CN112491530B - Chua's chaotic signal generator based on FPGA - Google Patents

Chua's chaotic signal generator based on FPGA Download PDF

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CN112491530B
CN112491530B CN202011344984.7A CN202011344984A CN112491530B CN 112491530 B CN112491530 B CN 112491530B CN 202011344984 A CN202011344984 A CN 202011344984A CN 112491530 B CN112491530 B CN 112491530B
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chua
chaotic
zeiss
fpga
signal generator
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CN112491530A (en
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余飞
沈辉
张梓楠
黄园媛
蔡烁
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Changsha University of Science and Technology
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    • 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

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Abstract

Compared with a typical Chua's system, the improved Chua's system provided by the invention uses a composite hyperbolic tangent three-time nonlinear function to generate a three-scroll chaotic attractor, and numerical values show richer dynamics such as coexistence of a plurality of attractors, transient periods, 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 existing Chua's system, the method for generating the multi-scroll Chua's chaotic signal by constructing the analog circuit provides a mode of realizing the improved Chua's system based on the FPGA, so that the effect error is not influenced by factors such as temperature, component aging and the like of the traditional Chua's analog circuit.

Description

Chua's chaotic signal generator based on FPGA
Technical Field
The invention relates to the technical field of Chua's system, in particular to an improved Chua's system and a Chua's chaotic signal generator based on an FPGA.
Background
In recent years, chaotic dynamics has become a very rich content and widely applied research field. Secure communication, image encryption, memory, system synchronization, random number generator, complex network, etc. are all chaos and intensive studies on its application.
The multi-scroll chaotic circuit has important application in the fields of chaotic communication, information safety, image encryption and the like, and has become a hotspot in chaotic theory research. In order to produce a multi-scroll chaotic attractor in some simple chaotic system, some nonlinear 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 Chua's circuit (Chua's system) established in 1983 by Chua's professor. This is a precedent for studying chaos phenomenon by using electronic circuits. The typical Chua's system can generate double-scroll chaotic attractors, and how to improve the typical Chua's system to obtain richer dynamic behaviors is an important point of current chaotic research. In addition, at present, two methods for realizing chaos in an analog circuit are mainly adopted for hardware implementation of a multi-scroll chaotic system (comprising a typical Chua's system), namely discrete element circuit design and chip integrated circuit design, and the chaotic system is extremely sensitive to an initial value due to the fact that discrete elements are easily influenced by environmental factors such as temperature, aging and the like, so that the effect of realizing the chaotic system by utilizing the discrete element circuit is very limited.
Disclosure of Invention
The present invention aims to solve at least one of the technical problems existing in 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 also provided with a mode based on FPGA, so that the result error is not influenced by factors such as temperature, component aging and the like of a traditional Chua's analog circuit.
In a first aspect of the present invention, an improved zeiss system is provided, and the state equation of the zeiss system is:
Wherein x is the system variable of the Chua's system, α is the system parameter, β is the system parameter, h (x) =ax-btanh (x 3), a is a constant value, and b is the 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's system, the Chua's system uses a composite hyperbolic tangent three-time nonlinear function, can generate a three-scroll chaotic attractor, and numerically shows richer dynamics such as coexistence of a plurality of attractors, transient periods, intermittent chaos, attractor offset and the like through a bifurcation diagram, a Lyapunov exponent, 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 by a digital circuit (based on FPGA), so that the result error is not influenced by factors such as temperature, component aging and the like of a traditional Chua's analog circuit.
According to some embodiments of the invention, the initial value of the zeiss system is [0.5, 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=0.01.
In a second aspect of the present invention, there is provided a zeiss chaotic signal generator based on an FPGA, including:
the Chua's oscillator is designed on an FPGA development board by the improved Chua's system of 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 that the Chua's oscillator generates 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 prior Chua's system, the invention provides a mode of realizing the improved Chua's system based on the FPGA by constructing an analog circuit to generate the multi-scroll Chua's chaotic signal, so that the effect error of the traditional Chua's analog circuit due to temperature, component aging and other factors is not required to be considered.
According to some embodiments of the invention, the zeiss oscillator is generated by an RK4 algorithm and generates the chaotic signal by the RK4 algorithm.
According to some embodiments of the invention, the FPGA development board is model number ZYNQ-XC7Z020.
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 foregoing and/or additional aspects and advantages of the invention will become apparent and may be better understood from the following description of embodiments taken in conjunction with the accompanying drawings in which:
fig. 1 is a phase diagram of a zeiss system simulated by using MATLAB and using different control parameters according to an embodiment of the present invention;
Fig. 2 is a bifurcation diagram and a lyapunov exponent spectrum of the present zeiss system according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of the number and location of balance 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 co-existing attractor of the zeiss system under the variation of the parameter b according to the embodiment of the present invention;
fig. 5 is a schematic diagram of transient transition behavior in the zeiss system according to an embodiment of the present invention;
FIG. 6 is a schematic diagram of one to three scroll chaotic attractor offsets at three different values of k 1 according to an embodiment of the present invention;
FIG. 7 is a schematic diagram of a two-dimensional projection of a one-to-three-scroll asymmetric chaotic attractor after constant excitation is added according to an embodiment of the present invention;
fig. 8 is a schematic structural diagram of a zeiss chaotic signal generator based on an FPGA according to an embodiment of the present invention;
fig. 9 is a simulation timing diagram of a zeiss oscillator according to an embodiment of the present invention;
fig. 10 is a schematic diagram of generating a zeiss attractor by a zeiss chaotic signal generator based on an FPGA according to an embodiment of the present invention;
Fig. 11 is a schematic diagram of an experimental apparatus according to an embodiment of the present invention.
Detailed Description
Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to like or similar elements or elements having like or similar functions throughout. The embodiments described below by referring to the drawings are illustrative only and are not to be construed as limiting the invention.
A first embodiment;
referring to fig. 1 to 7, there is provided an improved zeiss system, the state equation of which is:
Wherein x, y, z are system variables of Chua's system, alpha, beta are system parameters, h (x) is a composite hyperbolic tangent cubic nonlinear function, h (x) =ax-btanh (x 3), a is a constant value, and b is a parameter for controlling the number of scrolls. Compared with a typical Chua's system, the Chua's system provided in this embodiment uses a composite hyperbolic tangent three-time nonlinear function, so that the Chua's system has more abundant dynamic behavior than the typical Chua's system.
As an alternative embodiment, when the initial value of the zeiss system is [0.5, 0], the system parameters α=8, β=12, a=0.3, fig. 1 shows that MATLAB is used to simulate the phase diagram of the zeiss system with the control parameters b=0.4, b=0.5, and b=0.65, respectively, and the numerical simulation results of one to three scrolls zeiss attractors are realized through different parameter values b, specifically, fig. 1 (a) shows the phase diagram of one scroll on the x-y plane; FIG. 1 (b) shows a phase diagram of one wrap in the x-z plane; FIG. 1 (c) shows a phase diagram of one scroll 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 exponent spectra of the system (1) are respectively shown in fig. 2, along with the increase of the parameter b, one to three scrolls of chaotic attractors are realized, and in particular, fig. 2 (a) is a bifurcation diagram of a state variable x; fig. 2 (b) is the corresponding lyapunov exponent profile. The results demonstrate that the Chua's system is capable of producing a triple scroll Chua's attractor by setting the appropriate initial parameters and then adjusting the appropriate parameters b, as compared to a typical Chua's system.
Solving the balance point of the Chua's system: let the left of equation (1) be zero, then there is:
When b=0.6, the curve of the complex hyperbolic tangent cubic function has five determined balance points, four of which are symmetrical with respect to the origin of coordinates. Fig. 3 shows a smooth curve of h (x) =0.3 x-0.6tanh (x 3), the five equilibrium points determined are:
Formula (1) at equilibrium point The above jacobian matrix is:
Therefore, the characteristic equation of the balance point is:
in formula (5):
Obviously, these five equilibrium points, which are globally unstable saddle points, do not meet Routh-Hurwitz (laus-hellvetz stability criterion) conditions. The characteristic values of the five balance points can be obtained:
compared with three balance points generated by the conventional typical Chua's system, the Chua's system can generate five balance points under proper parameters, can generate richer dynamic behaviors, for example, can be applied to the field of information encryption, and can improve the encryption security.
When the initial value is [0.5, 0], and the system parameters are α=8, β=12, and a=0.3, the coexistence phenomenon of multiple attractors with different attractors of different subtypes, different topologies or different periods in the Chua's system is illustrated by the phase plane track: in the range of 0.3.ltoreq.b.ltoreq.0.37, coexistence of a pair of period 1 attractors, a pair of period 2 attractors, a pair of period 4 attractors, and a pair of multicycle attractors is shown in fig. 4 (a), (b), (c), and (d), respectively; FIGS. 4 (e) and (f) show the attractors coexisting with cycle 1 and a wrap in the neighborhood of 0.39.ltoreq.b.ltoreq.0.41, respectively; in the neighborhood of 0.42.ltoreq.b.ltoreq.0.64, FIGS. 4 (g) and (h) show two coexisting triple vortex attractors of different amplitudes, respectively; FIGS. 4 (i) and (j) show the behavior of a two-scroll chaotic attractor and a pair of multiple attractors coexisting in the 0.65.ltoreq.b.ltoreq.1 neighborhood for cycle 2, respectively. When the initial value is selected to be [0.5, 0], the system parameters α=8, β=12, a=0.3, b= 0.7257, the [0s,2000s ] time domain waveform is shown from fig. 5 (a), the time domain waveform of the state variable y of fig. 5 (a) with intermittent chaotic behavior. When the remaining parameters are unchanged, b=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, transient behavior of the Chua's system with a transient period of steady state chaos can be proved, and the behavior is related to the change of system parameters; by capturing these phenomena, the Chua's system is proved to have rich dynamic behaviors.
As an alternative embodiment, the system variable may add a corresponding offset variable to achieve chaotic attractor offset. Let the offset variable of the variable x be k 1, the offset variable of the variable y be k 2, and the offset variable of the variable z be k 3, the state variables x, y, z may be replaced by x+k 1,y+k2,z+k3, respectively. The offset variable can be lifted freely to any position, thereby becoming a bipolar signal or a unipolar signal. The zeiss system can enable different attractors to be lifted to any position of the balance plane by setting the appropriate value of the system parameter b and then adding an offset variable to the state variable. Providing an embodiment, consider the offset lifting state variable x, the state equation is as follows:
when the initial values are [0.5, 0], the system parameters α=8, β=12, a=0.3, fig. 6 shows the offset of the parameters b=0.4, b=0.5, and b=0.65 one to three-scroll chaotic attractors. The offset variable allows the chaotic signal x to be tuned from a bipolar signal to a unipolar signal. Specifically, fig. 6 (a) shows one scroll projected onto the x-z plane when b=0.4; fig. 6 (b) shows three scrolls projected onto the x-z plane when b=0.5; fig. 6 (c) shows that when b=0.65, the two scrolls are projected onto the x-z plane. The leftmost scroll in fig. 6 (a), (b), (c) corresponds to k 1 = -1; the middle scroll corresponds to k 1 =0; the rightmost wrap corresponds to k 1 =1.
As an alternative embodiment, since the attractors of the existing typical zeiss system are symmetrical with respect to the origin of coordinates, all attractors occur either as a single symmetrical attractor as mutually symmetrical pairs. In order to break the symmetry of the attractor, a constant excitation c is added in the Chua's system, so that the symmetry of the attractor can be broken, and a state equation after c is added can be described as follows:
When the initial value is selected to be [0.5, 0], the system parameter α=8, β=12, a=0.3, this embodiment makes c=0.01, fig. 7 shows an asymmetric chaotic attractor of one to three scrolls of the parameters b=0.4, b=0.5, and b=0.58, and specifically, fig. 7 (a) shows an asymmetric one-scroll chaotic attractor when b=0.4; fig. 7 (b) shows an asymmetric three-scroll chaotic attractor when b=0.5; fig. 7 (c) shows an asymmetric two-scroll chaotic attractor when b=0.58.
Compared with a typical Chua's system, the improved Chua's system uses a composite hyperbolic tangent three-time nonlinear function, can generate a three-scroll chaotic attractor, and numerical values show rich dynamics behaviors such as coexistence of a plurality of attractors, transient periods, intermittent chaos, attractor offset and the like through a bifurcation diagram, a Lyapunov exponent, 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 by a digital circuit (based on an FPGA development board), so that the result error is not influenced by factors such as temperature, component aging and the like of a traditional Chua's analog circuit.
A second embodiment;
Referring to fig. 8 to 11, there is provided a multi-scroll zeiss chaotic signal generator designed based on an FPGA for enabling the improved zeiss system to output chaotic signals, wherein a XilinxVirtex-6 (ZYNQ-XC 7Z 020) FPGA development board is used to build the overall architecture of the chaotic signal generator, the multi-scroll zeiss chaotic signal generator includes: a zeiss oscillator, a control unit and an analog converter (DAC), wherein:
and carrying out numerical modeling on the Chua's system on the FPGA by adopting an RK4 algorithm to generate a Chua's oscillator, and generating a chaotic signal by the Chua's oscillator through the RK4 algorithm.
The control unit is electrically connected with the Chua's oscillator, and is a mole state machine responsible for managing and scheduling the 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's oscillator into an analog form that can be shown in an oscilloscope.
Preferably, the control unit controls the Chua's oscillator to generate a chaotic signal (x-y) with a 32-bit word length, the analog converter converts the chaotic signal (x-y) with the 32-bit word length into an analog form, and the analog converter can output a real-time chaotic signal by repeating the process.
Compared with the Euler, heun, RK5 Butcher and other algorithms, the RK4 algorithm is adopted in the embodiment, so that better results can be produced, and the error range is smaller. The process of solving the equation set 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)
Where Δh is the discretized step size in the numerical solution, the Δh value is taken to be 0.001, let x K,yK,zK be initially defined as x 0＝0.5,y0＝0,z0 =0.
Since the hyperbolic tangent function consists of an infinite exponential progression, the piecewise linear (PWL) method is introduced as an option, with similar transitions between upper and lower saturation regions, to better approximate the hyperbolic tangent function. Thus a piecewise linear function like tanh () can be expressed by:
Where l=2, β=1 and θ=0.25, β and θ are used to determine the slope and gain of piecewise linear function H S (z) between-l+.x+.l.
A set of experimental results is provided below:
Wherein the Chua's oscillator and control unit are created with an Xilinx IP core generator whose 3 output signals conform to the 32 byte IEEE754-1985 floating point standard in Verilog HDL on an FPGA and are synthesized by a Vivado2018.3 platform.
The following table 1 counts chip resource usage information of the chaotic signal realized on the FPGA through the RK-4 algorithm. The multi-scroll Chua's chaotic signal generator operates at a clock frequency of up to 67.95MHz with a minimum operating period of 14.716ns. Fig. 9 shows simulation results of the vivado2018.3 platform when the zeiss oscillator is running. Fig. 10 shows a zeiss's attractor generated by a multi-scroll zeiss's 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 model has good consistency.
TABLE 1
The embodiment designs a multi-scroll Chua's chaotic signal generator based on an FPGA based on 32-byte IEEE754-1985 floating point number standard, performs digital implementation based on the FPGA on the Chua's system provided by the embodiment, and proves that the system is suitable for generating chaotic behaviors through experimental observation of chaotic attractors. Compared with the existing Chua's system, the embodiment provides a manner of realizing the improved Chua's system based on the FPGA by constructing the analog circuit to generate the multi-scroll Chua's chaotic signal, so that the effect error is not influenced by factors such as temperature, component aging and the like of the traditional Chua's analog circuit.
In the description of the present specification, reference to the terms "one embodiment," "some embodiments," "illustrative embodiments," "examples," "specific examples," or "some examples," etc., means 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, schematic representations of the above terms do not necessarily refer to the same embodiments or examples. 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 present invention have been shown and described, it will be understood by those of ordinary skill in the art that: many changes, modifications, substitutions and variations may be made to the embodiments without departing from the spirit and principles of the invention, the scope of which is defined by the claims and their equivalents.

Claims (5)

1. The Chua's chaotic signal generator based on the FPGA is characterized by comprising the following components:
The Chua's oscillator is designed on an FPGA development board by an improved Chua's system and is used for generating chaotic signals; the state equation of the improved Chua's system is:
Wherein x, y, z are system variables of the Chua's system, α, β are system parameters, h (x) =ax-b tan h (x 3), a is a constant value, and b is a parameter for controlling the number of scrolls; the initial value of the state equation of the modified zeiss system is [0.5, 0], α=8, β=12, a=0.3; h (x) is a composite hyperbolic tangent cubic nonlinear function;
The control unit is connected with the Chua's oscillator and used for managing and scheduling the Chua's oscillator so that the Chua's oscillator generates 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.
2. The zeiss chaotic signal generator based on the FPGA according to claim 1, wherein the zeiss oscillator is generated through an RK4 algorithm and generates chaotic signals through the RK4 algorithm.
3. The zeiss chaotic signal generator based on the FPGA according to claim 1, wherein the model of the FPGA development board is ZYNQ-XC7Z020.
4. The zeiss chaotic signal generator based on the FPGA according to claim 1, wherein the system variable adds a corresponding offset variable.
5. The FPGA-based zeiss chaotic signal generator of claim 1, wherein the modified zeiss system further comprises a constant excitation c, and a state equation is:
The c=0.01.
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