytical method of nonlinear kinetics, and after find improving, four-dimensional system can produce the attractor of complex topology structure more and have permanent Liapunov exponent characteristic.This chaotic signal with permanent Liapunov exponent characteristic can be for secure communication to improve the fail safe of information encryption.Indicate that it is at radar, secure communication, the fields such as the electronic countermeasures value that has a wide range of applications.
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 (4)
1. a four dimensional chaos system with permanent Liapunov exponent, its feature comprises: according to a three-dimensional chaotic system, by introducing partial differential equation, a new four dimensional chaos system has been proposed, by numerical simulation and power spectrumanalysis, studied the chaotic behavior of this system, the analysis showed that improved system can produce the more attractor of complex topology structure, and there is constant Liapunov exponent characteristic.
2. three-dimensional chaotic system according to claim 1, is characterized in that, described three-dimensional chaotic system institute corresponding equation is:
Wherein
for state variable, parameter
for arithmetic number.
3. four dimensional chaos system according to claim 1, it is characterized in that: the four dimensional chaos system institute corresponding equation after described improvement is:
Wherein, parameter
for arithmetic number.
4. four dimensional chaos system according to claim 1, is characterized in that: the feature with permanent Liapunov exponent.
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104092532A (en) * | 2014-08-03 | 2014-10-08 | 王忠林 | Balance-point-free hyper-chaos system based on three-dimensional chaos system, and analogue circuit |
| CN107623567A (en) * | 2017-09-30 | 2018-01-23 | 湖南科技大学 | A Chaotic Circuit with Constant Lyapunov Exponent Spectrum |
| CN109936436A (en) * | 2019-03-20 | 2019-06-25 | 湖南理工学院 | A Robust Chaos Mapping System for Image Data Encryption and Its Complexity Optimal Control Method |
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| CN103138915A (en) * | 2013-03-22 | 2013-06-05 | 王少夫 | Four-dimensional hyper-chaos system with fast attractors and slow attractors |
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| CN102843230A (en) * | 2012-09-17 | 2012-12-26 | 郑州轻工业学院 | Mathematical model of four-dimensional autonomous hyper-chaos system and achieving circuit of mathematical model |
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Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104092532A (en) * | 2014-08-03 | 2014-10-08 | 王忠林 | Balance-point-free hyper-chaos system based on three-dimensional chaos system, and analogue circuit |
| CN104092532B (en) * | 2014-08-03 | 2015-05-20 | 徐振峰 | Balance-point-free hyper-chaos system based on three-dimensional chaos system, and analogue circuit |
| CN107623567A (en) * | 2017-09-30 | 2018-01-23 | 湖南科技大学 | A Chaotic Circuit with Constant Lyapunov Exponent Spectrum |
| CN109936436A (en) * | 2019-03-20 | 2019-06-25 | 湖南理工学院 | A Robust Chaos Mapping System for Image Data Encryption and Its Complexity Optimal Control Method |
| CN109936436B (en) * | 2019-03-20 | 2022-06-21 | 湘潭大学 | Robust chaotic mapping system for image data encryption and complexity optimization control method thereof |
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Application publication date: 20131211 |