CN104363088A - 3-D grid multi-scroll chaotic attractor generation method based on step function - Google Patents
3-D grid multi-scroll chaotic attractor generation method based on step function Download PDFInfo
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Abstract
The invention provides a 3-D grid multi-scroll chaotic attractor generation method based on a step function. The method includes the following steps that (1) a 3-D grid multi-scroll chaotic system and a step function expression are determined; (2) models of chaotic attractors on the x-axis, the y-axis and the z-axis are built according to the step function; (3) balance points and turning points of the chaotic attractors are calculated according to the effective work range of an active device and the number of the chaotic attractors. Compared with existing literature, the method overcomes the defect that in the prior art, only a 3-D grid multi-scroll chaotic attractor simulation result can be shown, but balance points and turning points of 3-D grid multi-scroll chaotic attractors cannot be described. The 3-D grid multi-scroll chaotic attractors can be conveniently analyzed by means of the method, and the simulation result is more visual.
Description
Technical field
The invention belongs to non linear system, specifically relate to a kind of 3-D grid multi scroll chaotic attactors production method based on step function.
Background technology
The structure of multi scroll chaotic attactors and dynamic behavior have more complexity than the Lip river hereby chaos attractor that falls, and make its extensive use in the communications.As " the grid multiscroll chaotic circuit design and implimentation based on current transmission device ", Wang Chunhua, Acta Physica Sinica, the 61st volume the 21st phase, 210507-2-2 page, on November 5th, 2012; Propose and realize 3-D grid multi scroll chaotic attactors with saturation function, " study based on the grid multiscroll chaotic circuit of homophase second generation current transmission device ", woods is willing to, Acta Physica Sinica, the 61st volume the 24th phase, 240503-2-2 page, on December 20th, 2012, propose step function and realize 3-D grid multi scroll chaotic attactors, but all the balance point of 3-D grid multi scroll chaotic attactors and breakover point are not described.
Summary of the invention
The technical problem to be solved in the present invention is, overcome prior art and can only provide 3-D grid multi scroll chaotic attactors simulation result, the defect of 3-D grid multi scroll chaotic attactors balance point and breakover point can not be described, a kind of 3-D grid multi scroll chaotic attactors production method based on step function is provided, it is by adopting step function to be nonlinear function, take step function as model, set up 3-D grid multi scroll chaotic attactors at x, y, the width in z tri-directions and period-luminosity relation, draw the computing formula of 3-D grid multi scroll chaotic attactors balance point and breakover point, make analysis 3-D grid multi scroll chaotic attactors more convenient.
The technical solution adopted for the present invention to solve the technical problems is:
Based on the 3-D grid multi scroll chaotic attactors production method of step function, comprise the steps:
(1) 3-D grid multi-scroll chaotic system and step function expression formula is determined; (2) chaos attractor is set up at x-axis, y-axis, z-axis model according to step function; (3) chaos attractor is calculated at the axial balance point of x, y, z three and breakover point according to the efficient working range of active device and chaos attractor number, with Matlab, numerical simulation is carried out to 3-D grid multi-scroll chaotic system, observe the phasor of 3-D view and X-Y, X-Z, Y-Z respectively.
In described step (1), 3-D grid multi-scroll chaotic system is:
(1)，
In formula, x, y, z are state variable,
,
,
be variable x, y, z are respectively to the derivative of time t, and a is chaos system parameter and is arithmetic number, and f (y), f (z) are step function, and step function is expressed as follows:
(2),
or
(3),
In formula (2), (3), y is
statevariable, A is system adjustable parameter and for arithmetic number, and N is chaos attractor number and is variable for positive integer, i and is positive integer, and odd represents odd number, and even represents even number,
, sgn (u) represents sign function, and u represents variable, replaces y with z, can obtain f (z); As a=0.7, through type (1) can produce 3-D grid multi scroll chaotic attactors.
In described step (2), setting up chaos attractor according to step function at the concrete grammar of x-axis, y-axis, z-axis model is: set the dynamic range of x-axis direction chaos attractor as DR
x, x-axis direction chaos attractor number is N
x, y-axis direction chaos attractor number is N
y, z-axis direction chaos attractor number is N
z, and N
x=N
z, x-axis direction chaos attractor width is W
x, y-axis direction chaos attractor width is W
y, z-axis direction chaos attractor width is W
z, and W
x=DR
x/ N
x, W
y=DR
x/ (N
xn
y), W
y=W
z, the cycle of x-axis direction chaos attractor is A
x, the cycle of y-axis direction chaos attractor is A
y, the cycle of z-axis direction chaos attractor is A
z, and A
x=W
x/ 2, A
y=W
y/ 2, A
z=A
y.
In described step (3), calculate chaos attractor according to the efficient working range of active device and chaos attractor number as follows in the method for x, y, z three axial balance point Ep and breakover point Bp:
(a) x-axis coordinate direction, the expression formula of balance point Ep is as follows:
(4),
Or
(5),
Ep in formula (4), (5)
j+1represent the balance point of jth+1 chaos attractor, Ep
jrepresent the balance point of a jth chaos attractor, j is variable and is positive integer, N
xfor the number of x-axis direction chaos attractor, N
yfor the number of y-axis direction chaos attractor, A
xfor the cycle of x direction chaos attractor, Ep
0represent the balance point of origin position chaos attractor.
Breakover point Bp is expressed as follows:
(6),
(7),
Bp in formula (6), (7)
j+1represent the breakover point of jth+1 chaos attractor, j is variable and is positive integer, N
xfor the number of x-axis direction chaos attractor, A
xfor the cycle of x direction chaos attractor.
(b) y axial coordinate direction, the expression formula of balance point Ep is as follows:
(8)；
Or
(9)；
Ep in formula (8), (9)
j+1represent the balance point of jth+1 chaos attractor, Ep
jrepresent the balance point of a jth chaos attractor, j is variable and is positive integer, N
yfor the number of y-axis direction chaos attractor, A
yfor the cycle of y direction chaos attractor, Ep
0represent the balance point of origin position chaos attractor.
Breakover point Bp
'sbe expressed as follows:
(10)；
Or
(11)；
Bp in formula (10), (11)
j+1represent the breakover point of jth+1 chaos attractor, j is variable and is positive integer, N
yfor the number of y-axis direction chaos attractor, A
yfor the cycle of y direction chaos attractor.
(c) z axial coordinate direction, the expression formula of balance point Ep is as follows:
(12)；
Or
(13)；
Ep in formula (12), (13)
j+1represent the balance point of jth+1 chaos attractor, Ep
jrepresent the balance point of a jth chaos attractor, j is variable and is positive integer, N
zfor the number of z-axis direction chaos attractor, A
zfor the cycle of z direction chaos attractor, Ep
0represent the balance point of origin position chaos attractor.
Breakover point Bp is expressed as follows:
(14)；
Or
(15);
Bp in formula (14), (15)
j+1represent the breakover point of jth+1 chaos attractor, j is variable and is positive integer, N
zfor the number of z-axis direction chaos attractor, A
zfor the cycle of z direction chaos attractor.
The present invention, compared with existing document, overcomes prior art and can only provide 3-D grid multi scroll chaotic attactors simulation result, can not describe the defect of 3-D grid multi scroll chaotic attactors balance point and breakover point.Use the present invention, can analyze 3-D grid multi scroll chaotic attactors easily, simulation result is more directly perceived.
Accompanying drawing explanation
Fig. 1 is step function figure;
Fig. 2 (a) is the 3-D view of 5 × 4 grid multi scroll chaotic attactors;
Fig. 2 (b) is that the X-Y of 5 × 4 grid multi scroll chaotic attactors exports phasor;
Fig. 2 (c) is that the X-Z of 5 × 4 grid multi scroll chaotic attactors exports phasor;
Fig. 2 (d) is that the Y-Z of 5 × 4 grid multi scroll chaotic attactors exports phasor;
Fig. 3 (a) is the 3-D view of 4 × 3 grid multi scroll chaotic attactors;
Fig. 3 (b) is that the X-Y of 4 × 3 grid multi scroll chaotic attactors exports phasor;
Fig. 3 (c) is that the X-Z of 4 × 3 grid multi scroll chaotic attactors exports phasor;
Fig. 3 (d) is that the Y-Z of 4 × 3 grid multi scroll chaotic attactors exports phasor.
Embodiment
Below in conjunction with accompanying drawing, subordinate list and embodiment, the invention will be further described.
Based on the 3-D grid multi scroll chaotic attactors production method of step function, comprise the steps: that (1) determines 3-D grid multi-scroll chaotic system and step function expression formula; (2) chaos attractor is set up at x-axis, y-axis, z-axis model according to step function; (3) chaos attractor is calculated at the axial balance point of x, y, z three and breakover point according to the efficient working range of active device and chaos attractor number.
The present invention passes through the calculating of 3-D grid multi scroll chaotic attactors balance point and breakover point, and available Matlab carries out numerical simulation to 3-D grid multi-scroll chaotic system, then can observe the phasor of 3-D view and X-Y, X-Z, Y-Z respectively.
In described step (1), 3-D grid multi-scroll chaotic system is:
(1)，
In formula, x, y, z are state variable,
,
,
be variable x, y, z are respectively to the derivative of time t, and a is chaos system parameter and is arithmetic number, and f (y), f (z) are step function, and step function is expressed as follows:
(2),
or
(3),
In formula (2), (3), y is
statevariable, A is system adjustable parameter and for arithmetic number, and N is chaos attractor number and is variable for positive integer, i and is positive integer, and odd represents odd number, and even represents even number,
, sgn (u) represents sign function, and u represents variable, replaces y with z, can obtain f (z); As a=0.7, through type (1) can produce 3-D grid multi scroll chaotic attactors.
In described step (2), setting up chaos attractor according to step function at the concrete grammar of x-axis, y-axis, z-axis model is: set the dynamic range of x-axis direction chaos attractor as DR
x, x-axis direction chaos attractor number is N
x, y-axis direction chaos attractor number is N
y, z-axis direction chaos attractor number is N
z, and N
x=N
z, x-axis direction chaos attractor width is W
x, y-axis direction chaos attractor width is W
y, z-axis direction chaos attractor width is W
z, and W
x=DR
x/ N
x, W
y=DR
x/ (N
xn
y), W
y=W
z, the cycle of x-axis direction chaos attractor is A
x, the cycle of y-axis direction chaos attractor is A
y, the cycle of z-axis direction chaos attractor is A
z, and A
x=W
x/ 2, A
y=W
y/ 2, A
z=A
y.
In described step (3), calculate chaos attractor according to the efficient working range of active device and chaos attractor number as follows in the method for x, y, z three axial balance point Ep and breakover point Bp:
(a) x-axis coordinate direction, the expression formula of balance point Ep is as follows:
(4),
Or
(5),
Ep in formula (4), (5)
j+1represent the balance point of jth+1 chaos attractor, Ep
jrepresent the balance point of a jth chaos attractor, j is variable and is positive integer, N
xfor the number of x-axis direction chaos attractor, N
yfor the number of y-axis direction chaos attractor, A
xfor the cycle of x direction chaos attractor, Ep
0represent the balance point of origin position chaos attractor.
Breakover point Bp is expressed as follows:
(6),
(7),
Bp in formula (6), (7)
j+1represent the breakover point of jth+1 chaos attractor, j is variable and is positive integer, N
xfor the number of x-axis direction chaos attractor, A
xfor the cycle of x direction chaos attractor.
(b) y axial coordinate direction, the expression formula of balance point Ep is as follows:
(8)；
Or
(9)；
Ep in formula (8), (9)
j+1represent the balance point of jth+1 chaos attractor, Ep
jrepresent the balance point of a jth chaos attractor, j is variable and is positive integer, N
yfor the number of y-axis direction chaos attractor, A
yfor the cycle of y direction chaos attractor, Ep
0represent the balance point of origin position chaos attractor.
Breakover point Bp
'sbe expressed as follows:
(10)；
Or
(11)；
Bp in formula (10), (11)
j+1represent the breakover point of jth+1 chaos attractor, j is variable and is positive integer, N
yfor the number of y-axis direction chaos attractor, A
yfor the cycle of y direction chaos attractor.
(c) z axial coordinate direction, the expression formula of balance point Ep is as follows:
(12)；
Or
(13)；
Ep in formula (12), (13)
j+1represent the balance point of jth+1 chaos attractor, Ep
jrepresent the balance point of a jth chaos attractor, j is variable and is positive integer, N
zfor the number of z-axis direction chaos attractor, A
zfor the cycle of z direction chaos attractor, Ep
0represent the balance point of origin position chaos attractor.
Breakover point Bp is expressed as follows:
(14)；
Or
(15)
Bp in formula (14), (15)
j+1represent the breakover point of jth+1 chaos attractor, j is variable and is positive integer, N
zfor the number of z-axis direction chaos attractor, A
zfor the cycle of z direction chaos attractor.
Application example:
(1) 3-D grid multi-scroll chaotic system and step function expression formula is determined; (2) chaos attractor is set up in the width of x-axis, y-axis, z-axis and cycle according to step function; (3) calculate chaos attractor in the balance point of x-axis, y-axis, z-axis and breakover point parameter, with Matlab, numerical simulation is carried out to 3-D grid multi-scroll chaotic system, observe the phasor of 3-D view and X-Y, X-Z, Y-Z respectively.
The parameters relationship of step function and chaos attractor as can be seen from Figure 1.If the dynamic range DR=10.8V of chaos attractor, y-axis direction chaos attractor number N
y=5, z-axis direction chaos attractor number N
z=4, W, Bp and Ep can be calculated by formula (1), (2), (3), (5), (7), (8), (10), (13), (15), its result is as shown in table 1, and its Numerical Simulation Results as shown in Figure 2.
From Fig. 2 and table 1, chaos attractor balance point is consistent with simulation result with the calculated value of breakover point; In like manner, y-axis direction chaos attractor number N
y=4, z-axis direction chaos attractor number N
zwhen=3, W can be calculated, B by formula (1), (2), (3), (5), (6), (9), (11), (12), (14)
pand E
p, its result is as shown in table 2, and its Numerical Simulation Results as shown in Figure 3.
From Fig. 3 and table 2, chaos attractor balance point is consistent with simulation result with the calculated value of breakover point.
The balance point of comprehensive above-mentioned calculating chaos attractor and breakover point and Matlab simulation result show, the 3-D grid multi scroll chaotic attactors production method based on step function is feasible.
Claims (4)
1., based on the 3-D grid multi scroll chaotic attactors production method of step function, it is characterized in that, comprise the steps:
(1) 3-D grid multi-scroll chaotic system and step function expression formula is determined; (2) chaos attractor is set up at x-axis, y-axis, z-axis model according to step function; (3) chaos attractor is calculated at the axial balance point of x, y, z three and breakover point according to the efficient working range of active device and chaos attractor number, with Matlab, numerical simulation is carried out to 3-D grid multi-scroll chaotic system, observe the phasor of 3-D view and X-Y, X-Z, Y-Z respectively.
2. the 3-D grid multi scroll chaotic attactors production method based on step function according to claim 1, is characterized in that, in described step (1), 3-D grid multi-scroll chaotic system is:
(1)，
In formula, x, y, z are state variable,
,
,
be variable x, y, z are respectively to the derivative of time t, and a is chaos system parameter and is arithmetic number, and f (y), f (z) are step function, and step function is expressed as follows:
(2),
or
(3),
In formula (2), (3), y is state variable, and A is system adjustable parameter and for arithmetic number, and N is chaos attractor number and is variable for positive integer, i and is positive integer, and odd represents odd number, and even represents even number,
, sgn (u) represents sign function, and u represents variable, replaces y with z, can obtain f (z); As a=0.7, through type (1) can produce 3-D grid multi scroll chaotic attactors.
3. the 3-D grid multi scroll chaotic attactors production method based on step function according to claim 1, it is characterized in that, in described step (2), setting up chaos attractor according to step function at the concrete grammar of x-axis, y-axis, z-axis model is: set the dynamic range of x-axis direction chaos attractor as DR
x, x-axis direction chaos attractor number is N
x, y-axis direction chaos attractor number is N
y, z-axis direction chaos attractor number is N
z, and N
x=N
z, x-axis direction chaos attractor width is W
x, y-axis direction chaos attractor width is W
y, z-axis direction chaos attractor width is W
z, and W
x=DR
x/ N
x, W
y=DR
x/ (N
xn
y), W
y=W
z, the cycle of x-axis direction chaos attractor is A
x, the cycle of y-axis direction chaos attractor is A
y, the cycle of z-axis direction chaos attractor is A
z, and A
x=W
x/ 2, A
y=W
y/ 2, A
z=A
y.
4. the 3-D grid multi scroll chaotic attactors production method based on step function according to claim 1, it is characterized in that, in described step (3), calculate chaos attractor according to the efficient working range of active device and chaos attractor number as follows in the method for x, y, z three axial balance point Ep and breakover point Bp:
(a) x-axis coordinate direction, the expression formula of balance point Ep is as follows:
(4),
Or
(5),
In formula (4), (5), Ep
j+1represent the balance point of jth+1 chaos attractor, Ep
jrepresent the balance point of a jth chaos attractor, j is variable and is positive integer, N
xfor the number of x-axis direction chaos attractor, N
yfor the number of y-axis direction chaos attractor, A
xfor the cycle of x direction chaos attractor, Ep
0represent the balance point of origin position chaos attractor;
Breakover point Bp is expressed as follows:
(6),
(7),
In formula (6), (7), Bp
j+1represent the breakover point of jth+1 chaos attractor, j is variable and is positive integer, N
xfor the number of x-axis direction chaos attractor, A
xfor the cycle of x direction chaos attractor;
(b) y axial coordinate direction, the expression formula of balance point Ep is as follows:
(8)；
Or
(9)；
In formula (8), (9), Ep
j+1represent the balance point of jth+1 chaos attractor, Ep
jrepresent the balance point of a jth chaos attractor, j is variable and is positive integer, N
yfor the number of y-axis direction chaos attractor, A
yfor the cycle of y direction chaos attractor, Ep
0represent the balance point of origin position chaos attractor;
Breakover point Bp
'sbe expressed as follows:
(10)；
Or
(11)；
In formula (10), (11), Bp
j+1represent the breakover point of jth+1 chaos attractor, j is variable and is positive integer, N
yfor the number of y-axis direction chaos attractor, A
yfor the cycle of y direction chaos attractor;
(c) z axial coordinate direction, the expression formula of balance point Ep is as follows:
(12)；
Or
(13)；
In formula (12), (13), Ep
j+1represent the balance point of jth+1 chaos attractor, Ep
jrepresent the balance point of a jth chaos attractor, j is variable and is positive integer, N
zfor the number of z-axis direction chaos attractor, A
zfor the cycle of z direction chaos attractor, Ep
0represent the balance point of origin position chaos attractor;
Breakover point Bp is expressed as follows:
(14)；
Or
(15),
Bp in formula (14), (15)
j+1represent the breakover point of jth+1 chaos attractor, j is variable and is positive integer, N
zfor the number of z-axis direction chaos attractor, A
zfor the cycle of z direction chaos attractor.
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107248898A (en) * | 2017-05-27 | 2017-10-13 | 佛山科学技术学院 | Multiline message chaos encryption, decryption method and its device based on many plunging breaker signals |
| CN117424787A (en) * | 2023-11-20 | 2024-01-19 | 常州大学 | Chaotic orbit folding method and system based on complex operation |
-
2014
- 2014-11-21 CN CN201410671677.8A patent/CN104363088A/en active Pending
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107248898A (en) * | 2017-05-27 | 2017-10-13 | 佛山科学技术学院 | Multiline message chaos encryption, decryption method and its device based on many plunging breaker signals |
| CN107248898B (en) * | 2017-05-27 | 2019-06-04 | 佛山科学技术学院 | Multiline message chaos encrypting method and its device based on multireel wave signal |
| CN117424787A (en) * | 2023-11-20 | 2024-01-19 | 常州大学 | Chaotic orbit folding method and system based on complex operation |
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Application publication date: 20150218 |