A universal simplified chaotic circuit design method

Abstract

The present invention relates to a simplified circuit implementation of a chaotic system, and in particular to a circuit implementation of a universal simplified chaotic system. For a system in which the equation does not contain a corresponding negative linear term, a voltage follower isolation method can be used to implement the system, or a common-mode proportional circuit compensation method can be used to implement the system. For a system in which the equation contains a corresponding negative linear term, a direct method can be used to implement the system, or a voltage follower compensation method can be used to implement the system. For a system in which the equation contains a corresponding positive linear term, the common-mode proportional circuit compensation method can be used to implement the system. A method for implementing a chaotic system that uses only one operational amplifier and can implement the system without a corresponding negative term or with a corresponding positive term is proposed. The method has universality and can be used to implement various chaotic systems.

Description

Technical Field

[0001] The invention relates to a simplified circuit design of a chaotic system, and in particular to a design method of a universal simplified chaotic circuit. Background Art

[0002] Using circuits to realize chaotic systems is the basis for applying chaotic systems to engineering practice. The main method to realize chaotic system circuits is to use multipliers to realize multiplication operations. Resistors and capacitors cooperate with operational amplifiers to realize addition, subtraction, integration, inversion and other operations. For example, Wang Fanzhen et al. used 9 operational amplifiers to realize a three-dimensional chaotic system in the 8th issue of "Acta Physica Sinica" in 2006. Niu Yujian et al. used 12 operational amplifiers to realize a four-dimensional hyperchaotic system in the 15th volume of "Commun Nonlinear Sci Numer Simulat" in 2010. Li Qingdu et al. used 4 operational amplifiers to realize a hyperchaotic system in the 42nd volume of "INTERNATIONAL JOURNALOF CIRCUIT THEORY AND APPLICATIONS" in 2014. Generally, at least three or four operational amplifiers are required to realize a three-dimensional chaotic system. The number of devices used is relatively large, and the frequency cannot be too high, which affects the functional application of the chaotic system. American scholar Jonathan In 2007, N.Blakely et al. proposed a method in Chaos, Vol. 17, 023112, which does not use an op amp and realizes the method proposed by American scholar Lorenz in J.Atmos.Sci.》Volume 20, increased the frequency of chaotic systems. Recently, Wu Junyao et al. proposed a method for a modifiable Lorenz system in Volume 69, Issue 3 of IEEE TRANSACTIOS ON CIRCUITS AND SYSTEM-II:EXPRESS BRIEFS, 2022. These two methods can only realize systems with equation forms such as Lorenz type, that is, each equation of the system contains negative linear terms (e.g., the first equation contains -x, the second equation contains -y, and the third equation contains -z). For chaotic systems that do not contain all negative linear terms in each equation of the system, they cannot be realized. The Chen system proposed by Chen Guanrong et al. in Issue 9 of International Journal of Bifrucation and Choas in 1999 and the Chen system proposed by Lv Jinhu et al. in International Journal of Bifrucation in 2002 The second equation of the Lü system proposed in the 3rd issue of "Chaos, Solitons and Fractals" by Liu Chongxin et al. in 2004 and the second equation of the Yang-Cheng system proposed by Yang Qigui et al. in the 5th issue of "International Journal of Bifrucation and Choas" in 2008 do not contain y. For these four types of systems, the existing methods can only be implemented by using at least 3 or 4 operational amplifiers. The present invention proposes a method that can implement the above five types of systems using only one operational amplifier. The method is universal and can be used to implement other types of chaotic systems. Summary of the invention

[0003] The technical problem to be solved by the present invention is to provide a design method and implementation of a universal simplified chaotic system circuit. The present invention adopts the following technical means to achieve the purpose of the invention:

[0004] 1. A design method and implementation of a universal simplified chaotic system circuit, characterized in that: for systems containing corresponding negative linear terms in the equation, it can be implemented by a direct method or by a voltage follower compensation method; for systems that do not contain corresponding negative linear terms in the equation, it can be implemented by a voltage follower isolation method, by an in-phase proportional circuit compensation method, or by a negative resistance circuit compensation method; for systems that contain corresponding positive linear terms in the equation, it can be implemented by an in-phase proportional circuit compensation method or by a negative resistance circuit construction method.

[0005] 2. A design method and implementation of a universal simplified chaotic circuit, characterized in that: for a system containing a corresponding negative linear term in the equation, it can be implemented by a direct method or a voltage follower method;

[0006] The equation of the Lorenz chaotic system is:

[0007] a, b, c are positive numbers

[0008] The second equation of equation i contains -y, and the system i is realized by the direct method. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The multiplier integrates the output of the second equation through a resistor on a capacitor, and the multiplier integrates the output of the third equation through a resistor and connects the resistor in parallel with the capacitor to obtain the output of the third equation.

[0009] The second equation of equation i contains -y. The voltage follower method is used to implement system i. The output of the second equation is connected to the output of the first equation through a resistor, and then the output of the first equation is obtained by integrating the second equation through a capacitor. The output of the second equation is connected to an input of a multiplier through a voltage follower. The multiplier integrates the capacitor through a resistor to obtain the output of the second equation. The multiplier integrates the capacitor through a resistor and connects the resistor in parallel with the capacitor to obtain the output of the third equation.

[0010] 3. A design method and implementation of a universal simplified chaotic circuit, characterized in that: for a system that does not contain a corresponding negative linear term in the equation, a voltage follower isolation method is used for implementation, an in-phase proportional circuit compensation method is used for implementation, and a negative resistance circuit compensation method is used for implementation;

[0011] Adding a coefficient d to -y in the second equation of the Lorenz chaotic system equation, we can get the chaotic system equation as follows:

[0012] When d = 0, system ii is Yang-Chen

[0013] a, b, c are positive numbers

[0014] The second equation of equation iii does not contain y. The second equation is realized by isolation method using a voltage follower. The output of the second equation is connected to the output of the first equation through a voltage follower and a resistor, and then the output of the first equation is obtained by integrating the multiplier on the capacitor through the resistor to obtain the output of the second equation. The multiplier is integrated on the capacitor through the resistor and connected in parallel with the capacitor through the resistor to obtain the output of the third equation.

[0015] The second equation of equation iii does not include y. The second equation is realized by using the in-phase proportional circuit compensation method. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The output of the second equation is connected to an input terminal of a multiplier through an in-phase proportional circuit. The multiplier integrates the capacitor through a resistor to obtain the output of the second equation. The multiplier integrates the capacitor through a resistor and is connected in parallel with the capacitor through a resistor to obtain the output of the third equation.

[0016] The second equation of equation iii does not include y. The second equation is implemented by constructing a negative resistance circuit compensation method. The output of the second equation is connected to the output of the first equation through a resistor, and then the output of the first equation is obtained by integrating the capacitor. The output of the second equation is connected to an input terminal of the multiplier through a negative resistance circuit. The multiplier integrates the capacitor through the resistor to obtain the output of the second equation. The multiplier integrates the capacitor through the resistor and is connected in parallel with the capacitor through the resistor to obtain the output of the third equation.

[0017] 4. A design method and implementation of a universal simplified chaotic circuit, characterized in that: for a system containing a corresponding positive linear term in the equation, it can be implemented by an in-phase proportional circuit compensation method or by constructing a negative resistance method;

[0018] Adding a coefficient -d to -y in the second upper equation of the Lorenz chaotic system equation, we can get the Chen type and Lü type chaotic system equations as follows:

[0019]

[0020] When d>0, b<0, system iv is a Chen type system, when d>0, b=0, system iv is a Lü type system, when d>0, b>0, system iv is a quasi-Chen type and Lü type system v;

[0021] a, c, d are positive numbers,

[0022] The second equation of equation v contains +y, and the system v is realized by using the in-phase proportional circuit compensation method. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The output of the second equation is connected to an input terminal of a multiplier through an in-phase proportional circuit. The multiplier integrates the output of the second equation on the capacitor through a resistor, and the multiplier integrates the output of the second equation on the capacitor through a resistor and connects the resistor in parallel with the capacitor to obtain the output of the third equation.

[0023] The second equation of equation v contains +y, and the system v is realized by constructing a negative resistance circuit. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The output of the second equation is connected to an input terminal of a multiplier through a negative resistance circuit. The multiplier integrates the capacitor through a resistor to obtain the output of the second equation. The multiplier integrates the capacitor through a resistor and is connected in parallel with the capacitor through a resistor to obtain the output of the third equation.

[0024] The beneficial effect of the present invention is that a method for realizing a chaotic system without corresponding negative terms or including corresponding positive terms is proposed by using only one operational amplifier. The method is universal and can be used to realize various chaotic systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 Circuit diagram, numerical simulation and circuit simulation diagram for the direct method to realize chaotic system i.

[0026] Figure 2 This is the circuit diagram, numerical simulation and circuit simulation diagram of chaotic system i realized by voltage follower method.

[0027] Figure 3 Circuit diagram, numerical simulation and circuit simulation diagram for realizing chaotic system III using voltage follower isolation method.

[0028] Figure 4 Circuit diagram, numerical simulation and circuit simulation diagram for realizing chaotic system III using the in-phase proportional circuit method.

[0029] Figure 5 Circuit diagram, numerical simulation and circuit simulation diagram for realizing chaotic system III using negative resistance circuit method.

[0030] Figure 6 Circuit diagram, numerical simulation and circuit simulation diagram of chaotic system v realized by in-phase proportional circuit method.

[0031] Figure 7 This is the circuit diagram, numerical simulation and circuit simulation diagram for realizing chaotic system v using the negative resistance circuit method. DETAILED DESCRIPTION

[0032] The present invention is described in further detail below in conjunction with the accompanying drawings and preferred embodiments. Figure 1-Figure 7 .

[0033] 1. For systems containing corresponding negative linear terms in the equation, the direct method or the voltage follower method can be used to implement it;

[0034] The equation of the Lorenz chaotic system is:

[0035] a, b, c are positive numbers

[0036] The second equation of equation i contains -y, and the system i is realized by the direct method. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The multiplier integrates the output of the second equation through a resistor on a capacitor, and the multiplier integrates the output of the third equation through a resistor and connects the resistor in parallel with the capacitor to obtain the output of the third equation.

[0037] according to Figure 1 The circuit diagram in , and taking R2 = 100R1, we get:

[0038]

[0039] Take the parameters a=10, b=40, c=8 / 3, and select the appropriate scale change to obtain the numerical simulation diagram and circuit simulation diagram as shown in Figure 1 As shown in , the numerical simulation diagram is consistent with the circuit simulation diagram, so the designed circuit can realize the system.

[0040] The second equation of equation i contains -y, and system i is implemented using the voltage follower method. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The output of the second equation is connected to an input terminal of a multiplier through a voltage follower, and the multiplier integrates the capacitor through a resistor to obtain the output of the second equation. The multiplier integrates the capacitor through a resistor and is connected in parallel with the capacitor through a resistor to obtain the output of the third equation.

[0041] according to Figure 2 The circuit diagram in , and taking R2 = 100R1, we get:

[0042]

[0043] Take the parameters a=10, b=40, c=8 / 3, and select the appropriate scale change to obtain the numerical simulation diagram and circuit simulation diagram as shown in Figure 5 As shown in , the numerical simulation diagram is consistent with the circuit simulation diagram, so the designed circuit can realize the system.

[0044] 2. For systems that do not contain corresponding negative linear terms in the equation, the voltage follower isolation method can be used to implement it, or the in-phase proportional circuit compensation method can be used to implement it;

[0045] Adding a coefficient d to -y in the second equation of the Lorenz chaotic system equation, we can get the chaotic system equation as follows:

[0046] When d = 0, system ii is Yang-Chen

[0047] a, b, c are positive numbers

[0048] The second equation of equation iii does not contain y. The second equation is realized by isolation method using a voltage follower. The output of the second equation is connected to the output of the first equation through a voltage follower and a resistor, and then the output of the first equation is obtained by integrating the multiplier on the capacitor through the resistor to obtain the output of the second equation. The multiplier is integrated on the capacitor through the resistor and connected in parallel with the capacitor through the resistor to obtain the output of the third equation.

[0049] See also Figure 3 From the circuit diagram, we can see that:

[0050]

[0051] Take the parameters a=10, b=40, c=8 / 3, and select the appropriate scale change to obtain the numerical simulation diagram and circuit simulation diagram as shown in Figure 3 As shown in , the numerical simulation diagram is consistent with the circuit simulation diagram, so the designed circuit can realize system iii.

[0052] The second equation of equation iii does not include y. The second equation is implemented using the in-phase proportional circuit compensation method. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The output of the second equation is connected to an input end of the multiplier through a in-phase proportional circuit. The multiplier integrates the capacitor through a resistor to obtain the output of the second equation. The multiplier integrates the capacitor through a resistor and is connected in parallel with the capacitor through a resistor to obtain the output of the third equation.

[0053] See also Figure 4 From the circuit diagram, we can see that:

[0054]

[0055] Where f(y) is the output of the in-phase proportional circuit, Take R2=100R1, we can get:

[0056]

[0057] Take the parameters a=10, b=40, c=8 / 3, and select the appropriate scale change to obtain the numerical simulation diagram and circuit simulation diagram as shown in Figure 4 As shown in , the numerical simulation diagram is consistent with the circuit simulation diagram, so the designed circuit can realize system iii.

[0058] The second equation of equation iii does not contain y. The second equation is realized by using a negative resistance circuit compensation method. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The output of the second equation is connected to an input terminal of a multiplier through a negative resistance circuit. The multiplier integrates the output of the second equation on the capacitor through a resistor. The multiplier integrates the output of the second equation on the capacitor through a resistor and is connected in parallel with the capacitor through a resistor to obtain the output of the third equation.

[0059] See also Figure 5 From the circuit diagram, we can see that:

[0060]

[0061] Where f(R) is the output of the negative resistance circuit, Take R5=R6=100kΩ, R2=100R1, and we can get: f(R)=-R7;

[0062]

[0063] Take the parameters a=10, b=40, c=8 / 3, and select the appropriate scale change to obtain the numerical simulation diagram and circuit simulation diagram as shown in Figure 3 As shown in , the numerical simulation diagram is consistent with the circuit simulation diagram, so the designed circuit can realize system iii.

[0064] 3. For systems containing corresponding positive linear terms in the equation, the compensation method can be implemented by using the in-phase proportional circuit method, or by constructing a negative resistance circuit;

[0065] In the second square, by adding a coefficient -d to -y in the second equation of the Lorenz chaotic system equation, the Chen type and Lü type chaotic system equations can be obtained as follows:

[0066]

[0067] When d>0, b<0, system iv is a Chen type system, when d>0, b=0, system iv is a Lü type system, when d>0, b>0, system iv is a quasi-Chen type and Lü type system v;

[0068] a, c, d are positive numbers,

[0069] The second equation of equation v contains +y, and the system v is realized by using the in-phase proportional circuit compensation method. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The output of the second equation is connected to an input terminal of the multiplier through a in-phase proportional circuit, and the multiplier integrates the capacitor through a resistor to obtain the output of the second equation. The multiplier integrates the capacitor through a resistor and is connected in parallel with the capacitor through a resistor to obtain the output of the third equation.

[0070] according to Figure 6 The circuit diagram in , and taking d = 1, we get:

[0071]

[0072] Where f(y) is the output of the in-phase proportional circuit, Take R2=100R1, we can get:

[0073]

[0074] Take the parameters a=10, b=40, c=8 / 3, and select the appropriate scale change to obtain the numerical simulation diagram and circuit simulation diagram as shown in Figure 6 As shown in , the numerical simulation diagram is consistent with the circuit simulation diagram, so the designed circuit can realize the system.

[0075] The second equation of equation v contains +y, and the system v is realized by constructing a negative resistance circuit. The output of the second equation is connected to the output of the first equation through a resistor, and then integrated through a capacitor to obtain the output of the first equation. The output of the second equation is connected to an input terminal of a multiplier through a negative resistance circuit. The multiplier integrates the capacitor through a resistor to obtain the output of the second equation. The multiplier integrates the capacitor through a resistor and is connected in parallel with the capacitor through a resistor to obtain the output of the third equation.

[0076] according to Figure 7 The circuit diagram in , and taking d = 2, we get:

[0077]

[0078] Where f(R) is the output of the negative resistance circuit, Taking R5=R6=100kΩ,R2=100R1, we can get: f(R)=-R7

[0079]

[0080] Take the parameters a=10, b=40, c=8 / 3, and select the appropriate scale change to obtain the numerical simulation diagram and circuit simulation diagram as shown in Figure 7 As shown in , the numerical simulation diagram is consistent with the circuit simulation diagram, so the designed circuit can realize the system.

[0081] Of course, the above description is not a limitation of the present invention, and the present invention is not limited to the above examples. Changes, modifications, additions or substitutions made by ordinary technicians in this technical field within the essential scope of the present invention also fall within the protection scope of the present invention.

Claims

1. A general simplified chaotic circuit design method, characterized in that: For systems that do not contain corresponding negative linear terms in the equation, the compensation method is implemented by using the in-phase proportional circuit method or the negative resistance circuit method; Among them, the system that does not contain the corresponding negative linear term in the equation is the Yang-Chen system: a, b, c are positive numbers The implementation using the in-phase proportional circuit compensation method includes: the second equation of the system i does not contain y, the second equation is implemented using the in-phase proportional circuit compensation method, the output of the second equation is connected to the output of the first equation through the second resistor, and then the output of the first equation is obtained by the capacitance integration of the second resistor and the first capacitor; the output of the second equation is connected to an input end of the first multiplier through the in-phase proportional circuit, the output of the first multiplier is connected to the output of the second equation through the first resistor, the output of the second equation is obtained by the capacitance integration of the output of the first multiplier on the first resistor and the second capacitor and the capacitance integration of the second resistor and the second capacitor; the output of the third equation is obtained by the capacitance integration of the output of the second multiplier on the third resistor and the third capacitor and the parallel connection of the fourth resistor and the third capacitor; The implementation using the negative resistance circuit compensation method includes: the second equation of the system i does not contain y, the second equation is implemented by constructing a negative resistance circuit compensation method, the output of the second equation is connected to the output of the first equation through the second resistor, and then the output of the first equation is obtained by the capacitance integration of the second resistor and the first capacitor; the output of the second equation is connected to an input end of the first multiplier, the output of the first multiplier is connected to the output of the second equation through the first resistor, the output of the second equation is obtained by the capacitance integration of the output of the first multiplier on the first resistor and the second capacitor, the capacitance integration of the second resistor and the second capacitor, and the parallel connection of the negative resistance circuit and the second capacitor; the output of the third equation is obtained by the capacitance integration of the output of the second multiplier on the third resistor and the third capacitor, and the parallel connection of the fourth resistor and the third capacitor.

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