CN115412248B - Safe data transmission method based on precise time synchronization chaotic system - Google Patents
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Abstract
The invention discloses a safe data transmission method based on a precise time synchronization chaotic system, which designs a safe communication scheme with precise time synchronization convergence performance; by utilizing the direction symbol function and the proportion maintaining characteristic, all non-zero error states are converged to an expected value at the same time with high precision, and a unique limited time synchronization stable control strategy is designed; further, a fixed time synchronous stable control strategy is designed, wherein the upper bound of the synchronous stable time is irrelevant to the initial state of the error system; aiming at a typical Chua's oscillator and a Wang's hyper-chaotic system, a limited time synchronization sliding mode controller and a fixed time synchronization sliding mode controller are designed, and two chaotic encryption safety communication mechanisms with precise time synchronization performance are respectively provided based on the limited time synchronization sliding mode controller and the fixed time synchronization sliding mode controller. The invention has the characteristics of strong data security, high transmission precision, low system energy consumption and the like, and is suitable for being applied to various safety communication tasks of large data volume network transmission.
Description
Technical Field
The invention provides a safe data transmission method based on a precise time synchronization chaotic system, relates to a time synchronization and fixed time synchronization control algorithm of the chaotic system, is suitable for a safe communication task of large-data-volume network transmission, and belongs to the field of communication safety.
Background
Chaotic systems refer to a class of nonlinear dynamics systems that are deterministic but exhibit random irregular motion. Because of its unpredictability and uncertainty, chaotic systems are widely used in various fields, such as biology, information technology, and economics. The research of the chaotic system is also of great significance to the safety communication scheme. The chaotic signal generated by the chaotic system has high randomness and complexity, so that a chaotic signal source which can be applied to encryption and decryption processes is a key part for realizing chaotic communication. Generally, a chaotic secure communication system mainly comprises two synchronous chaotic systems, namely a transmitting system and a receiving system. The message signal is encrypted in the sender (also denoted as encryptor) and then decrypted at the receiver (also called decryptor). In addition, the synchronization of the chaotic system must be analyzed to investigate the problem of chaotic secure communication.
In recent years, intensive research into synchronous chaotic systems has become a hotspot in the field of secure communication. A typical zeiss oscillator is the most representative chaotic circuit and contains a nonlinear term described by a piecewise linear function. In addition, hyperchaotic systems with multiple positive lyapunov indices were first proposed in 1979. This means that the hyper-chaotic system can improve the safety of signal transmission by generating complex dynamic characteristics. Hyper-chaotic systems are widely used as a master system and a slave system in a safe transmission mechanism.
The data transmission accuracy can be improved by calculating the error convergence time of the two-party chaotic system, which requires a finite/fixed time synchronous control algorithm. A series of studies have shown that with a finite/fixed time control algorithm, all elements of the system can converge to an equilibrium state within a bounded settling time, either associated with or not associated with the initial state. Although the finite/fixed time control method has a certain advantage in implementing synchronization, in a specific practical task, all elements are generally required to converge to a desired value at the same time, and then the finite/fixed time stability of each system state in which each system state converges independently cannot meet the actual requirement. For example, secure data transmission strategies require that errors between encrypted and decrypted signals be closely synchronized to zero at the same time, otherwise packet losses and data transmission errors occur, ultimately resulting in reduced reliability and security.
Recent researches show that the system states of a general power system can synchronously achieve convergence at the same time, namely synchronous stability. In particular, this particular "time synchronization stability" may be expressed as a fusion between "limited time stability" and "proportional hold feature", but this is a special type of time synchronization stability.
In conclusion, the system error can be ensured to be synchronously and precisely converged to the original point in the safe data transmission method based on the chaotic synchronous system. However, it is desirable that not only the error can reach zero, but also the convergence path is shorter and the power consumption is lower, which is of great importance for secure data transmission.
Disclosure of Invention
The invention solves the technical problems: in order to overcome the defects of the existing synchronization algorithm and integrate time synchronization property, limited/fixed time control and sliding mode control methods in a secure data transmission method for encrypting by using a chaotic synchronization system, the secure data transmission method based on the precise time synchronization chaotic system is provided, so that the convergence process of the synchronous chaotic system is quicker and smoother, the convergence path is shorter, and the convergence energy consumption is lower, so that the safety and the accuracy of data transmission are improved.
The invention discloses a safe data transmission method based on a precise time synchronization chaotic system, which comprises the following steps:
S1: establishing a secret communication model based on a precise time synchronization chaotic system, wherein the secret communication model encrypts transmission data by utilizing a random chaotic signal extracted from the precise time synchronization chaotic system; the basic characteristics of the secret communication model are as follows: the sender and the receiver which participate in secret communication respectively have one chaotic system, precise time synchronization is realized between the two chaotic systems, a new system formed by the two chaotic systems is the precise time synchronization chaotic system, and after synchronization, the sender and the receiver extract completely consistent state vectors from the respective chaotic systems, so that reversible data encryption and decryption are realized;
S2: based on the precise time synchronization chaotic system provided in the step S1, a precise time synchronization stability method is provided, and the precise time synchronization chaotic system realizes the time synchronization requirement involved in the time synchronization process; the precise time synchronization stability method comprises time synchronization stability and fixed time synchronization stability, wherein the fixed time synchronization stability further improves the precision and efficiency in the synchronization process on the basis of the time synchronization stability;
s3: according to the method for time synchronization stability and fixed time synchronization stability provided in S2, in order to control the synchronization process of the precise time synchronization chaotic system to meet the requirements of time synchronization and fixed time synchronization, a unit direction vector is utilized to respectively design a sliding mode surface meeting the requirements of time synchronization and fixed time synchronization;
s4: based on the time synchronization and fixed time synchronization sliding mode surface designed in the step S3, aiming at the chaotic system in space laser communication, a control law meeting the requirements of time synchronization stability and fixed time synchronization stability is designed, and the control law is utilized to complete time synchronization in the chaotic system time synchronization process in a secret communication model.
Further, in the step S1,
When precise time synchronization is realized between two chaotic systems, namely, precise time synchronization is realized between two chaotic systems respectively belonging to a sender and a receiver, the formula of the secret communication model is as follows:
The encryption function E (-) and the decryption function D (-) are:
wherein m (t) is a source message signal, x (t) is a chaotic signal extracted from a chaotic system of the transmitter, For the transmitted encrypted signal, y (t) is the chaotic signal extracted from the own chaotic system by the receiver, and is/areFor decrypting the resulting message signal.
Further, in the step S2,
The time synchronization stability is achieved as follows:
(1) If the process of converging the controlled chaotic system to the original point meets the requirement of time synchronization stabilization, the process is required to meet the requirement of:
i, the chaotic system is stable in limited time, namely can converge to an original point in limited time;
Setting T: N 0 → {0} → (0, ++) as a stable time function of the chaotic system, wherein D 0 is an open neighborhood of the origin, satisfying the following:
where x i is the ith component of x;
In this condition, if N 0＝D0 =r, R is the real number domain, then it is called global time synchronization stable;
The fixed time synchronization stability method is described as follows:
(2) If the process of converging a controlled chaotic system to the original point meets the requirement of synchronous stabilization of fixed time, the process needs to meet the following requirements:
i, the chaotic system is globally time-synchronous and stable;
II, a constant T m irrelevant to the initial state of the chaotic system exists, and the following conditions are satisfied:
in the above formula, the chaotic system should satisfy the following general mathematical form, maintaining equilibrium at the origin:
where x is a time-dependent n-dimensional column vector, written as x (t), x (0) =x 0 denotes that the initial value at time 0 is x 0, f is a function mapped from n-dimensional vector space to n-dimensional vector space, Meaning that x derives time, the state vector of a chaotic system will remain motionless at 0 after reaching 0, and the process of moving from initial value to 0 is called convergence.
In step S3, a slip-form surface satisfying the time synchronization and the fixed time synchronization is designed by using the unit direction vector, respectively, as follows:
(1) The slip form surface designed to meet time synchronization is as follows:
Wherein, And/>The form of the sliding mode surface can be controlled for the sliding mode surface parameter, e is a state vector of a controlled chaotic system, and the formula only defines the derivative of the sliding mode surface s 1;
(2) The slip form surface designed to meet the fixed time synchronization is as follows:
Wherein, And/>For the slip plane parameters, the form of the slip plane can be controlled, e is the state vector of the controlled chaotic system, and the formula only defines the derivative of the slip plane s 2.
In equations (7) and (8), function sig n (·) finds the unit vectors that are parallel to each other for the variables in brackets. When the function has an exponent p, i.e. in the form of a power exponent, the function is expanded as follows:
Further, in the step S4, the control law time synchronization control law required by the synchronization stability and the fixed time synchronization stability is as follows, and the specific form is given here by using a typical zeiss oscillator and a Wang hyperchaotic system in a common chaotic system.
For a typical zeiss oscillator:
(1) The time synchronization control law of a typical zeiss oscillator is as follows:
wherein ζ 1,ξ2,α,α1 is an adjustable control parameter, s 3 is a slip-form surface defined by a derivative satisfying formula (7);
(2) The fixed time synchronization control law of a typical zeiss oscillator is as follows:
wherein ζ 3,ξ4,ξ5,ξ6,p,p1,g,g1 is an adjustable control parameter, s 4 is a slip-form surface defined by a derivative satisfying formula (8);
In formulas (10) and (11), e= [ e 2,e3]T ] is an error vector, which is obtained by subtracting state vectors of two chaotic systems contained in the precise time-synchronous chaotic system, and the error still forms the chaotic system, and the form is as follows:
F= [ F 2,F3]T ] in formulas (10) and (11) is a shorthand substitution of a complex variable, where F 2＝ζ(e1-e2+e3),F3＝-ζγe2;
X 1,x2,x3 in the formula (12) is a required state variable of a chaotic system which participates in a precise time synchronization chaotic system and is positioned on a transmitting side, wherein the chaotic system is a Chua's oscillator in a common laser chaotic system, and the derivative of the chaotic system meets the following conditions:
Wherein, the definition of each variable is: As a smooth bounded function on a real number domain, ζ > 0 is a time scale factor of a typical Chua's oscillator, γ, σ is a selected normal number, and here σ=9.35, γ=14.35 is selected to enable the typical Chua's oscillator to present a chaos phenomenon; u 1,u2 is a control variable that applies control to the rate of change of the 2 nd, 3 rd components of a typical zeiss oscillator to achieve synchronization;
Y 1,y2,y3 in the formula (13) is a state variable of the chaotic system at the receiving party, which participates in the time synchronization chaotic system required in the step S1, and the chaotic system is also a zeiss oscillator, and the derivative of the chaotic system meets the following conditions:
Wherein, An adjustable parameter for a second Chua's oscillator for adjusting the synchronization rate with the Chua's oscillator of equation (12);
For the Wang's hyper-chaotic system:
(1) The Wang's hyper-chaotic system time synchronization control law is as follows:
Wherein η 1,η2,α,α1 is an adjustable control parameter, s 3 is a sliding mode surface satisfying formula (7), and Δf is random disturbance;
(2) The fixed time synchronization control law of the Wang's hyper-chaotic system is as follows:
Wherein η 3,η4,η5,η6,α,α1,β,β1 is an adjustable control parameter, s 6 is a sliding mode surface satisfying formula (8), and Δf is random disturbance;
In formulas (15) and (16), e= [ e 1,e2,e3,e4]T ] is an error vector, which is obtained by subtracting state vectors of two chaotic systems contained in the precise time-synchronous chaotic system, and the error still forms the chaotic system, and the form is as follows:
L= [ L 1,L2,L3,L4]T ] in equations (15) and (16) is a shorthand substitution of complex variables, where each component is ：L1＝a(e2-e1),L2＝be1-k(e1e3+x1e3+e1x3)+e4, L4＝-de1；
X 1,x2,x3,x4 in the formula (17) is a required state variable of a chaotic system which participates in a precise time synchronization chaotic system and is positioned on a transmitting side, wherein the chaotic system is a Wang hyperchaotic system in a common laser chaotic system, and the derivative of the chaotic system meets the following conditions:
a, b, k, c, l, d are selected constant parameters; selecting a=10, b=40, k=1, c=25, l=4, d=20 can make the Wang hyperchaotic system exhibit hyperchaotic property.
Y 1,y2,y3,y4 in the formula (17) is a state variable of the chaotic system which is required in the step S1 and participates in the time synchronization chaotic system and is positioned at the receiving party, the chaotic system is also a Wang hyper-chaotic system, and the derivative of the chaotic system meets the following conditions:
Where u 1,u2,u3,u4 is the control quantity, the rate of change of each state component of the system is influenced to achieve synchronization, and the meaning of the remaining variables is the same as equation (18).
Compared with the prior art, the invention has the advantages that:
(1) Compared with the prior safety communication strategy, the time synchronization stability and the fixed time synchronization stability provided by the invention have stricter requirements, so that the convergence process of the precise time synchronization chaotic system is more precise and rapid.
(2) Compared with the control method based on the traditional sign function, the control energy consumption of the method can be obviously reduced, and the control performance can be effectively improved.
(3) The invention uses the sliding mode control method to control the synchronous process, and the method has the characteristics of strong robustness, high safety and low energy consumption, and is suitable for being applied to the communication network with a plurality of nodes, frequent communication and higher safety requirement.
(4) The two chaotic systems selected by the invention are two chaotic systems which are common in laser chaotic systems and are easy to realize, can be easily realized by a semiconductor laser, are suitable for space laser communication scenes due to the excellent property of chaos, and provide strong safety for the space laser communication scenes.
Drawings
FIG. 1 is a flow chart of an implementation of the method of the present invention;
FIG. 2 is a schematic diagram of a data transmission strategy of the method of the present invention;
FIG. 3 is a simulation result of the state of the error system using the control method of the present invention;
FIG. 4 is a simulation result of the synchronization performance of the control algorithm of the control method of the present invention;
FIG. 5 is a simulation result of the ratio maintenance between the components of the control method of the present invention;
FIG. 6 is a simulation of the output of an encrypted signal using the control method of the present invention;
fig. 7 shows the result of a comparison simulation of a decrypted signal and a source signal using the control method of the present invention.
Detailed Description
The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is apparent that the described embodiments are merely examples and are not intended to limit the present invention.
As shown in fig. 1 and 2, the method of the present invention is implemented as follows.
The first step: establishing a secret communication model based on a precise time synchronization chaotic system, wherein the secret communication model encrypts transmission data by utilizing a chaotic signal with randomness extracted from the precise time synchronization chaotic system; the basic features of the secret communication model are: the sender and the receiver participating in secret communication respectively have one chaotic system, precise time synchronization is realized between the two chaotic systems, a new system formed by the two chaotic systems is the precise time synchronization chaotic system, and after synchronization, the sender and the receiver extract completely consistent state vectors from the respective chaotic systems, so that reversible data encryption and decryption are realized.
The precise time synchronization chaotic system in the model is composed of two chaotic systems, wherein one chaotic system selects a common laser chaotic system Wang's hyperchaotic system, and the derivative of a state vector x= [ x 1,x2,x3,x4]T ] of the chaotic system meets the following conditions:
Wherein a, b, k, c, l, d are selected constant parameters; selecting a=10, b=40, k=1, c=25, l=4, d=20 can make the Wang hyperchaotic system exhibit hyperchaotic property.
In order to realize time synchronization, the other chaotic system also selects the Wang's hyper-chaotic system, and the derivative of the state vector y= [ y 1,y2,y3,y4]T ] satisfies the following conditions:
Where u 1,u2,u3,u4 is the control quantity, the rate of change of each state component of the system is influenced to achieve synchronization, and the meaning of the remaining variables is the same as equation (17).
The chaotic systems chosen here as examples have initial values of x (0) = [ -4,2, -2,3] T and y (0) = [ -1,6, -4,1] T.
The encryption functions required in this model are:
wherein m (t) is a source message signal, x (t) is a chaotic signal, For the transmitted encrypted signal, y (t) is the signal of the synchronous system,/>For decrypting the resulting message signal, x 1,x2,x3,x4 is the four components of the state vector x= [ x 1,x2,x3,x4]T of the wang hyperchaotic system.
The decryption function in the model is its inverse.
And a second step of: based on the precise time synchronization chaotic system provided in the first step, a precise time synchronization stability method is provided, and the precise time synchronization chaotic system meets the time synchronization requirement in the time synchronization process; the precise time synchronization stability method comprises time synchronization stability and fixed time synchronization stability, wherein the fixed time synchronization stability is used for further improving the precision and efficiency in the synchronization process on the basis of the time synchronization stability.
The required synchronization mode is determined to be precise fixed time synchronization stability, namely the synchronization process of the two chaotic systems needs to meet the fixed time synchronization stability, and the requirements are provided as follows:
i, the chaotic system is globally time-synchronous and stable;
II, a constant T m irrelevant to the initial state of the chaotic system exists, and the following conditions are satisfied:
Wherein, T is N 0 → {0} → (0, ++) is the stable time function of the chaotic system, wherein D 0 is an open neighborhood of the origin, x 0 is an initial state vector of the chaotic system.
And a third step of: according to the method for stabilizing the fixed time synchronization provided in the second step, in order to control the synchronization process of the precise time synchronization chaotic system to meet the requirement of the fixed time synchronization, a unit direction vector is utilized to design a sliding mode surface meeting the fixed time synchronization.
Designed by design criteria as the following slip plane:
Wherein, each parameter is selected as the value E is the state vector of the controlled system, here obtained by subtracting equations (1) and (2).
Fourth step: based on the fixed time synchronization sliding mode surface designed in the third step and the chaotic system in space laser communication, a control law meeting the requirement of fixed time synchronization stability is designed, and the time synchronization in the chaotic system time synchronization process in the secret communication model is completed by utilizing the given control law.
First, the formulas (1) and (2) are subtracted to obtain a new error system, namely:
to simplify the expression, the result is made compact, this will give e＝[e1,e2,e3,e4]T,u＝[u1,u2,u3,u4]T,L＝[L1,L2,L3,L4]T,L1＝a(e2-e1),L2＝be1-k(e1e3+x1e3+e1x3)+e4, L 4＝-de1, the formula (6) can be rewritten into a chaotic system form, namely:
Wherein Δf= [ Δf 1,Δf2,Δf3,Δf3]T ] is random disturbance, noise in an actual environment is simulated by random value in a certain range, and u is a control quantity.
Based on the simplified formula (7), a controller is designed for the error system, and the fixed time synchronization control law based on the Wang hyperchaotic system can be used as described above:
Wherein, each measurement value is: η 3＝1,η4＝8,η5＝1,η6＝8,α＝5/7,α1＝5/7,β＝11/9,β1 =11/9.
According to the flow, the fixed time synchronization stability can be replaced by the time synchronization stability, and the Wang's hyper-chaotic system can be replaced by a typical Chua's oscillator, so that the control law meeting the self requirements can be obtained similarly.
Based on the implementation method, the simulation result of a communication strategy which adopts the Wang hyper-chaotic system as a chaotic signal source and has a synchronous target of fixed time synchronization stability can be obtained. Fig. 1 shows a flow chart of the system when the user is the main body, and fig. 2 shows a flow chart of the system when the message to be sent is the main body. As shown in fig. 3, fig. 3 shows the change of each system state component of the error system in the synchronization process, and all errors synchronously converge to 0 in 2 seconds. Fig. 4 shows the simulation results of the control algorithm, and it can be seen that the four control components converge synchronously to a small neighborhood of the origin within 2 seconds, and that the amplitudes of the components coincide with the initial disturbance. Fig. 5 shows the nature of the ratio maintenance between the components in the error system, where the vibration occurs near the synchronous stabilization point because the calculation of the direction sign function jumps as it approaches 0. Fig. 6 shows the output of the reversible encryption function, and the source signal characteristic is masked by the chaotic signal compared with the source signal shown in fig. 7. Fig. 7 shows a comparison of the decrypted signal and the source signal, it can be seen that after about 1.1 seconds the source signal and the decrypted signal have been completely identical, fulfilling the expected communication requirements.
The simulation result fully shows that the invention can use the chaotic signal to encrypt and transmit data in the communication of two nodes, and achieves the aim of considering the safety, the communication efficiency and the accuracy by utilizing the high-precision finite time state synchronous convergence control.
It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the present invention also include such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.
Claims (4)
1. The safe data transmission method based on the precise time synchronization chaotic system is characterized by comprising the following steps of:
S1: establishing a secret communication model based on a precise time synchronization chaotic system, wherein the secret communication model encrypts transmission data by utilizing a random chaotic signal extracted from the precise time synchronization chaotic system; the basic characteristics of the secret communication model are as follows: the sender and the receiver which participate in secret communication respectively have one chaotic system, precise time synchronization is realized between the two chaotic systems, a new system formed by the two chaotic systems is the precise time synchronization chaotic system, and after synchronization, the sender and the receiver extract completely consistent state vectors from the respective chaotic systems, so that reversible data encryption and decryption are realized;
S2: based on the precise time synchronization chaotic system provided in the step S1, a precise time synchronization stability method is provided, and the precise time synchronization chaotic system realizes the time synchronization requirement involved in the time synchronization process; the precise time synchronization stability method comprises time synchronization stability and fixed time synchronization stability, wherein the fixed time synchronization stability further improves the precision and efficiency in the synchronization process on the basis of the time synchronization stability;
s3: according to the method for time synchronization stability and fixed time synchronization stability provided in S2, in order to control the synchronization process of the precise time synchronization chaotic system to meet the requirements of time synchronization and fixed time synchronization, a unit direction vector is utilized to respectively design a sliding mode surface meeting the requirements of time synchronization and fixed time synchronization;
S4: based on the time synchronization and fixed time synchronization sliding mode surface designed in the step S3, aiming at the chaotic system in space laser communication, designing a control law meeting the requirements of time synchronization stability and fixed time synchronization stability, and utilizing the control law to complete time synchronization in the chaotic system time synchronization process in a secret communication model;
In the step S1 of the above-mentioned process,
When precise time synchronization is realized between two chaotic systems, namely, precise time synchronization is realized between two chaotic systems respectively belonging to a sender and a receiver, the formula of the secret communication model is as follows:
(1)
Encryption function And decryption function/>The method comprises the following steps of:
(2)
(3)
Wherein, For source message signal,/>For chaotic signals extracted from own chaotic system by a sender,/>For transmitted encrypted signals,/>For the chaotic signal extracted from own chaotic system by a receiver,/>For decrypting the resulting message signal.
2. The method according to claim 1, characterized in that: in the step S2 of the above-mentioned process,
The time synchronization stability is achieved as follows:
(1) If the process of converging the controlled chaotic system to the original point meets the requirement of time synchronization stabilization, the process is required to meet the requirement of:
i, the chaotic system is stable in limited time, namely can converge to an original point in limited time;
II. Design As a stable time function of a chaotic system, wherein/>，/>On an open neighborhood that is the origin, the following is satisfied:
(4)
In the form of Is the system state quantity/>(1 /)A component;
In this condition, if ，/>The real number domain is called global time synchronization stabilization;
The fixed time synchronization stability method is described as follows:
(2) If the process of converging a controlled chaotic system to the original point meets the requirement of synchronous stabilization of fixed time, the process needs to meet the following requirements:
i, the chaotic system is globally time-synchronous and stable;
II, there is a constant which is irrelevant to the initial state of the chaotic system The method comprises the following steps:
(5)
in the above formula, the chaotic system should satisfy the following general mathematical form, maintaining equilibrium at the origin:
(6)
Wherein, For a time-dependent/>Wiener column vector, write/>，/>The initial value at time 0 is/>，Is a slave/>Space mapping of dimension vectors to/>Function of the dimension vector space,/>Representation/>Deriving time, a state vector of a chaotic system keeps moving at 0 after reaching 0, and a process of moving from an initial value to 0 is called convergence.
3. The method according to claim 1, characterized in that: in the step S3, the slip-form surfaces satisfying the time synchronization and the fixed time synchronization are respectively designed by using the unit direction vectors as follows:
(1) The slip form surface designed to meet time synchronization is as follows:
(7)
Wherein, And/>For the parameters of the sliding mode surface, the form of the sliding mode surface can be controlled,/>Is the state vector of the controlled chaotic system, and the formula only defines the sliding mode surface/>Is a derivative of (2);
(2) The slip form surface designed to meet the fixed time synchronization is as follows:
(8)
Wherein, And/>For the parameters of the sliding mode surface, the form of the sliding mode surface can be controlled,/>Is the state vector of the controlled chaotic system, and the formula only defines the sliding mode surface/>Is a derivative of (2);
in equations (7) and (8), the function Solving unit vectors parallel to the direction of variables in brackets; when the function has an index/>And in the case of a power exponent form, the method is developed as follows:
(9)。
4. The method according to claim 1, characterized in that: in the step S4, the control law time synchronization control law required by the time synchronization stability and the fixed time synchronization stability is as follows, and the specific form is given by using a typical zeiss oscillator and a Wang hyperchaotic system in a common chaotic system;
For a typical zeiss oscillator:
(1) The time synchronization control law of a typical zeiss oscillator is as follows:
(10)
Wherein, Is an adjustable control parameter,/>A slip-form surface defined for satisfying the derivative of equation (7);
(2) The fixed time synchronization control law of a typical zeiss oscillator is as follows:
(11)
Wherein, Is an adjustable control parameter,/>A slip-form surface defined for satisfying the derivative of equation (8);
in equations (10) and (11), The error vector is obtained by subtracting state vectors of two chaotic systems contained in the precise time synchronization chaotic system, and the error still forms the chaotic system, and the form is as follows:
(12)
In formulas (10) and (11) Is a shorthand substitution of complex variables, wherein/>，；
In the formula (12)In order to participate in the state variable of the chaotic system of the precise time synchronization chaotic system, which is a Chua's oscillator in the common laser chaotic system, the derivative of the Chua's oscillator meets the following conditions:
(13)
Wherein, the definition of each variable is: as a smooth bounded function over the real number domain,/> Is the time scale factor of a typical Chua's oscillator,/>Is a selected normal number, here selected/>The chaos phenomenon of a typical Chua's oscillator is caused; /(I)Is a control variable, which applies control to the rate of change of the 2 nd and 3 rd components of a typical zeiss oscillator to achieve synchronization;
in formula (13) In order to participate in the state variable of the chaotic system of the time synchronization chaotic system at the receiving party, the chaotic system is also a Chua's oscillator, and the derivative of the chaotic system meets the following conditions:
(14)
Wherein the method comprises the steps of An adjustable parameter for a second Chua's oscillator for adjusting the synchronization rate with the Chua's oscillator of equation (12);
For the Wang's hyper-chaotic system:
(1) The Wang's hyper-chaotic system time synchronization control law is as follows:
(15)
Wherein, Is an adjustable control parameter,/>To satisfy the slip form face of equation (7)/>Is a random disturbance;
(2) The fixed time synchronization control law of the Wang's hyper-chaotic system is as follows:
(16)
Wherein, Is an adjustable control parameter,/>To satisfy the sliding mode surface of formula (8)/>Is a random disturbance;
in equations (15) and (16), The error vector is obtained by subtracting state vectors of two chaotic systems contained in the precise time synchronization chaotic system, and the error still forms the chaotic system, and the form is as follows:
(17)
in formulas (15) and (16) Is a shorthand substitution of complex variables, wherein each component is:, />,/>, />；
In the formula (17) In order to participate in the state variable of the chaotic system of the precise time synchronization chaotic system which is positioned at the transmitting side, the chaotic system is a Wang hyperchaotic system in the common laser chaotic system, and the derivative of the Wang hyperchaotic system meets the following conditions:
(18)
Is a selected constant parameter; select/> The Wang's hyper-chaotic system can be enabled to present hyper-chaos;
In the formula (17) In order to participate in the state variable of the chaotic system of the time synchronization chaotic system which is positioned at the receiving party, the chaotic system is also a Wang hyperchaotic system, and the derivative of the chaotic system meets the following conditions:
(19)
Wherein, To control the amount, the rate of change of each state component of the system is influenced to achieve synchronization, and the meaning of the remaining variables is the same as equation (18).
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| CN105306193A (en) * | 2015-11-16 | 2016-02-03 | 郑州轻工业学院 | Finite time synchronization sliding-mode control method of complex chaotic system with unknown parameters |
| CN112036035A (en) * | 2020-08-31 | 2020-12-04 | 郑州轻工业大学 | Point multiplication function projection synchronization method based on bowknot type laser complex chaotic system |
| CN114254753A (en) * | 2021-12-17 | 2022-03-29 | 中国人民解放军海军工程大学 | Method for constructing heterogeneous chaotic system finite time synchronization model and designing controller |
| US11424906B1 (en) * | 2021-11-02 | 2022-08-23 | King Abdulaziz University | Secure encryption and decryption method and system |
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| CN105306193A (en) * | 2015-11-16 | 2016-02-03 | 郑州轻工业学院 | Finite time synchronization sliding-mode control method of complex chaotic system with unknown parameters |
| CN112036035A (en) * | 2020-08-31 | 2020-12-04 | 郑州轻工业大学 | Point multiplication function projection synchronization method based on bowknot type laser complex chaotic system |
| US11424906B1 (en) * | 2021-11-02 | 2022-08-23 | King Abdulaziz University | Secure encryption and decryption method and system |
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