CN109495239A - A kind of double generalized synchronization methods for the chaos system that the risk management based on self adaptive control is unknown - Google Patents
A kind of double generalized synchronization methods for the chaos system that the risk management based on self adaptive control is unknown Download PDFInfo
- Publication number
- CN109495239A CN109495239A CN201811306436.8A CN201811306436A CN109495239A CN 109495239 A CN109495239 A CN 109495239A CN 201811306436 A CN201811306436 A CN 201811306436A CN 109495239 A CN109495239 A CN 109495239A
- Authority
- CN
- China
- Prior art keywords
- chaos
- response
- parameter
- unknown
- double
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Pending
Links
Classifications
-
- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/001—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols using chaotic signals
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
- G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
- G05B13/0205—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric not using a model or a simulator of the controlled system
- G05B13/024—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric not using a model or a simulator of the controlled system in which a parameter or coefficient is automatically adjusted to optimise the performance
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
- G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
- G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
- G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
Abstract
The present invention provides a kind of double generalized synchronization methods of chaos system that the risk management based on self adaptive control is unknown, specifically it is based on lyapunov Theory of Stability, two chaos systems are given first realizes the bisynchronous adequate condition of broad sense, then by the way that suitable controller and parameter update law is arranged, it ensure that chaos system can be asymptotically stable double synchronous；The scheme proposed can also accurately identify the unknown parameter of system during realizing Synchronization of Chaotic Systems.The generalized synchronization of chaos system has been extended to dual system from a single chaos system by the present invention, due to combining generalized synchronization and bisynchronous advantage, therefore compared with traditional generalized synchronization, the reliability and security during secret communication is greatly improved.
Description
Technical field
The invention belongs to fields of communication technology, are related to a kind of chaos system that the risk management based on self adaptive control is unknown
Double generalized synchronization methods of system.
Background technique
Chaology originates from early 20th century, is developed in the 1970s, now still in the ascendant.In chaos
In theoretical developments, various chaos phenomenons are constantly found, and various analysis methods and criterion are also put forward one after another.Chaology by
In its scientific and naturality combination, it is widely studied and applied in science, engineering and mathematical region.In recent years, it mixes
The research of ignorant synchronization has become a hot spot.Chaotic Synchronous is in physics and engineering system, secret communication, chemical reaction, life
Object medicine, information processing, social science and other many fields have a wide range of applications.
Chaos system has high susceptibility, pseudo-randomness and long-term unpredictability to initial value and control parameter, this
So that chaotic signal has natural concealment, it is suitable as the carrier of secret communication.In general, chaotic secret communication is
In transmitting terminal, information is illustrated as having the waveform or code stream of chaotic characteristic；It is extensive from the signal received in receiving end
It appears again correct information.Chaotic secret communication require transmitting terminal and receiving end Synchronization of Chaotic Systems, therefore, Chaotic Synchronous at
For the critical issue and important theoretical basis of chaotic secret communication.
So-called Chaotic Synchronous is referred to for two chaos systems from different primary condition, with pushing away for time
It moves, their track can gradually reach unanimity, and such as identical, completely opposite or two states keep the relationship of certain function.
Therefore the conventional method of research chaos system is the error for defining chaos system first, is then synchronized for error system design
Controller makes error system asymptotically stability.In addition, in many cases, the parameter of chaos system be it is uncertain, need
The uncertain parameter of system is adjusted in synchronizing process, this wheel synchronization type is referred to as adaptive synchronicity, this is also current
Realize one of method most general used by Chaotic Synchronous.
Traditional Chaotic Synchronous mode, including fully synchronized (track of response system and drive system is completely the same), phase
Bit synchronization (response system and the track of drive system are completely opposite), Projective Synchronization (track of response system and drive system it
Between meet certain proportionate relationship) and generalized synchronization (between response system and the track of drive system meet a specific letter
Number relationship) etc..Currently, existing Synchronization of Chaotic Systems problem encountered has the following aspects:
(1) even more complicated generalized synchronization in the existing method of synchronization, when being applied in secret communication, quilt
Decoding the probability come out also can be very big.Although currently existing multistage many communication schemes such as synchronizes, for
It is proposed that more reliable and safer Chaotic Synchronous scheme is still extremely urgent；
(2) in view of under actual physical environment and engineering background, the parameter of some systems will not be accurate in advance
Ground is known, therefore the synchronization for studying the chaos system with uncertain parameter has very big realistic meaning；
(3) existing a large amount of Chaotic Synchronous work is primarily directed in a drive system and a response system, individually
For chaos system when being applied to synchronous, the difficulty when information content and decoding that can carry is all relatively small, how to improve this
A little performances are also the problem of being considered of the invention.
Therefore, the present invention considers the safety and reliability of secret communication, proposes more valuable Chaotic Synchronous side
Case.
Summary of the invention
It is a kind of based on the complete of self adaptive control it is an object of the invention in view of the problems of the existing technology, provide
Double generalized synchronization methods of the chaos system of unknown parameters.
For this purpose, the present invention adopts the following technical scheme:
A kind of double generalized synchronization methods for the chaos system that the risk management based on self adaptive control is unknown, including it is as follows
Step:
(1) for the chaos drive system of following two unknown parameters:
Wherein, x=(x1,x2,...,xm)T∈RmWith y=(y1,y2,...,yn)T∈RnRespectively two drive systems
State vector, and f (x) ∈ RmWith g (y) ∈ RnThe continuous vector function of respectively m peacekeeping n dimension, F (x) ∈ Rm×pFor function square
Battle array, α ∈ RpFor unknown parameter vector；Similarly, G (y) ∈ Rn×lFor Jacobian matrix, β ∈ RlFor unknown parameter vector；It is above-mentioned
The response system of drive system is as follows:
Wherein, X=(X1,X2,...,Xm)T∈RmWith Y=(Y1,Y2,...,Yn)T∈RnRespectively two response systems
State vector,WithRespectively indicate the estimated value of unknown parameter α and β, u1=(u11,u12,...,u1m)T∈Rm
And u2=(u21,u22,...,u2n)T∈RnFor controller；For above-mentioned drive system and response system, a given vector reflects
It penetrates The vector is mapped as a continuous differentiation function and drive system and response system can be made to reach same
Step meets:
Wherein | | | | it is European norm；In the present invention, vector mappingContinuous differential letter can be arbitrary
Number, such as SIN function, cosine function, square；
(2) step (1) drive system is rewritten as such as ordering system form:
Wherein, ε=(x y)T, φ (ε)=(f (x) g (y))T,Λ=(α β)T；It is similar
Step (1) response system is rewritten as such as ordering system form by ground:
Wherein, η=(X Y)T, φ (η)=(f (X) g (Y))T,
(3) definition of the bisynchronous margin of error of chaos system is given below:
Transmitting terminal is set by drive system, sets receiving end for response system, in transmitting terminal, two drive systems
Linear coupling are as follows:
In receiving end, the linear coupling of two response systems are as follows:
δ2=AX+BY=(a1,...,am)(X1,...,Xm)+(b1,...,bn)(Y1,...,Yn)
=(a1,...,am,b1,...,bn)(X1,...,Xm,Y1,...,Yn)=C η
Wherein A=(a1,a2,...,am) and B=(b1,b2,...,bn) it is coupling parameter, C=(A, B), thus double synchronizations
The margin of error are as follows: es=Ce, whereinBy double synchronous error esInject response system in, when response system with
When drive system reaches synchronous, then error e will become 0, not have signal injection in response system at this time, namely realize double
Generalized synchronization；Correspondingly, error dynamics system are as follows:
In formula,For mappingJacobian matrix.Present invention aim to design a controller u, its energy
Enough in the case where system parameter is totally unknown, it is come generalized synchronization drive system (4) and response by adaptive control technology
The state of system (5).Corresponding parameter update law is proposed simultaneously, to identify system unknown parameter；
(4) design of adaptive controller and the selection of parameter update law
Controller is selected as:
Parameter update law is selected as:
Wherein,It is the estimation to unknown parameter Λ, Em+nIt is the unit column vector of a m+n row；In above controller
Under the action of parameter update law, double synchronous error esBecome 0, there is no signal injection, drive system in response system at this time
Reach Global Asymptotic generalized synchronization with response system；Meanwhile it enablingThen unknown parameter Λ can pass throughIt is estimated
Out, detailed process is as follows:
Formula (7) substitution formula (6) is obtained:
Construct Lyapunov function are as follows:
In formula, V (t) >=0, according to the controller and parameter update law that formula (7) and formula (8) are given, to formula (10) derivation
:
Wherein P=Em+nC will such as realize that double broad sense of above-mentioned chaos system are same according to Lyapunov Theory of Stability
Step it is necessary to selecting suitable matrix P to come so thatAllowFor negative definite, as long as therefore select suitable matrix A=
(a1,a2,...,am) and B=(b1,b2,...,bn) make matrix P for negative definite matrix, the double of above-mentioned chaos system can be realized
Generalized synchronization.
The beneficial effects of the present invention are:
(1) generalized synchronization of chaos system dual system has been extended into from a single chaos system, due to combining
Generalized synchronization and bisynchronous advantage, therefore compared with traditional generalized synchronization, during greatly improving secret communication
Reliability and security；
(2) scheme proposed is suitable for the chaos system of any two pairs not same orders, and corresponding processing means are based on Lee
Ya Punuofu Theory of Stability selects coupling parameter, and dual system carried out specific linear coupling；
(3) controller and parameter update law proposed, it is with ensure that chaos system energy Asymptotic Stability double synchronous；It is mentioned
Scheme out can also accurately identify the unknown parameter of system during realizing Synchronization of Chaotic Systems.
Detailed description of the invention
Fig. 1 is the schematic illustration of traditional chaos system generalized synchronization；
Fig. 2 is the schematic illustration of the double generalized synchronizations of chaos system of the present invention；
Fig. 3 is Lorenz chaos system phasor；
Fig. 4 is Chen chaos system phasor；
Fig. 5 a is drive system and response system x3And x1State trajectory；
Fig. 5 b is drive system and response system y3With-y1State trajectory；
Fig. 5 c is drive system and response system z3And y1z1State trajectory；
Fig. 5 d is drive system and response system x4WithState trajectory；
Fig. 5 e is drive system and response system y4And y1+y2State trajectory；
Fig. 5 f is drive system and response system z4And z1z2State trajectory；
Fig. 6 a e between drive system and response system1、e2And e3State error；
Fig. 6 b e between drive system and response system4、e5And e6State error；
Fig. 7 a is the identification effect figure of unknown parameter a, b and c；
Fig. 7 b is the identification effect figure of unknown parameter d, e and f.
Specific embodiment
A pair Lorenz and Chen chaos system is chosen below illustrates that the present invention program's is effective as drive system
Property, wherein the phasor of Lorenz and Chen chaos system is shown in Fig. 3 and Fig. 4 respectively.Specific step is as follows for its double generalized synchronization:
(1) nonlinear differential equation of Lorenz and Chen chaos system is as follows:
Drive (1): Lorenz system
Drive (2): Chen system
Corresponding response system is as follows:
Response(1):
Response(2):
Wherein, a, b, c, d, e, f are unknown parameters,It is the estimation to unknown parameter, u=(u1,
u2,u3,u4,u5,u6)TIt is controller to be determined.Below by designing efficient adaptive controller and parameter adaptive
Rule, to realize double generalized synchronizations of the Lorenz and Chen system under totally unknown parameter.Regard formula (12) and (13) as one
A drive system, formula (14) and (15) regard a response system as, then following matrix form can be expressed as:
In numerical simulation, definition mapping
Then
Controller is selected as:
Parameter update law is selected as:
Controller and parameter update law are constructed by above formula, then
Wherein, Double synchronous error amounts after drive system and response system linear coupling are es=a1e1+
a2e2+a3e3+b1e4+b2e5+b3e6。
Numerical simulation has been carried out to double stationary problems of Lorenz and Chen system below.By the parameter of Lorenz system
It is set as: a=10, b=8/3, c=28；The parameter of Chen system is set as: d=35, e=3, f=35.It can be with from Fig. 3 and Fig. 4
Find out, the two systems all enter chaos state.In addition, coupling parameter is set as: ai=(- 1, -2, -3), bi=(- 4 ,-
5, -6), wherein i=1,2,3, then P is negative definite matrix.The primary condition of drive system (12) and (13) are as follows: x1(0)=1, y1
(0)=5, z1(0)=10, x2(0)=1.21, y2And z (0)=302(0)=0.05.Response system (14) and (15) it is initial
Condition are as follows: x3(0)=0.53, y3(0)=1.02, z3(0)=28.3, x4(0)=0.78, y4(0)=22.2, z4(0)=
6.32.So the initial error of error system is are as follows: e1(0)=- 0.47, e2(0)=- 3.98, e3(0)=18.3, e4(0)
=-0.43, e5(0)=- 7.8, e6(0)=6.27.The estimated value of the initial parameter at drive system end is chosen as follows: Fig. 5 a to Fig. 5 f gives driving system
It unites (12), the state trajectory of (13) and response system (14), (15).It can be seen that after 10s, drive system and sound
The curve of system is answered to essentially coincide.Fig. 6 a and Fig. 6 b give error signal e1, e2, e3And e4, e5, e6State trajectory, can be with
See, over time, error signal finally converges to 0.Show in the case where unknown parameters, drive system and sound
Answering realizes synchronization between system.Fig. 7 a and Fig. 7 b give the estimation of unknown parameter, show as t → ∞, unknown parameter
Estimated valueSuccessively converge to a=10, b=8/3, c=28, d=35, e=
3, f=28.It is achieved that the double generalized synchronizations and parameter identification of chaos system.
Claims (1)
1. a kind of double generalized synchronization methods for the chaos system that the risk management based on self adaptive control is unknown, which is characterized in that
Include the following steps:
(1) for the chaos drive system of following two unknown parameters:
Wherein, x=(x1,x2,...,xm)T∈RmWith y=(y1,y2,...,yn)T∈RnThe state arrow of respectively two drive systems
Amount, and f (x) ∈ RmWith g (y) ∈ RnThe continuous vector function of respectively m peacekeeping n dimension, F (x) ∈ Rm×pFor Jacobian matrix, α ∈ Rp
For unknown parameter vector；Similarly, G (y) ∈ Rn×lFor Jacobian matrix, β ∈ RlFor unknown parameter vector；Above-mentioned drive system
Response system is as follows:
Wherein, X=(X1,X2,...,Xm)T∈RmWith Y=(Y1,Y2,...,Yn)T∈RnThe state arrow of respectively two response systems
Amount,WithRespectively indicate the estimated value of unknown parameter α and β, u1=(u11,u12,...,u1m)T∈RmAnd u2=
(u21,u22,...,u2n)T∈RnFor controller；For above-mentioned drive system and response system, a vector mapping is given The vector is mapped as a continuous differentiation function and drive system can be made to reach synchronous with response system, i.e.,
Meet:
Wherein | | | | it is European norm；
(2) step (1) drive system is rewritten as such as ordering system form:
Wherein, ε=(x y)T, φ (ε)=(f (x) g (y))T,Λ=(α β)T；Similarly, will
Step (1) response system is rewritten as such as ordering system form:
Wherein, η=(X Y)T, φ (η)=(f (X) g (Y))T, U=(u1
u2)T；
(3) definition of the bisynchronous margin of error of chaos system is given below:
Set transmitting terminal for drive system, set receiving end for response system, in transmitting terminal, two drive systems it is linear
Coupling are as follows:
In receiving end, the linear coupling of two response systems are as follows:
δ2=AX+BY=(a1,...,am)(X1,...,Xm)+(b1,...,bn)(Y1,...,Yn)
=(a1,...,am,b1,...,bn)(X1,...,Xm,Y1,...,Yn)=C η
Wherein A=(a1,a2,...,am) and B=(b1,b2,...,bn) it is coupling parameter, C=(A, B), thus bisynchronous mistake
Residual quantity are as follows: es=Ce, whereinBy double synchronous error amount esIt injects in response system, when response system and drives
When dynamic system reaches synchronous, then error e will become 0, not have signal injection in response system at this time, namely realize double broad sense
It is synchronous；Correspondingly, error dynamics system are as follows:
In formula,For mappingJacobian matrix；
(4) design of adaptive controller and the selection of parameter update law
Controller is selected as:
Parameter update law is selected as:
Wherein,It is the estimation to unknown parameter Λ, Em+nIt is the unit column vector of a m+n row；In above controller and parameter
Under the action of adaptive law, double synchronous error esBecome 0, there is no signal injection, drive system and response in response system at this time
System reaches Global Asymptotic generalized synchronization；Meanwhile it enablingThen unknown parameter Λ can pass throughIt is estimated, specifically
Process is as follows:
Formula (7) substitution formula (6) is obtained:
Construct Lyapunov function are as follows:
In formula, V (t) >=0 obtains formula (10) derivation according to the controller and parameter update law that formula (7) and formula (8) are given:
Wherein P=Em+nC will such as realize double generalized synchronizations of above-mentioned chaos system, just according to Lyapunov Theory of Stability
Select suitable matrix P to come so thatAllowFor negative definite, as long as therefore selecting suitable matrix A=(a1,
a2,...,am) and B=(b1,b2,...,bn) make matrix P for negative definite matrix, double broad sense that above-mentioned chaos system can be realized are same
Step.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201811306436.8A CN109495239A (en) | 2018-11-05 | 2018-11-05 | A kind of double generalized synchronization methods for the chaos system that the risk management based on self adaptive control is unknown |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201811306436.8A CN109495239A (en) | 2018-11-05 | 2018-11-05 | A kind of double generalized synchronization methods for the chaos system that the risk management based on self adaptive control is unknown |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CN109495239A true CN109495239A (en) | 2019-03-19 |
Family
ID=65693682
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201811306436.8A Pending CN109495239A (en) | 2018-11-05 | 2018-11-05 | A kind of double generalized synchronization methods for the chaos system that the risk management based on self adaptive control is unknown |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN109495239A (en) |
Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US6212239B1 (en) * | 1998-01-09 | 2001-04-03 | Scott T. Hayes | Chaotic dynamics based apparatus and method for tracking through dropouts in symbolic dynamics digital communication signals |
| CN105450388A (en) * | 2015-12-08 | 2016-03-30 | 郑州轻工业学院 | Compound-combined type synchronization method based on five chaotic systems |
| CN105785763A (en) * | 2016-03-24 | 2016-07-20 | 郑州轻工业学院 | Finite time combination synchronization sliding mode control method for composite chaotic systems with uncertain parameters |
| CN206237417U (en) * | 2016-08-01 | 2017-06-09 | 西南大学 | A kind of secret signalling based on electro-optical feedback ring laser |
| CN107086903A (en) * | 2017-06-16 | 2017-08-22 | 郑州轻工业学院 | The simplification synchronous method of driving chaos system and response chaos system is realized using two inputs |
-
2018
- 2018-11-05 CN CN201811306436.8A patent/CN109495239A/en active Pending
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US6212239B1 (en) * | 1998-01-09 | 2001-04-03 | Scott T. Hayes | Chaotic dynamics based apparatus and method for tracking through dropouts in symbolic dynamics digital communication signals |
| CN105450388A (en) * | 2015-12-08 | 2016-03-30 | 郑州轻工业学院 | Compound-combined type synchronization method based on five chaotic systems |
| CN105785763A (en) * | 2016-03-24 | 2016-07-20 | 郑州轻工业学院 | Finite time combination synchronization sliding mode control method for composite chaotic systems with uncertain parameters |
| CN206237417U (en) * | 2016-08-01 | 2017-06-09 | 西南大学 | A kind of secret signalling based on electro-optical feedback ring laser |
| CN107086903A (en) * | 2017-06-16 | 2017-08-22 | 郑州轻工业学院 | The simplification synchronous method of driving chaos system and response chaos system is realized using two inputs |
Non-Patent Citations (2)
| Title |
|---|
| HASSAN SALARIEH 等: "Dual synchronization of chaotic systems via time-varying gain proportional feedback", 《CHAOS,SOLITONS AND FRACTALS》 * |
| 方洁: "混沌修正函数投影同步控制与应用研究", 《中国博士学位论文全文数据库 基础科学辑》 * |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Nishimura et al. | Environmental viral genomes shed new light on virus-host interactions in the ocean | |
| Ardao | Assimilation and transformation of positivism in Latin America | |
| Al-Shayeb et al. | Borgs are giant extrachromosomal elements with the potential to augment methane oxidation | |
| CN111095210A (en) | Storing shared blockchain data based on error correction coding | |
| CN105227176A (en) | A kind of mixed type PUF circuit | |
| CN103886235A (en) | Face image biological key generating method | |
| CN110035063A (en) | A kind of intelligent appliance data ciphering method neural network based | |
| Wang et al. | Non-Noether symmetries of Hamiltonian systems with conformable fractional derivatives | |
| Fan et al. | Cramér moderate deviations for the elephant random walk | |
| CN106817217B (en) | A kind of dynamic password Encryption Algorithm of interrupted aging | |
| CN109495239A (en) | A kind of double generalized synchronization methods for the chaos system that the risk management based on self adaptive control is unknown | |
| Xi | Some further results for finite‐time stability of impulsive nonlinear systems | |
| Lei et al. | Evolution of the genetic code | |
| Lamosa et al. | Occurrence of 1-glyceryl-1-myo-inosityl phosphate in hyperthermophiles | |
| Hogan et al. | Evolution of IESs and scrambling in the actin I gene in hypotrichous ciliates | |
| Mullineaux et al. | The rolex and the hourglass: a simplified circadian clock in prochlorococcus? | |
| CN106548094A (en) | A kind of physics unclonable function circuit of utilization monostable timing deviation | |
| Dalmasso et al. | Complete genome sequence of the hyperthermophilic and piezophilic archeon Thermococcus piezophilus CDGST, able to grow under extreme hydrostatic pressures | |
| Gao et al. | Adaptive impulsive synchronization of uncertain delayed chaotic system with full unknown parameters via discrete‐time drive signals | |
| Mohamed et al. | Comparative transcriptomic analyses of Chromera and Symbiodiniaceae | |
| Wang et al. | The birhythmicity increases the diversity of p53 oscillation induced by DNA damage | |
| Valdespino-Díaz et al. | Regulatory evolution of the phoH ancestral gene in Salmonella enterica serovar Typhimurium | |
| CN103034814B (en) | A kind of data access method | |
| Zhang et al. | TQ relations for the integrable two-species asymmetric simple exclusion process with open boundaries | |
| Banks | Magnetosphere, ionosphere and atmosphere interactions |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| RJ01 | Rejection of invention patent application after publication |
Application publication date: 20190319 |
|
| RJ01 | Rejection of invention patent application after publication |