CN112491530A - Improved Chua's system and FPGA-based Chua's chaotic signal generator thereof - Google Patents
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Abstract
The invention discloses an improved Chua's system and a Chua's chaotic signal generator based on an FPGA (field programmable gate array), compared with a typical Chua's system, the improved Chua's system provided by the invention uses a composite hyperbolic tangent cubic nonlinear function, can generate a three-scroll chaotic attractor, and numerically shows richer dynamic behaviors of coexisting a plurality of attractors, a transient period, intermittent chaos, attractor offset and the like through a bifurcation diagram, a Lyapunov index, a phase diagram and a time domain diagram; and after the constant excitation is added, the symmetrical characteristic of the chaotic attractor is destroyed by the constant excitation. Compared with the conventional Chua's system which generates a multi-scroll Chua's chaotic signal by building an analog circuit, the invention provides a mode for realizing the improved Chua's system based on the FPGA, so that the influence of the traditional Chua's analog circuit on result errors caused by factors such as temperature, component aging and the like is not required to be considered.
Description
Technical Field
The invention relates to the technical field of Chua's systems, in particular to an improved Chua's system and a Chua's chaotic signal generator based on an FPGA (field programmable gate array).
Background
In recent years, chaotic dynamics has become a research field with abundant content and wide application. Secure communication, image encryption, memories, system synchronization, random number generators, complex networks, and the like are all chaos and the application thereof is deeply studied.
The multi-scroll chaotic circuit has important application in the fields of chaotic communication, information security, image encryption and the like, and becomes a hot point of chaotic theory research. In order to generate a multi-scroll chaotic attractor in some simple chaotic systems, some non-linear functions are required, including piecewise linear functions, sawtooth waves, triangular waves, step waves, saturation functions, polynomial functions, trigonometric functions, and absolute value functions. There are many circuits with chaotic behavior, the simplest of which is the Chua's circuit (Chua's system) established by professor Chua in 1983. This is a precedent for using electronic circuits to study chaos. A typical Chua's system can generate a double-vortex chaotic attractor, and how to improve the typical Chua's system to obtain richer dynamic behaviors is a key point of the current chaos research. In addition, the hardware implementation of the prior multi-scroll chaotic system (including a typical Chua's system) mainly adopts two methods for implementing chaos in an analog circuit, namely a discrete element circuit design and a chip integrated circuit design, and because the discrete element is easily influenced by environmental factors such as temperature, aging and the like, the chaotic system is extremely sensitive to an initial value, the effect of implementing the chaotic system by using the discrete element circuit is very limited.
Disclosure of Invention
The present invention is directed to solving at least one of the problems of the prior art. Therefore, the invention provides an improved Chua's system and a Chua's chaotic signal generator based on the FPGA, and compared with the existing typical Chua's system, the improved Chua's system can generate richer dynamic behaviors; the improved Chua's system is realized based on the FPGA, so that the influence of the traditional Chua's analog circuit on result errors caused by factors such as temperature, aging of components and the like is not required to be considered.
In a first aspect of the present invention, an improved zeiss system is provided, wherein the state equation of the zeiss system is:
wherein x, y, z are system variables of the Chua's system, α, β are system parameters, and h (x) ax-b tan h (x)3) Where a is a constant value and b is a parameter for controlling the number of scrolls.
According to the embodiment of the invention, at least the following technical effects are achieved:
compared with a typical Chua system, the Chua system uses a composite hyperbolic tangent cubic nonlinear function, can generate a three-scroll chaotic attractor, and numerically shows richer dynamic behaviors such as coexistence of a plurality of attractors, a transient period, intermittent chaos, attractor offset and the like through a bifurcation diagram, a Lyapunov index, a phase diagram and a time domain diagram; and after the constant excitation is added, the symmetrical characteristic of the chaotic attractor is destroyed by the constant excitation.
The Chua's system can be realized through a digital circuit (realized based on an FPGA), so that the influence of the traditional Chua's analog circuit on result errors caused by factors such as temperature, aging of components and the like does not need to be considered.
According to some embodiments of the invention, the zeiss system has an initial value of [0.5,0,0], the α -8, the β -12 and the a-0.3.
According to some embodiments of the invention, the system variable adds a corresponding offset variable.
According to some embodiments of the invention, the zeiss system further comprises a constant excitation c, the state equation being:
according to some embodiments of the invention, c is 0.01.
In a second aspect of the present invention, there is provided a Chua's chaotic signal generator based on an FPGA, comprising:
the Chua's oscillator is designed on an FPGA development board by the improved Chua's system in the first aspect of the invention and is used for generating chaotic signals;
the control unit is connected with the Chua's oscillator and used for managing and scheduling the Chua's oscillator so as to enable the Chua's oscillator to generate chaotic signals;
and the analog converter is connected with the control unit and is used for converting the generated digital signal into an analog signal.
According to the embodiment of the invention, at least the following technical effects are achieved:
compared with the conventional Chua's system which generates a multi-scroll Chua's chaotic signal by building an analog circuit, the invention provides a mode for realizing the improved Chua's system on the basis of the FPGA, so that the influence of the traditional Chua's analog circuit on result errors caused by factors such as temperature, component aging and the like is not required to be considered.
According to some embodiments of the invention, the Chua's oscillator is generated by the RK4 algorithm and generates a chaotic signal by the RK4 algorithm.
According to some embodiments of the invention, the FPGA development board is model ZYNQ-XC7Z 020.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
fig. 1 is a phase diagram for simulating a zai system by using MATLAB according to an embodiment of the present invention and using different control parameters, respectively;
FIG. 2 is a diagram of a bifurcation and Lyapunov index spectra of the Chua system according to an embodiment of the present invention;
FIG. 3 is a diagram illustrating the number and location of equilibrium points of a curve of a composite hyperbolic tangent cubic function according to an embodiment of the present invention;
fig. 4 is a schematic diagram of a zeiss system coexisting attractor under a change of a parameter b according to an embodiment of the present invention;
fig. 5 is a schematic diagram of transient transition behavior in the zeiss system according to the embodiment of the present invention;
FIG. 6 shows an embodiment of the present invention providing three different k1Under the value, a schematic diagram of one to three scroll chaotic attractors shift;
FIG. 7 is a schematic diagram of a two-dimensional projection of one-to-three scroll asymmetric chaotic attractors with constant excitation added according to an embodiment of the present invention;
fig. 8 is a schematic structural diagram of a Chua's chaotic signal generator based on an FPGA according to an embodiment of the present invention;
FIG. 9 is a timing diagram of a Chua's oscillator simulation provided by an embodiment of the present invention;
fig. 10 is a schematic diagram of a zeiss attractor generated by the FPGA-based zeiss chaotic signal generator according to the embodiment of the present invention;
fig. 11 is a schematic diagram of an experimental apparatus provided in an embodiment of the present invention.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the accompanying drawings are illustrative only for the purpose of explaining the present invention, and are not to be construed as limiting the present invention.
A first embodiment;
referring to fig. 1 to 7, an improved zeiss system is provided, the state equations of which are:
wherein x, y and z are system variables of the Chua's system, alpha and beta are system parameters, h (x) is a composite hyperbolic tangent cubic nonlinear function, and h (x) ax-b tan h (x)3) A is a constant value, and b is a parameter for controlling the number of scrolls. Compared with the classicThe zeiss system provided by the embodiment uses a composite hyperbolic tangent cubic nonlinear function, so that the zeiss system has richer dynamic behavior than a typical zeiss system.
As an alternative embodiment, when the initial value of the zeiss system is [0.5,0,0], the system parameter α is 8, β is 12, and a is 0.3, fig. 1 shows a phase diagram of the zeiss system simulated using MATLAB using the control parameters b is 0.4, b is 0.5, and b is 0.65, respectively, and the numerical simulation results of the one to three scrolls of the zeiss attractors are achieved by different parameter values b, specifically, fig. 1(a) shows a phase diagram of one scroll on the x-y plane; FIG. 1(b) shows a phase diagram of a scroll wrap in the x-z plane; FIG. 1(c) shows a phase diagram of a scroll wrap in the y-z plane; FIG. 1(d) shows a phase diagram of three scrolls in the x-y plane; FIG. 1(e) shows a phase diagram of three scrolls in the x-z plane; FIG. 1(f) shows a phase diagram of three scrolls in the y-z plane; FIG. 1(g) shows a phase diagram of two scrolls in the x-y plane; FIG. 1(h) shows a phase diagram of two scrolls in the x-z plane; FIG. 1(i) shows a phase diagram of two scrolls in the y-z plane. The bifurcation diagram and the lyapunov exponential spectrum of the system (1) are respectively shown in fig. 2, with the increase of the parameter b, one-to-three scroll chaotic attractors are realized, and specifically, fig. 2(a) is a bifurcation diagram of a state variable x; FIG. 2(b) is the corresponding Lyapunov exponential spectrum. The results in the figure demonstrate that compared to a typical zeiss system, the present zeiss system is able to generate a three-scroll zeiss attractor by setting the appropriate initial parameters and then adjusting the appropriate parameter b.
Solving the balance point of the Chua system: let equation (1) have zero on the left, then there is:
when b is 0.6, the curve of the composite hyperbolic tangent cubic function has five defined equilibrium points, four of which are symmetric with respect to the origin of coordinates. Fig. 3 shows that h (x) is 0.3x-0.6tan h (x)3) The determined five balance points are respectively:
formula (1) at equilibrium point
The jacobian matrix above is:
thus, the characteristic equation for the equilibrium point is:
P(λ)＝λ3+δ1λ2+δ2λ+δ3＝0 (5)
in formula (5):
obviously, these five equilibrium points do not satisfy the Routh-Hurwitz (laus-herwitz stability criterion) condition, which are globally unstable saddle points. The eigenvalues of the five equilibrium points can thus be found:
compared with the three balance points generated by the existing typical Chua system, the Chua system can generate five balance points under appropriate parameters and can generate richer dynamic behaviors, and the encryption safety can be improved when the Chua system is applied to the field of information encryption.
When the initial value is [0.5,0,0], the system parameter α is 8, β is 12, and a is 0.3, the coexistence phenomenon of multiple attractors with different attractor types, different topologies, or different periods in the current chua system is illustrated by the phase plane trajectory: in the range of 0.3. ltoreq. b.ltoreq.0.37, the coexistence of a pair of period 1 attractors, a pair of period 2 attractors, a pair of period 4 attractors, a pair of multi-period attractors is shown in FIGS. 4(a), (b), (c), and (d), respectively; FIGS. 4(e) and (f) show attractors coexisting with cycle 1 and a wrap in the neighborhood of 0.39. ltoreq. b.ltoreq.0.41, respectively; FIGS. 4(g) and (h) show two coexisting triple scroll attractors of different amplitudes, respectively, in the neighborhood of 0.42. ltoreq. b.ltoreq.0.64; FIGS. 4(i) and (j) show the behavior of a two-scroll chaotic attractor and a pair of multiple attractors with period 2 coexisting in the neighborhood of 0.65. ltoreq. b.ltoreq.1, respectively. When the initial value is selected to be [0.5,0,0], the system parameter α is 8, β is 12, a is 0.3, and b is 0.7257, the time domain waveform of [0s, 2000s ] is shown in fig. 5(a), and the time domain waveform of the state variable y having the intermittent chaotic behavior in fig. 5 (a). When the remaining parameters are unchanged, b is 0.78, fig. 5(b) shows that the time-domain waveform at [0s, 1500s ] has two different time intervals [0s, 900s ] and [900s, 1500s ]. Therefore, the Chua's system can be proved to have transient behavior of a steady chaotic transient period, and the behavior is related to the change of system parameters; by capturing these phenomena, the Chua's system was demonstrated to have abundant kinetic behavior.
As an alternative embodiment, the system variable may add a corresponding shift variable to achieve chaotic attractor shifting. Let the offset variable of variable x be k1Offset variable of variable y is k2Offset variable of variable z is k3Then the state variables x, y, z can be respectively x + k1,y+k2,z+k3And (4) substituting. The offset variable can be lifted freely to any position, becoming a bipolar signal or a unipolar signal. The zeiss system enables different attractors to be lifted to any position of the balance plane by setting the appropriate values of the system parameter b and then by adding an offset variable to the state variables. In one embodiment, considering the offset lifting state variable x, the state equation is as follows:
when the initial value is [0.5, 0]]When the system parameter α is 8, β is 12, and a is 0.3, fig. 6 shows the offset of the one-to-three scroll chaotic attractor with the parameters b being 0.4, b being 0.5, and b being 0.65. The offset variable allows the chaotic signal x to be tuned from a bipolar signal to a unipolar signal. Specifically, fig. 6(a) shows a scroll projected onto the x-z plane when b is 0.4; fig. 6(b) shows that when b is 0.5, the three scrolls are projected onto the x-z plane; fig. 6(c) shows that when b is 0.65, the two scrolls project onto the x-z plane. In FIG. 6(a), (b), and (c), the leftmost wrap corresponds to k1-1; middle scroll corresponds to k10; rightmost scroll corresponds to k1＝1。
As an alternative embodiment, since the attractors of the existing typical zeiss system are symmetric with respect to the origin of the coordinates, all attractors occur either as a single symmetric attractor or as pairs of mutual symmetry. In order to break the symmetry of the attractor, a constant excitation c is added in the Chua system, so that the symmetry of the attractor can be broken, and a state equation after the c is added can be described as follows:
when the initial value is selected as [0.5,0,0], the system parameter α is 8, β is 12, and a is 0.3, this embodiment makes c be 0.01, fig. 7 shows the asymmetric chaotic attractor of one to three scrolls with parameters b being 0.4, b being 0.5, and b being 0.58, specifically, fig. 7(a) shows the asymmetric chaotic attractor of one scroll when b is 0.4; fig. 7(b) shows that when b is 0.5, the asymmetric three-scroll chaotic attractor; fig. 7(c) shows that when b is 0.58, the asymmetric two-scroll chaotic attractor.
The embodiment provides an improved Chua's system, compared with a typical Chua's system, the Chua's system uses a composite hyperbolic tangent cubic nonlinear function, can generate a three-scroll chaotic attractor, and numerically shows abundant dynamic behaviors of coexistence of a plurality of attractors, a transient period, intermittent chaos, attractor offset and the like through a bifurcation diagram, a Lyapunov index, a phase diagram and a time domain diagram; and after the constant excitation is added, the symmetrical characteristic of the chaotic attractor is destroyed by the constant excitation. The Chua's system can be realized through a digital circuit (realized based on an FPGA development board), so that influence on result errors caused by factors such as temperature, component aging and the like of the traditional Chua's analog circuit is not required to be considered.
A second embodiment;
referring to fig. 8 to 11, a multi-scroll zeiss chaotic signal generator designed based on an FPGA is provided, which is used for enabling the above-mentioned improved zeiss system to output chaotic signals, wherein a XilinxVirtex-6(ZYNQ-XC7Z020) FPGA development board is used to build the overall architecture of the chaotic generator, and the multi-scroll zeiss chaotic signal generator includes: a zeiss oscillator, a control unit and an analog converter (DAC), wherein:
and performing numerical modeling on the Chua's system on the FPGA by adopting an RK4 algorithm to generate a Chua's oscillator, wherein the Chua's oscillator generates chaotic signals through an RK4 algorithm.
The control unit is electrically connected with the Chua's oscillator, and the control unit is a Moore state machine and is responsible for managing and scheduling different operations and functions of the Chua's oscillator.
The analog converter is connected to the control unit for converting the digital signal generated by the zeiss oscillator into an analog form that can be shown in an oscilloscope.
Preferably, the control unit controls the Chua's oscillator to generate the chaotic signal (x-y) with the word length of 32 bits, the analog converter converts the chaotic signal (x-y) with the word length of 32 bits into an analog form, the process is repeated, and the analog converter can output the real-time chaotic signal.
Compared with the algorithms such as Euler, Heun and RK5 Butcher, the RK4 algorithm adopted in the embodiment can produce better results and has smaller error range. The process of solving the system of equations using the RK4 algorithm is as follows:
Kx1＝Δh[α(yK-(axK-btanh(xK 3)))] (11)
Kx4＝Δh[α(yK-(a(xK+Kx3)-btanh((xK+Kx3)3)))] (14)
Ky1＝Δh[xK+1-yK+zK] (16)
Ky4＝Δh[xK+1-(yK+Ky3)+zK] (19)
Kz1＝Δh[-βyK+1] (21)
Kz2＝Δh[-βyK+1] (22)
Kz3＝Δh[-βyK+1] (23)
Kz4＝Δh[-βyK+1] (24)
wherein, Δ h is the discretization step length in the numerical solution, the value of Δ h is 0.001, let xK,yK,zKWith the initial condition of x0＝0.5,y0＝0,z0＝0。
Since the hyperbolic tangent function consists of an infinite exponential series, a piecewise linear (PWL) method is introduced as an option, and the PWL method has similar transition between an upper saturation region and a lower saturation region, so that the hyperbolic tangent function can be better approximated. A piecewise linear function like tanh () can be represented by:
where L2, β 1 and θ 0.25, β and θ are used to determine the piecewise linear function HS(z) slope and gain between-L ≦ x ≦ L.
One set of experimental results is provided below:
the Chua's oscillator and the control unit are created by a Xilinx IP core generator, 3 output signals of the generator conform to the 32-byte IEEE754-1985 floating point number standard in Verilog HDL on FPGA, and the generator is synthesized through a Vivado2018.3 platform.
The following table 1 counts chip resource use information for realizing chaotic signals on the FPGA through the RK-4 algorithm. The multi-scroll Chua's chaotic signal generator runs at a clock frequency of 67.95MHz, and the minimum running period is 14.716 ns. Fig. 9 shows the simulation result of the zeiss oscillator in operation for the vivado2018.3 platform. Fig. 10 shows a chua attractor generated by a multi-scroll chua chaotic signal generator, specifically, fig. 10(a) shows a periodic attractor phase diagram, fig. 10(b) shows a one-scroll attractor phase diagram, fig. 10(c) shows a three-scroll attractor phase diagram, and fig. 10(d) shows a two-scroll attractor phase diagram. FIG. 11 shows an experimental apparatus of this example. Experimental results show that the phase diagram obtained based on the MATLAB and FPGA models has good consistency.
TABLE 1
The embodiment is based on a 32-byte IEEE754-1985 floating point number standard, a multi-scroll Chua's chaotic signal generator is designed based on an FPGA, FPGA-based digital implementation is carried out on the Chua's system provided by the embodiment, and experimental observation on a chaotic attractor proves that the chaotic attractor is suitable for generating chaotic behaviors. Compared with the conventional Chua's system which generates a multi-scroll Chua's chaotic signal by building an analog circuit, the embodiment provides a mode for realizing the improved Chua's system based on the FPGA, so that the influence of the traditional Chua's analog circuit on result errors caused by factors such as temperature, aging of components and the like is not required to be considered.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an illustrative embodiment," "an example," "a specific example," or "some examples" or the like mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
While embodiments of the invention have been shown and described, it will be understood by those of ordinary skill in the art that: various changes, modifications, substitutions and alterations can be made to the embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.
Claims (8)
1. An improved Chua's system, characterized in that the equation of state of the Chua's system is:
wherein x, y, z are system variables of the Chua's system, α, β are system parameters, and h (x) ax-btan (x)3) Where a is a constant value and b is a parameter for controlling the number of scrolls.
2. An improved zeiss system as claimed in claim 1, characterized in that the initial values of the zeiss system are [0.5,0,0], the α -8, the β -12 and the a-0.3.
3. An improved zeiss system as claimed in claim 1, characterized in that the system variables add corresponding offset variables.
4. An improved zeiss system as claimed in claim 1, characterized in that the zeiss system further comprises a constant excitation c, the equation of state being:
5. the improved zeiss system as claimed in claim 4, characterized in that c ═ 0.01.
6. A Chua's chaotic signal generator based on FPGA is characterized by comprising:
the Chua's oscillator is designed on an FPGA development board by the improved Chua's system of any one of claims 1 to 5 and is used for generating chaotic signals;
the control unit is connected with the Chua's oscillator and used for managing and scheduling the Chua's oscillator so as to enable the Chua's oscillator to generate chaotic signals;
and the analog converter is connected with the control unit and is used for converting the generated chaotic signal into an analog signal.
7. The FPGA-based Chua's chaotic signal generator according to claim 6, wherein the Chua's oscillator is generated through an RK4 algorithm and generates chaotic signals through an RK4 algorithm.
8. The Chua's chaotic signal generator based on the FPGA of claim 6 is characterized in that the model of the FPGA development board is ZYNQ-XC7Z 020.
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Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN114614967A (en) * | 2022-02-23 | 2022-06-10 | 北京电子科技学院 | FPGA system adopting controllable grid multi-scroll chaotic system |
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