Chaotic system construction based on nonlinear switching excitation

Abstract

The invention relates to a nonlinear switching excitation-based chaotic system construction, which comprises the steps of method description, system design and circuit construction, and is characterized in that a nonlinear switching excitation signal is introduced into a nonlinear system, and the amplitude and frequency of the nonlinear switching excitation signal are adjusted, so that a nonlinear power system can generate a chaotic phenomenon. According to the invention, the construction of the chaotic system with a wide application range and low cost can be realized, and the controllability of the chaotic system is improved. Then, a four-dimensional periodic nonlinear dynamic system is constructed based on the method, a circuit corresponding to the system is constructed, and the reliability and feasibility of the method are verified; the method has wide application prospects in the fields of nonlinear dynamics, chaotic system technologies and the like, and a new method is provided for scientific research and engineering technology development.

Description

Technical Field

[0001] The present invention relates to the technical field of nonlinear dynamics and chaotic systems, and in particular to a method for constructing a chaotic system based on nonlinear switching excitation. Background Art

[0002] Nonlinear systems, as carriers of complex dynamical behavior, are widely present in numerous disciplines, including natural sciences, engineering, and even social sciences, and exhibit promising application prospects. Chaotic systems, in particular, have attracted extensive research due to their extreme sensitivity to initial conditions and nonlinear characteristics such as quasi-random motion. In-depth research into the dynamical behavior and control of chaotic systems is crucial for promoting technological innovation and development in related fields.

[0003] Chaosization, as a core research direction in nonlinear science, aims to make a system exhibit a chaotic state or enhance its chaotic characteristics through specific means. Methods to achieve chaosization include using the results of mapping and cascade mapping to construct a new system or introducing constant terms or special operators into a specific system.

[0004] Although existing methods can effectively implement the chaotic process of a system, they still face the problems of high cost and mathematical difficulty when applied to high-dimensional systems. In addition, the scope of application of different methods is significantly system-dependent, which restricts the engineering application of chaotic technology. Therefore, it is necessary to study universal chaotic theories and methods applicable to complex systems. Summary of the Invention

[0005] To address the shortcomings of existing chaotic generation methods in high-dimensional systems, this paper proposes a broadly applicable and cost-effective method for constructing chaotic systems based on nonlinear switching excitation. By introducing a nonlinear switching excitation signal and then adjusting its amplitude and frequency, the proposed method achieves superior chaotic generation and generation. The accuracy of the proposed method was verified by constructing a periodic nonlinear system, and its physical reliability was verified by building a circuit.

[0006] In order to solve the above problems, the present invention proposes a method for constructing a chaotic system based on nonlinear switching excitation, which is as follows:

[0007] S1, consider an n-dimensional nonlinear dynamic system, whose state equation can be expressed as:

[0008]

[0009] x is the state variable of the system, is the coefficient matrix of the system, satisfying The stability condition, f : is a smooth nonlinear function vector, is a state variable function.

[0010] S2, introduces a nonlinear switching excitation signal into the above n-dimensional nonlinear dynamic system After that, the system transforms into a non-autonomous system:

[0011]

[0012] The coupling term G Designed as a single-dimensional injection mode, the system can be written as follows:

[0013]

[0014] Among them, is the state variable of the system, are the coefficients of the system, is a function of system variables, is 0 or 1, .in , r is the signal amplitude, D is the duty cycle, f is the signal frequency.

[0015] S3, adjust the amplitude of the nonlinear switching excitation signal r and frequency f It can cause chaotic phenomena in nonlinear dynamic systems.

[0016] The method for constructing a chaotic system based on nonlinear switching excitation is characterized in that: a four-dimensional periodic nonlinear dynamic system is constructed based on the method, and a nonlinear switching excitation signal is introduced. The specific expression of the post-system is:

[0017]

[0018] in, Introducing a functional equation with nonlinear switching excitation properties into the system; are system parameters, where r It has a regulating effect on the dynamic behavior of the system, and the nonlinear switching excitation signal plays a key role in the chaotic state induction process through its parameter coupling mechanism; when the initial value of the system is , parameter is , amplitude r and frequency f The process of change from small to large, the hierarchical transition of the system from stable equilibrium state to limit cycle periodic motion and then to chaotic motion state, affects the generation and evolution path of chaotic attractors.

[0019] A chaotic system based on nonlinear switching excitation is characterized by: designing a circuit for the chaotic system, wherein a 555 timer, a resistor, a capacitor, a 10V DC power supply and a diode constitute the time-varying circuit. item, r Indicates the coefficient related to the 555 timer. The specific physical meaning of this coefficient is: when the 555 timer is in the high level output state, the voltage value of its output pin In the circuit design, an operational amplifier is combined with a capacitor to realize the function of reverse integration, and is combined with a resistor to realize the function of an inverter. The nonlinear term in the circuit is realized by a multiplier. Using Kirchhoff's voltage law, the specific expression of the chaotic system based on the nonlinear time-varying oscillator described in claim 2 can be obtained, and the corresponding circuit equation is:

[0020]

[0021] Based on the circuit equation, the system consists of four channel circuits: the first channel circuit consists of a DC voltage source VCC, a DC voltage source VEE, operational amplifiers U1, U2, U3, resistors R1, R2, R3, R4, R5, R6, capacitor C1 and analog ground; the second channel circuit consists of a DC voltage source VCC, a DC voltage source VEE, operational amplifiers U4, U5, U6, resistors R7, R8, R9, R10, R11, R12, R13, multiplier A1, capacitor C2 and analog ground; the third channel circuit consists of a DC voltage source VCC, a DC voltage source VEE, operational amplifiers U7, U8, U9, resistors R14, R15, R16, R17, R18, R19, multipliers A2, A3, capacitor C3 and analog ground; the fourth channel circuit is composed of DC voltage sources VCC, VCC1, DC voltage source VEE, multipliers A4, A5, 555 timer A6, operational amplifiers U10, U11, U12, resistors R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, diodes D1, D2, capacitors C4, C5, C6 and analog ground;

[0022] In the first channel circuit, one end of resistors R1 and R2 is connected to the inverting input terminal of the operational amplifier U1 and is also connected to one end of the resistor R3, the other end of the resistor R1 is connected to the output terminal of the operational amplifier U3, the other end of the resistor R2 is connected to the output terminal of the operational amplifier U5, the other end of the resistor R3 is connected to one end of the resistor R4 and the output terminal of the operational amplifier U1, the other end of the resistor R4 is connected to one end of the capacitor C1 and the inverting input terminal of the operational amplifier U2, the other end of the capacitor C1 is connected to the output terminal of the operational amplifier U2 and one end of the resistor R5, the other end of the resistor R5 is connected to one end of the resistor R6 and the inverting input terminal of the operational amplifier U3, and the other end of the resistor R6 is connected to the output terminal of the operational amplifier U3;

[0023] In the second channel circuit, one input terminal of the multiplier A1 is connected to the output terminal of the operational amplifier U3, the other input terminal of the multiplier A1 is connected to the output terminal of the operational amplifier U11, one end of the resistors R7, R8, and R9 are connected to the inverting input terminal of the operational amplifier U4 and are also connected to one end of the resistor R10, the other end of the resistor R7 is connected to the output terminal of the multiplier A1, the other end of the resistor R8 is connected to the output terminal of the operational amplifier U6, the other end of the resistor R9 is connected to the output terminal of the operational amplifier U2, the other end of the resistor R10 is connected to one end of the resistor R11 and the output terminal of the operational amplifier U4, the other end of the resistor R11 is connected to one end of the capacitor C2 and the inverting input terminal of the operational amplifier U5, the other end of the capacitor C2 is connected to the output terminal of the operational amplifier U5 and one end of the resistor R12, the other end of the resistor R12 is connected to one end of the resistor R13 and the inverting input terminal of the operational amplifier U6, and the other end of the resistor R13 is connected to the output terminal of the operational amplifier U6;

[0024] In the third channel circuit, two input terminals of the multiplier A2 are connected to the output terminal of the operational amplifier U5, the output terminal of the multiplier A2 is connected to one input terminal of the multiplier A3, the other input terminal of the multiplier A3 is connected to the output terminal of the operational amplifier U2, one end of the resistors R14 and R15 are connected to the inverting input terminal of the operational amplifier U7 and one end of the resistor R16, the other end of the resistor R14 is connected to the output terminal of the operational amplifier U9, the other end of the resistor R15 is connected to the output terminal of the multiplier A3, the other end of the resistor R16 is connected to one end of the resistor R17 and the output terminal of the operational amplifier U7, the other end of the resistor R17 is connected to one end of the capacitor C3 and the inverting input terminal of the operational amplifier U8, the other end of the capacitor C3 is connected to the output terminal of the operational amplifier U8 and one end of the resistor R18, the other end of the resistor R18 is connected to one end of the resistor R19 and the inverting input terminal of the operational amplifier U9, and the other end of the resistor R19 is connected to the output terminal of the operational amplifier U9;

[0025] In the fourth channel circuit, the VCC pin and the RST pin of the 555 timer A6 and one end of the resistor R28 are connected to the DC voltage source VCC1, the other end of the resistor R28 is connected to the DIS pin of the 555 timer A6, one end of the resistor R29 and the anode of the diode D1, the other end of the resistor R29 is connected to the cathode of the diode D2, the anode of the diode D2, the cathode of the diode D1 and one end of the capacitor C5 are connected to the THR and THI pins of the 555 timer A6, the other end of the capacitor C5 and one end of the capacitor C6 are connected to the GND pin of the 555 timer A6 and grounded, the other end of the capacitor C6 is connected to the CON pin of the 555 timer A6, one input end of the multiplier A4 is connected to the output end of the operational amplifier U5, the other input end of the multiplier A4 is connected to the output end of the operational amplifier U8, one input end of the multiplier A5 is connected to the output end of the operational amplifier U5, and the other input end of the multiplier A5 is connected to the operational amplifier U8. The output terminal of the amplifier U11 is connected, one end of the resistors R20, R21, R22, and R23 is connected to the inverting input terminal of the operational amplifier U10 and is also connected to one end of the resistor R24, the other end of the resistor R20 is connected to the output terminal of the operational amplifier U12, the other end of the resistor R21 is connected to the output terminal of the multiplier A4, the other end of the resistor R22 is connected to the output terminal of the multiplier A5, the other end of the resistor R22 is connected to the OUT pin of the 555 timer A6, the other end of the resistor R24 ​​is connected to one end of the resistor R25 and the output terminal of the operational amplifier U10, the other end of the resistor R25 is connected to one end of the capacitor C4 and the inverting input terminal of the operational amplifier U11, the other end of the capacitor C4 is connected to the output terminal of the operational amplifier U11 and one end of the resistor R26, the other end of the resistor R26 is connected to one end of the resistor R27 and the inverting input terminal of the operational amplifier U12, and the other end of the resistor R27 is connected to the output terminal of the operational amplifier U12;

[0026] The positive power supply terminal of the operational amplifiers U1, U2, U3, U4, U5, U6, U7, U8, U9, U10, U11, and U12 is connected to the DC voltage source VCC, the negative power supply terminal of the operational amplifiers U1, U2, U3, U4, U5, U6, U7, U8, U9, U10, U11, and U12 is connected to the DC voltage source VEE, and the ground terminals of the operational amplifiers U1, U2, U3, U4, U5, U6, U7, U8, U9, U10, U11, and U12 are grounded;

[0027] The beneficial effects of this invention lie in addressing the challenges of limited technical universality and high engineering implementation costs in chaotic transformation of complex systems. This invention proposes a broadly applicable and cost-effective method for constructing chaotic systems based on nonlinear switching excitation. This method effectively regulates the system's dynamic characteristics by introducing a nonlinear switching excitation signal and employing a design that facilitates adjustment of its amplitude and frequency. The accuracy of the method is further verified by constructing a periodic nonlinear system, and its physical reliability is verified by building a circuit. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 Construct an overall block diagram for a chaotic system based on nonlinear switching excitation.

[0029] Figure 2 A chaotic system based on nonlinear switching excitation is based on amplitude r Lyapunov exponent diagram of .

[0030] Figure 3 A chaotic system based on nonlinear switching excitation is based on amplitude r The bifurcation diagram of .

[0031] Figure 4 The phase diagram of a chaotic system based on nonlinear switching excitation varies with frequency. f Graph of changes in .

[0032] Figure 5 This is a circuit schematic diagram of a chaotic system based on nonlinear switching excitation.

[0033] Figure 6 This is a circuit implementation diagram of a chaotic system based on nonlinear switching excitation. DETAILED DESCRIPTION

[0034] The present invention will be further described below with reference to the accompanying drawings and examples.

[0035] like Figure 1 As shown in Figure 1, a chaotic system based on nonlinear switching excitation is constructed. The construction method is as follows: First, consider an n-dimensional nonlinear dynamic system, whose state equation can be expressed as:

[0036]

[0037] x is the state variable of the system, is the coefficient matrix of the system, satisfying The stability condition, f : is a smooth nonlinear function vector, is a state variable function.

[0038] S2, introduces a nonlinear switching excitation signal into the above n-dimensional nonlinear dynamic system After that, the system transforms into a non-autonomous system:

[0039]

[0040] The coupling term G Designed as a single-dimensional injection mode, the system can be written as follows:

[0041]

[0042]

[0043] Among them, is the state variable of the system, are the coefficients of the system, is a function of system variables, is 0 or 1, .in , r is the signal amplitude, D is the duty cycle, f is the signal frequency.

[0044] S3, adjust the nonlinear switching excitation signal to the appropriate amplitude based on the amplitude r and frequency f It can cause nonlinear dynamic systems to produce chaotic phenomena and switching systems to be chaotic systems.

[0045] Based on the chaotic system construction process based on nonlinear switching excitation, a four-dimensional periodic nonlinear dynamic system is constructed, and the nonlinear switching excitation signal is introduced. The specific expression of the post-system is:

[0046]

[0047] in, Introducing a functional equation with nonlinear switching excitation properties into the system; are system parameters, where r It has a regulating effect on the dynamic behavior of the system, and the nonlinear switching excitation signal plays a key role in the chaotic state induction process through its parameter coupling mechanism; when the initial value of the system is , parameter is , amplitude r and frequency f The process of change from small to large, the hierarchical transition of the system from stable equilibrium state to limit cycle periodic motion and then to chaotic motion state, affects the generation and evolution path of chaotic attractors.

[0048] Furthermore, if Figure 2 and Figure 3 As shown, the amplitude of the nonlinear switching excitation signal is verified r The impact on the chaotic characteristics of the system, Figure 2 and Figure 3 The amplitude-based r The Lyapunov exponent diagram and bifurcation diagram that continuously changes in the interval [0,10]. Figure 2 As shown, when When , the maximum Lyapunov exponent changes from negative to positive, indicating that the phase space trajectory changes from convergence to exponential divergence; Figure 3 As shown, the phase space trajectory presents a non-periodic dense filling pattern, which conforms to the geometric definition of chaotic attractor, indicating that the system enters a chaotic state. The above analysis shows that the amplitude r It has a regulatory effect on the system's dynamic behavior and plays a key role in the induction of chaos.

[0049] Furthermore, if Figure 4 As shown, the signal frequency is verified f In order to investigate the impact of the chaotic characteristics of the system, the corresponding system phase diagrams were drawn at different signal frequencies. When no nonlinear switching excitation signal was introduced, the system xyz plane phase diagram showed a stable limit cycle structure, and the system was in a periodic state. After the 1Hz signal was introduced, an attractor structure appeared in the system xyz plane phase diagram, breaking the stable limit cycle state when no signal was introduced. When the signal frequency was further increased to 20Hz and 80Hz, the period of the system xyz plane phase diagram doubled, and the chaotic characteristics of the system became more obvious. When the signal frequency was further increased to 130Hz, the system xyz plane phase diagram was not as dense as before at 80Hz, and the chaotic characteristics were reduced.

[0050] Furthermore, in order to verify the reliability of the constructed chaotic system at the physical level, the circuit design of the chaotic system was carried out. The 555 timer, resistors, capacitors, 10V DC power supply and diodes constitute the time-varying circuit. item, r Indicates the coefficient related to the 555 timer. The specific physical meaning of this coefficient is: when the 555 timer is in the high level output state, the voltage value of its output pin In the circuit design, an operational amplifier is combined with a capacitor to realize the function of reverse integration, and is combined with a resistor to realize the function of an inverter. The nonlinear term in the circuit is realized by a multiplier. Using Kirchhoff's voltage law, the specific expression of the chaotic system based on the nonlinear time-varying oscillator described in claim 2 can be obtained, and the corresponding circuit equation is:

[0051] Furthermore, since the four state variables of the system are initially , parameter is When the dynamic range is At this time, the value of the nonlinear cross product term of the above system will be relatively large, exceeding the operating voltage range of the integrated operational amplifier and multiplier. Therefore, firstly make proportional compression on the state variables x and z of the above system, so that , where the multiplier is AD633, the operational amplifier is TL085, and the voltage applied to the integrating capacitor is , and perform time scale transformation, is the time factor, let , the standardized equation is as follows:

[0052]

[0053] By comparing the dimensionless equations of the corresponding circuits listed, keeping the corresponding coefficients equal, let , , the remaining resistance values ​​can be calculated as: , , , , , , , , , , , , , , .

[0054] like Figure 5As shown in FIG, the chaotic system circuit based on nonlinear switching excitation includes four channel circuits: the first channel circuit is composed of a DC voltage source VCC, a DC voltage source VEE, operational amplifiers U1, U2, U3, resistors R1, R2, R3, R4, R5, R6, capacitor C1 and analog ground; the second channel circuit is composed of a DC voltage source VCC, a DC voltage source VEE, operational amplifiers U4, U5, U6, resistors R7, R8, R9, R10, R11, R12, R13, multiplier A1, capacitor C2 and analog ground; the third channel circuit is composed of a DC voltage source VCC, a DC voltage source VEE, operational amplifiers U4, U5, U6, resistors R7, R8, R9, R10, R11, R12, R13, multiplier A1, capacitor C2 and analog ground. The voltage source VEE, operational amplifiers U7, U8, U9, resistors R14, R15, R16, R17, R18, R19, multipliers A2, A3, capacitor C3 and analog ground are composed; the fourth channel circuit is composed of DC voltage sources VCC, VCC1, DC voltage source VEE, multipliers A4, A5, 555 timer A6, operational amplifiers U10, U11, U12, resistors R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, diodes D1, D2, capacitors C4, C5, C6 and analog ground;

[0055] In the first channel circuit of the chaotic system circuit based on nonlinear switching excitation, one end of the resistors R1 and R2 is connected to the inverting input terminal of the operational amplifier U1 and is also connected to one end of the resistor R3, the other end of the resistor R1 is connected to the output terminal of the operational amplifier U3, the other end of the resistor R2 is connected to the output terminal of the operational amplifier U5, the other end of the resistor R3 is connected to one end of the resistor R4 and the output terminal of the operational amplifier U1, the other end of the resistor R4 is connected to one end of the capacitor C1 and the inverting input terminal of the operational amplifier U2, the other end of the capacitor C1 is connected to the output terminal of the operational amplifier U2 and one end of the resistor R5, the other end of the resistor R5 is connected to one end of the resistor R6 and the inverting input terminal of the operational amplifier U3, and the other end of the resistor R6 is connected to the output terminal of the operational amplifier U3;

[0056] In the second channel circuit of the chaotic system circuit based on nonlinear switching excitation, one input end of the multiplier A1 is connected to the output end of the operational amplifier U3, the other input end of the multiplier A1 is connected to the output end of the operational amplifier U11, one end of the resistors R7, R8, and R9 is connected to the inverting input end of the operational amplifier U4 and is also connected to one end of the resistor R10, the other end of the resistor R7 is connected to the output end of the multiplier A1, the other end of the resistor R8 is connected to the output end of the operational amplifier U6, the other end of the resistor R9 is connected to the output end of the operational amplifier U2, the other end of the resistor R10 is connected to one end of the resistor R11 and the output end of the operational amplifier U4, the other end of the resistor R11 is connected to one end of the capacitor C2 and the inverting input end of the operational amplifier U5, the other end of the capacitor C2 is connected to the output end of the operational amplifier U5 and one end of the resistor R12, the other end of the resistor R12 is connected to one end of the resistor R13 and the inverting input end of the operational amplifier U6, and the other end of the resistor R13 is connected to the output end of the operational amplifier U6;

[0057] In the third channel circuit of the chaotic system circuit based on nonlinear switching excitation, two input ends of the multiplier A2 are connected to the output end of the operational amplifier U5, the output end of the multiplier A2 is connected to one input end of the multiplier A3, the other input end of the multiplier A3 is connected to the output end of the operational amplifier U2, one end of the resistors R14 and R15 is connected to the inverting input end of the operational amplifier U7 and is also connected to one end of the resistor R16, the other end of the resistor R14 is connected to the output end of the operational amplifier U9, the other end of the resistor R15 is connected to the output end of the multiplier A3, the other end of the resistor R16 is connected to one end of the resistor R17 and the output end of the operational amplifier U7, the other end of the resistor R17 is connected to one end of the capacitor C3 and the inverting input end of the operational amplifier U8, the other end of the capacitor C3 is connected to the output end of the operational amplifier U8 and one end of the resistor R18, the other end of the resistor R18 is connected to one end of the resistor R19 and the inverting input end of the operational amplifier U9, and the other end of the resistor R19 is connected to the output end of the operational amplifier U9;

[0058] In the fourth channel circuit of the chaotic system circuit based on nonlinear switching excitation, the VCC pin and the RST pin of the 555 timer A6 and one end of the resistor R28 are connected to the DC voltage source VCC1, the other end of the resistor R28 is connected to the DIS pin of the 555 timer A6, one end of the resistor R29 and the anode of the diode D1, the other end of the resistor R29 is connected to the cathode of the diode D2, the anode of the diode D2, the cathode of the diode D1 and one end of the capacitor C5 are connected to the THR and THI pins of the 555 timer A6, the other end of the capacitor C5 and one end of the capacitor C6 are connected to the GND pin of the 555 timer A6 and grounded, the other end of the capacitor C6 is connected to the CON pin of the 555 timer A6, one input end of the multiplier A4 is connected to the output end of the operational amplifier U5, the other input end of the multiplier A4 is connected to the output end of the operational amplifier U8, one input end of the multiplier A5 is connected to the output end of the operational amplifier U5, and the other end of the multiplier A5 is connected to the output end of the operational amplifier U5. One input terminal is connected to the output terminal of the operational amplifier U11, one end of resistors R20, R21, R22, and R23 is connected to the inverting input terminal of the operational amplifier U10 and is also connected to one end of the resistor R24, the other end of the resistor R20 is connected to the output terminal of the operational amplifier U12, the other end of the resistor R21 is connected to the output terminal of the multiplier A4, the other end of the resistor R22 is connected to the output terminal of the multiplier A5, the other end of the resistor R22 is connected to the OUT pin of the 555 timer A6, the other end of the resistor R24 ​​is connected to one end of the resistor R25 and the output terminal of the operational amplifier U10, the other end of the resistor R25 is connected to one end of the capacitor C4 and the inverting input terminal of the operational amplifier U11, the other end of the capacitor C4 is connected to the output terminal of the operational amplifier U11 and one end of the resistor R26, the other end of the resistor R26 is connected to one end of the resistor R27 and the inverting input terminal of the operational amplifier U12, and the other end of the resistor R27 is connected to the output terminal of the operational amplifier U12;

[0059] The positive power supply terminal of the operational amplifiers U1, U2, U3, U4, U5, U6, U7, U8, U9, U10, U11, and U12 is connected to the DC voltage source VCC, the negative power supply terminal of the operational amplifiers U1, U2, U3, U4, U5, U6, U7, U8, U9, U10, U11, and U12 is connected to the DC voltage source VEE, and the ground terminals of the operational amplifiers U1, U2, U3, U4, U5, U6, U7, U8, U9, U10, U11, and U12 are grounded.

[0060] like Figure 6 As shown, the xy, yz, and zw plane phase diagrams of the chaotic attractor are displayed by the system through an oscilloscope, which proves the feasibility of the proposed technical solution.

[0061] The specific implementation methods are only for illustrating the technical idea of ​​the present invention and cannot be used to limit the protection scope of the present invention. Any changes made on the basis of the technical solution in accordance with the technical idea proposed by the present invention are within the protection scope of the present invention.

Claims

1. A chaotic system based on nonlinear switching excitation, characterized by: include: S1, consider an n-dimensional nonlinear dynamic system, whose state equation can be expressed as: ; in, x is the state variable of the system, is the coefficient matrix of the system, satisfying The stability condition, f : is a smooth nonlinear function vector, is the state variable function; S2, introduces a nonlinear switching excitation signal into the above n-dimensional nonlinear dynamic system After that, the system transforms into a non-autonomous system: ; The coupling term G Designed as a single-dimensional injection mode, the system can be written as follows: ; Among them, is the state variable of the system, are the coefficients of the system, is a function of system variables, is 0 or 1, .in , r is the signal amplitude, D is the duty cycle, f is the signal frequency; S3, adjust the amplitude of the nonlinear switching excitation signal r and frequency f It can cause chaotic phenomena in nonlinear dynamic systems.

2. The chaotic system construction based on nonlinear switching excitation according to claim 1, characterized in that: Based on the method, a four-dimensional periodic nonlinear dynamic system was constructed, and a nonlinear switching excitation signal was introduced. The specific expression of the post-system is: ； in, Introducing a functional equation with nonlinear switching excitation properties into the system; are system parameters, where r It has a regulating effect on the dynamic behavior of the system, and the nonlinear switching excitation signal plays a key role in the chaotic state induction process through its parameter coupling mechanism; when the initial value of the system is , parameter is , amplitude r and frequency f The process of change from small to large, the hierarchical transition of the system from stable equilibrium state to limit cycle periodic motion and then to chaotic motion state, affects the generation and evolution path of chaotic attractors.

3. The chaotic system based on nonlinear switching excitation according to claim 2, characterized in that: Design the circuit of the chaotic system. 555 timer, resistor, capacitor, 10V DC power supply and diode constitute the time-varying circuit. item, r Indicates the coefficient related to the 555 timer. The specific physical meaning of this coefficient is: when the 555 timer is in the high level output state, the voltage value of its output pin In the circuit design, an operational amplifier is combined with a capacitor to realize the function of reverse integration, and is combined with a resistor to realize the function of an inverter. The nonlinear term in the circuit is realized by a multiplier. Using Kirchhoff's voltage law, the specific expression of the chaotic system based on the nonlinear time-varying oscillator described in claim 2 can be obtained, and the corresponding circuit equation is: ； Based on the circuit equation, the system consists of four channel circuits: the first channel circuit consists of a DC voltage source VCC, a DC voltage source VEE, operational amplifiers U1, U2, U3, resistors R1, R2, R3, R4, R5, R6, capacitor C1 and analog ground; the second channel circuit consists of a DC voltage source VCC, a DC voltage source VEE, operational amplifiers U4, U5, U6, resistors R7, R8, R9, R10, R11, R12, R13, multiplier A1, capacitor C2 and analog ground; the third channel circuit consists of a DC voltage source VCC, a DC voltage source VEE, operational amplifiers U7, U8, U9, resistors R14, R15, R16, R17, R18, R19, multipliers A2, A3, capacitor C3 and analog ground; the fourth channel circuit is composed of DC voltage sources VCC, VCC1, DC voltage source VEE, multipliers A4, A5, 555 timer A6, operational amplifiers U10, U11, U12, resistors R20, R21, R22, R23, R24, R25, R26, R27, R28, R29, diodes D1, D2, capacitors C4, C5, C6 and analog ground; In the first channel circuit, one end of resistors R1, R2, and R3 is connected to the inverting input terminal of the operational amplifier U1, the other end of resistor R1 is connected to the output terminal of the operational amplifier U3, the other end of resistor R2 is connected to the output terminal of the operational amplifier U5, the other end of resistor R3 and one end of resistor R4 are connected to the output terminal of the operational amplifier U1, the other end of resistor R4 is connected to the inverting input terminal of the operational amplifier U2, capacitor C1 is connected between the inverting input terminal and the output terminal of the operational amplifier U2, the operational amplifier U2 is connected to the inverting input terminal of the operational amplifier U3 via resistor R5, and resistor R6 is connected between the inverting input terminal and the output terminal of the operational amplifier U3; In the second channel circuit, the two input terminals of the multiplier A1 are connected to the output terminals of the operational amplifiers U3 and U11, respectively; one ends of the resistors R7, R8, R9, and R10 are connected to the inverting input terminal of the operational amplifier U4; the other end of the resistor R7 is connected to the output terminal of the multiplier A1; the other end of the resistor R8 is connected to the output terminal of the operational amplifier U6; the other end of the resistor R9 is connected to the output terminal of the operational amplifier U2; the other end of the resistor R10 is connected to one end of the resistor R11 and the output terminal of the operational amplifier U4; the other end of the resistor R11 is connected to the inverting input terminal of the operational amplifier U5; the capacitor C2 is connected between the inverting input terminal and the output terminal of the operational amplifier U5; the operational amplifier U5 is connected to the inverting input terminal of the operational amplifier U6 via the resistor R12; and the resistor R13 is connected between the inverting input terminal and the output terminal of the operational amplifier U6; In the third channel circuit, both input terminals of the multiplier A2 are connected to the output terminal of the operational amplifier U5, the output terminal of the multiplier A2 is connected to one input terminal of the multiplier A3, the other input terminal of the multiplier A3 is connected to the output terminal of the operational amplifier U2, one end of the resistors R14, R15, and R16 are connected to the inverting input terminal of the operational amplifier U7, the other end of the resistor R14 is connected to the output terminal of the operational amplifier U9, the other end of the resistor R15 is connected to the output terminal of the multiplier A3, the other end of the resistor R16 is connected to one end of the resistor R17 and the output terminal of the operational amplifier U7, the other end of the resistor R17 is connected to the inverting input terminal of the operational amplifier U8, the capacitor C3 is connected between the inverting input terminal and the output terminal of the operational amplifier U8, the operational amplifier U8 is connected to the inverting input terminal of the operational amplifier U9 via the resistor R18, and the resistor R19 is connected between the inverting input terminal and the output terminal of the operational amplifier U9; In the fourth channel circuit, the VCC pin and the RST pin of the 555 timer A6 and one end of the resistor R28 are connected to the DC voltage source VCC1, the other end of the resistor R28 is connected to the DIS pin of the 555 timer A6, one end of the resistor R29 and the anode of the diode D1, the other end of the resistor R29 is connected to the cathode of the diode D2, the anode of the diode D2, the cathode of the diode D1 and one end of the capacitor C5 are connected to the THR and THI pins of the 555 timer A6, the other end of the capacitor C5 and one end of the capacitor C6 are connected to the GND pin of the 555 timer A6 and grounded, the other end of the capacitor C6 is connected to the CON pin of the 555 timer A6, one input end of the multiplier A4 is connected to the output end of the operational amplifier U5, the other input end of the multiplier A4 is connected to the output end of the operational amplifier U8, one input end of the multiplier A5 is connected to the output end of the operational amplifier U5, and the multiplier A The other input terminal of 5 is connected to the output terminal of the operational amplifier U11, one end of resistors R20, R21, R22, R23, and R24 are connected to the inverting input terminal of the operational amplifier U10, the other end of resistor R20 is connected to the output terminal of the operational amplifier U12, the other end of resistor R21 is connected to the output terminal of the multiplier A4, the other end of resistor R22 is connected to the output terminal of the multiplier A5, the other end of resistor R22 is connected to the OUT pin of the 555 timer A6, the other end of resistor R24 ​​is connected to one end of resistor R25 and the output terminal of the operational amplifier U10, the other end of resistor R25 is connected to the inverting input terminal of the operational amplifier U11, capacitor C4 is connected across the inverting input terminal and output terminal of the operational amplifier U11, the operational amplifier U11 is connected to the inverting input terminal of the operational amplifier U12 via resistor R26, and resistor R27 is connected across the inverting input terminal and output terminal of the operational amplifier U12.

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