CN113162551A - Multi-frequency slow excitation Lorenz derivative system capable of generating novel complex clustering phenomenon - Google Patents
Multi-frequency slow excitation Lorenz derivative system capable of generating novel complex clustering phenomenon Download PDFInfo
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Abstract
The invention discloses a multi-frequency slow excitation Lorenz derivative system capable of generating a novel complex clustering phenomenon, which comprises input sine wave voltage sources V1 and V2, capacitors C1, C2 and C3, multipliers A1-A4, resistors R1-R18 and voltage comparators U1-U9. By adding new excitation into the single excitation system and adjusting the magnitude relation of the two excitation amplitudes, the system shows a completely new dynamic behavior. The system has two working states in total, and when the amplitude of the excitation 1 is greater than that of the excitation 2, the system is in a1 st working mode; when excitation 1 is less in magnitude than excitation 2, it is in the second mode of operation. The circuit shows brand-new dynamic behavior, provides a solution for scientific and engineering problems in systems with superior design and manufacturing performance, and promotes the development of a nonlinear theory.
Description
Technical Field
The invention belongs to a nonlinear circuit, and particularly relates to a method for adjusting circuit parameters to obtain two new clustering states by introducing new excitation on the basis of a single-excitation three-dimensional system to enable the system to show brand-new dynamic behaviors. The novel dynamic behavior provides a solution for the scientific and engineering problems in designing and manufacturing systems with excellent performance, and promotes the development of a nonlinear theory.
Background
Nonlinear dynamics relates to multiple subject fields such as mathematics and circuits. In practical activities, people try to adopt a linear model instead of a nonlinear model, and for a non-negligible nonlinear term, a linear approximation or a linear iteration processing method is generally adopted. However, the application of new materials and new technologies is more and more extensive, and nonlinear theory and method are urgently needed. The multi-scale problem is always the leading-edge problem of scientific and technical research at present due to the application value of theory and engineering, and the problem caused by the multi-scale coupling factor is worthy of further research. The clusterification was used to describe the phenomenon of alternating discharges between neurons in biology, characterized by the alternation of a resting state (near steady) -active state (spike oscillations).
A multi-frequency slow excitation Lorenz-derived system (LSMFSPEs) capable of generating novel complex clustering phenomena has more complex and dynamic behaviors and provides a larger free space for a nonlinear dynamic system compared with a cluster discovery image generated by only a single excitation. Therefore, the method has important significance for the research of the cluster oscillation system generated by LSMFSPEs and the hardware circuit thereof, provides a solution for the scientific and engineering problems in the system with excellent design and manufacture performance, and promotes the development of the nonlinear theory. The invention discloses a multi-frequency slow excitation Lorenz derivative system capable of generating a novel complex clustering phenomenon, and promotes the application of a circuit clustering oscillation system generated by LSMFSPEs in dynamics.
The basic Lorenz derivative system only has single excitation, and by adding multi-frequency slow parameter excitation, more complex dynamic behaviors are obtained, and bifurcations appear on non-zero equilibrium branches, namely a clustering oscillation circuit induced by 'turnover lagging fork bifurcations/subprof/homoclinic connection' in the 1 st working mode; on the basis, by adjusting circuit parameters, a2 nd working mode is obtained: a 'cascaded subphopf/homoclinic connection' clustered oscillation circuit induced by a delayed branch bifurcation point; furthermore, the amplitude of the excitation signal also induces a change in the tufting pattern.
Disclosure of Invention
The invention aims to solve the technical problem of realizing a multi-frequency slow excitation Lorenz derivative system capable of generating a novel complex clustering phenomenon.
In order to solve the technical problem, the invention provides a multi-frequency slow excitation Lorenz derivative system capable of generating a novel complex clustering phenomenon. 1(a) - (C), comprising input sine wave voltage sources V1, V2, capacitors C1, C2, C3, multipliers A1-A4, resistors R1-R18 and voltage comparators U1-U9.
The specific connection mode is as follows:
FIG. 1 (a): the second end of the resistor R1 is connected with the second end of the resistor 2, the first end of the resistor R3 and the terminal of the voltage comparator U1 '-'. The pin 6 of the voltage comparator U1 is connected with the second end of the resistor R3 and the first end of the resistor R4, and the pins 1, 3 and 5 of the voltage comparator U1 are grounded. The "-" terminal of the voltage comparator U2 is connected to the second terminal of the resistor R4 and the first terminal of the capacitor C1. A second terminal of the capacitor C1 is connected with a "6" pin of the voltage comparator U2 and a first terminal of the resistor R5. Pins 1, 3 and 5 of the voltage comparator U2 are grounded. The "-" terminal of the voltage comparator U3 is connected to the second terminal of the resistor R5 and the first terminal of the resistor R6. The second terminal of the resistor R6 is connected to pin "6" of the voltage comparator U3. Pins 1, 3 and 5 of the voltage comparator U3 are grounded.
FIG. 1 (b): the first terminal of the sine wave voltage source V1 is connected to the "A" terminal of the multiplier A3, and the second terminal of the sine wave voltage source V1 is grounded. The first terminal of the sine wave voltage source V2 is connected to the 'B' terminal of the multiplier A3, and the second terminal of the sine wave voltage source V2 is grounded. The 'C' end of the multiplier A3 is suspended, and the output end is connected with the 'X' end of the multiplier A2. The output end of the multiplier A1 is connected with the first end of the resistor R7, and the output end of the multiplier A2 is connected with the first end of the resistor R12. The second end of the resistor R7 is connected with the second end of the resistor R12, the first end of the resistor R11 and the terminal of the voltage comparator U4 '-'. The pin 6 of the voltage comparator U4 is connected with the second end of the resistor R11 and the first end of the resistor R8, and the pins 1, 3 and 5 of the voltage comparator U4 are grounded. The "-" terminal of the voltage comparator U5 is connected to the second terminal of the resistor R8 and the first terminal of the capacitor C2. A second terminal of the capacitor C2 is connected with a "6" pin of the voltage comparator U5 and a first terminal of the resistor R9. Pins 1, 3 and 5 of the voltage comparator U5 are grounded. The "-" terminal of the voltage comparator U6 is connected to the second terminal of the resistor R9 and the first terminal of the resistor R10. The second terminal of the resistor R10 is connected to pin "6" of the voltage comparator U6. Pins 1, 3 and 5 of the voltage comparator U6 are grounded.
③ FIG. 1 (c): the output end of the multiplier A4 is connected with the first end of the resistor R13. The second end of the resistor R13 is connected with the second end of the resistor R14, the first end of the resistor R15 and the terminal of the voltage comparator U7 '-'. The pin 6 of the voltage comparator U7 is connected with the second end of the resistor R15 and the first end of the resistor R6, and the pins 1, 3 and 5 of the voltage comparator U7 are grounded. The "-" terminal of the voltage comparator U8 is connected to the second terminal of the resistor R16 and the first terminal of the capacitor C3. A second terminal of the capacitor C3 is connected with a "6" pin of the voltage comparator U8 and a first terminal of the resistor R17. Pins 1, 3 and 5 of the voltage comparator U8 are grounded. The "-" terminal of the voltage comparator U9 is connected to the second terminal of the resistor R17 and the first terminal of the resistor R18. The second terminal of the resistor R18 is connected to pin "6" of the voltage comparator U9. Pins 1, 3 and 5 of the voltage comparator U6 are grounded.
The system used in the present invention is as follows:
wherein a and b are non-negative parameters, beta1 cos(w1t)、β2 cos(w2t) is a slow variable, w1,2(w1,2<<0) Small enough that there is a step difference between the excitation frequency and the system natural frequency, resulting in two time scale coupling effects.
The 1 st mode of operation is as shown in FIGS. 3(a) - (c) at β2 cos(w2t), the oscillation of the system exhibits more complex dynamics than without β1 cos(w1t) original systemThe system is totally different, and the delayed supercritical branch bifurcation behavior is more complex and has more interesting dynamic characteristics. The parameters of the two excitations satisfy the following relationship: w is a2＝nw1，β1>β2. Each tufting cycle consists of two phases, one being a quiescent state and the other being a spike state. Delayed branch splitting may produce a phenomenon similar to that of fig. 2 (a). The motion profiles in fig. 3(a) - (c) show different characteristics than the profile in fig. 2. In fig. 3, a distinct oscillation can be observed, the frequency of which varies with β2 cos(w2t) increases with increasing frequency.
Selecting a sine wave voltage source with circuit parameters of V1 ═ 1V, f ═ 0.8Hz, V2 ═ 0.3V, f ═ 0.8Hz, C1 ═ C2 ═ C3 ═ 10nF, R1 ═ R2 ═ 50k Ω, R3 ═ R4 ═ R7 ═ R8 ═ R11 ═ R12 ═ R13 ═ R14 ═ R15 ═ R16 ═ 100k Ω, R5 ═ R6 ═ R9 ═ R10 ═ R17 ═ R18 ═ 10k, and when the parameter is a ═ 2, b ═ 1, β ═ R1＝1.5，w1"delayed PB/supHB/homing connections simulated by the corresponding MATLAB values under this condition at 0.005The fast-slow analysis of the "tufting" phenomenon is shown in FIG. 2.
Selecting a sine wave voltage source with circuit parameters of V1 ═ 1V, f ═ 0.8Hz, V2 ═ 0.3V, f ═ 0.8Hz, C1 ═ C2 ═ C3 ═ 10nF, R1 ═ R2 ═ 50k Ω, R3 ═ R4 ═ R7 ═ R8 ═ R11 ═ R12 ═ R13 ═ R14 ═ R15 ═ R16 ═ 100k Ω, R5 ═ R6 ═ R9 ═ R10 ═ R17 ═ R18 ═ 10k, and when the parameter is a ═ 2, b ═ 1, β ═ R1＝1.5，w1＝0.005,w2The 1 st mode of operation of the MATLAB numerical simulation under this condition is as shown in fig. 3, at 0.025.
Fig. 4(a) - (f) are schematic diagrams of fast-slow analysis for the 1 st mode of operation. Wherein FIGS. 4(a) and (d) correspond to FIG. 3(a), i.e., w in the operation mode 120.025; FIGS. 4(b) and (e) correspond to FIG. 3(b), i.e., w in the operation mode 12Case 0.045; FIGS. 4(c) and (f) correspond to FIG. 3(c), i.e., w in the operation mode 12＝0.065，β2Case 0.4. The fast subsystem is
n is 5,9 and 13, γ is the control parameter, and the remaining parameters are consistent with fig. 3. The points marked by "+" are extreme points.
FIGS. 5(a) - (d) are views of w in FIG. 3(c)L＝w2Schematic representation of the situation. (a) Schematic representation of the "delayed PB/subphopf/HomoConnection" clustering phenomenon. (b) Beta is a1 cos(w1Graph of t) and g (t), illustrated at β2 cos(w2Under the influence of t), g (t) is at beta1cos(w1t) rapid oscillations with small amplitude. (c) Is the excitation beta in FIG. 3(c)2 cos(w2t) timing diagrams and motion trajectories. (d) Is the fast-slow analysis of fig. 4 (c).
Selecting a sine wave voltage source with circuit parameters of V1 ═ 1V, f ═ 0.8Hz, V2 ═ 0.3V, f ═ 0.8Hz, C1 ═ C2 ═ C3 ═ 10nF, R1 ═ R2 ═ 50k Ω, R3 ═ R4 ═ R7 ═ R8 ═ R11 ═ R12 ═ R13 ═ R14 ═ R15 ═ R16 ═ 100k Ω, R5 ═ R6 ═ R9 ═ R10 ═ R17 ═ R18 ═ 10k, and when the parameter is a ═ 2, b ═ 1, β ═ R1＝0.2，β2＝1.6，w1＝0.005,w2The 1 st mode of operation of the MATLAB numerical simulation under this condition is as shown in fig. 6, at 0.025.
Fig. 7(a) - (c) correspond to the fast-slow analysis diagrams of fig. 6 for the 2 nd operation mode (a) - (c), respectively. (d) Fast subsystem of (i) - (f)
Where n is 2,3 and 4, a bifurcation diagram is drawn. γ ═ cos (0.005t) is a slow variable. When n is 2,3 and 4, 2,3 and 4 independent "subphopf/homozygote" cluster oscillations induced by delayed branch junctions were observed, respectively.
FIG. 8, 2 nd mode of operation, (a) denotes w1＝0.005，w20.025 and (b) represents w1＝0.005，w20.05. At beta2 cos(w2t), 5 and 10 independent "subphopf/homed connection" cluster oscillations induced by delayed branch points were observed in each cycle, respectively.
The invention has the following beneficial effects:
(1) the multi-frequency slow excitation Lorenz derivative system capable of generating the novel complex clustering phenomenon enables the system to show more complex dynamic behaviors by introducing new excitation on the basis of the original single excitation system. The novel dynamic behavior provides a solution for the scientific and engineering problems in designing and manufacturing systems with excellent performance, and promotes the development of the nonlinear theory
(2) The multi-frequency slow excitation Lorenz derivative system capable of generating the novel complex clustering phenomenon adjusts the excitation parameter beta under the condition that the system parameter is fixed1，β2Can obtain two new hair cluster states and can be used as an adjustable signal generator
Drawings
Fig. 1(a) - (c) are schematic diagrams of a multi-frequency slow excitation Lorenz derivative system capable of generating a novel complex clustering phenomenon.
FIG. 2(a) is a fast-slow analysis of the "delayed PB/SupHB/HomoConnection" clustering phenomenon, 1.5cos (0.005t) is the slow variable, FIG. 2(b) is a timing diagram
Fig. 3(a) - (c) mode 1, a clustered oscillator circuit induced by "flip-flop hysteretic fork-like bifurcation/subphopf/homed connection". (a) Is w2In the case of 0.025, (b) is w2In the case of 0.045, (c) is w2＝0.065，β2Case 0.4. The remaining parameters are consistent with fig. 2.
FIGS. 4(a) - (f) schematic fast-slow analysis of the 1 st mode of operation. Wherein FIGS. 4(a) and (d) correspond to FIG. 3(a), i.e., w in the operation mode 120.025; FIGS. 4(b) and (e) correspond to FIG. 3(b), i.e., w in the operation mode 12Case 0.045; FIGS. 4(c) and (f) correspond to FIG. 3(c), i.e., w in the operation mode 12＝0.065，β2Case 0.4. The fast subsystem is
n is 5,9 and 13, γ is the control parameter, and the remaining parameters are consistent with fig. 3. The points marked by "+" are extreme points.
FIGS. 5(a) - (d) w in FIG. 3(c)L＝w2Schematic representation of (a). (a) Schematic representation of the "delayed PB/subphopf/HomoConnection" clustering phenomenon. (b) Beta is a1 cos(w1Graph of t) and g (t), illustrated at β2 cos(w2Under the influence of t), g (t) is at beta1 cos(w1t) rapid oscillations with small amplitude. (c) Excitation β in FIG. 3(c)2 cos(w2t) timing diagrams and motion trajectories. (d) Fast-slow analysis of fig. 4 (c).
Fig. 6(a) - (c) shows mode 2 operation, in which "cascaded subphopf/homed" burst oscillator circuits are induced by delayed branch points, and the corresponding parameter values w are 0.01,0.015 and 0.02. (d) - (f) correspond to FIGS. 6(a) - (c) w2＝nw1When n is 2,3,4, beta1 cos(w1t) and g (t) β1 cos(w1t)+β2 cos(w2t) of the first image. Obviously, in beta1 cos(w1One period of t), g (t) passing through branch point γ PB0 and superThe critical Hopf branches 2,3 and 4 times (n ═ 2,3, 4).
Fig. 7(a) - (c) correspond to the fast-slow analysis diagrams of fig. 6 for the 2 nd operation mode (a) - (c), respectively. (d) - (f) fast subsystem
Where n is 2,3 and 4, the bifurcation diagram. γ ═ cos (0.005t) is a slow variable. When n is 2,3 and 4, 2,3 and 4 independent "subphopf/homozygote" cluster oscillations induced by delayed branch junctions were observed, respectively.
FIG. 8, 2 nd mode of operation, (a) denotes w1＝0.005，w20.025 and (b) represents w1＝0.005，w20.05. At beta2 cos(w2t), 5 and 10 independent "subphopf/homed connection" cluster oscillations induced by delayed branch points were observed in each cycle, respectively.
Detailed Description
The invention will now be described in further detail with reference to the drawings and preferred examples, and with reference to figures 1 to 8.
Referring to fig. 1, the multi-frequency slow excitation Lorenz derivative system capable of generating a novel complex clustering phenomenon provided by the invention comprises input sine wave voltage sources V1 and V2, capacitors C1, C2 and C3, multipliers a1-a4, resistors R1-R18 and voltage comparators U1-U9.
The specific connection mode is as follows:
FIG. 1 (a): the second end of the resistor R1 is connected with the second end of the resistor 2, the first end of the resistor R3 and the terminal of the voltage comparator U1 '-'. The pin 6 of the voltage comparator U1 is connected with the second end of the resistor R3 and the first end of the resistor R4, and the pins 1, 3 and 5 of the voltage comparator U1 are grounded. The "-" terminal of the voltage comparator U2 is connected to the second terminal of the resistor R4 and the first terminal of the capacitor C1. The second terminal of the capacitor C1 is connected to the "6" pin of the voltage comparator U2 and the first terminal of the resistor R95. Pins 1, 3 and 5 of the voltage comparator U2 are grounded. The "-" terminal of the voltage comparator U3 is connected to the second terminal of the resistor R5 and the first terminal of the resistor R6. The second terminal of the resistor R6 is connected to pin "6" of the voltage comparator U3. Pins 1, 3 and 5 of the voltage comparator U3 are grounded.
FIG. 1 (b): the first terminal of the sine wave voltage source V1 is connected to the "A" terminal of the multiplier A3, and the second terminal of the sine wave voltage source V1 is grounded. The first terminal of the sine wave voltage source V2 is connected to the 'B' terminal of the multiplier A3, and the second terminal of the sine wave voltage source V2 is grounded. The 'C' end of the multiplier A3 is suspended, and the output end is connected with the 'X' end of the multiplier A2. The output end of the multiplier A1 is connected with the first end of the resistor R7, and the output end of the multiplier A2 is connected with the first end of the resistor R12. The second end of the resistor R7 is connected with the second end of the resistor R12, the first end of the resistor R11 and the terminal of the voltage comparator U4 '-'. The pin 6 of the voltage comparator U4 is connected with the second end of the resistor R11 and the first end of the resistor R8, and the pins 1, 3 and 5 of the voltage comparator U4 are grounded. The "-" terminal of the voltage comparator U5 is connected to the second terminal of the resistor R8 and the first terminal of the capacitor C2. A second terminal of the capacitor C2 is connected with a "6" pin of the voltage comparator U5 and a first terminal of the resistor R9. Pins 1, 3 and 5 of the voltage comparator U5 are grounded. The "-" terminal of the voltage comparator U6 is connected to the second terminal of the resistor R9 and the first terminal of the resistor R10. The second terminal of the resistor R10 is connected to pin "6" of the voltage comparator U6. Pins 1, 3 and 5 of the voltage comparator U6 are grounded.
③ FIG. 1 (c): the output end of the multiplier A4 is connected with the first end of the resistor R13. The second end of the resistor R13 is connected with the second end of the resistor R14, the first end of the resistor R15 and the terminal of the voltage comparator U7 '-'. The pin 6 of the voltage comparator U7 is connected with the second end of the resistor R15 and the first end of the resistor R6, and the pins 1, 3 and 5 of the voltage comparator U7 are grounded. The "-" terminal of the voltage comparator U8 is connected to the second terminal of the resistor R16 and the first terminal of the capacitor C3. A second terminal of the capacitor C3 is connected with a "6" pin of the voltage comparator U8 and a first terminal of the resistor R17. Pins 1, 3 and 5 of the voltage comparator U8 are grounded. The "-" terminal of the voltage comparator U9 is connected to the second terminal of the resistor R17 and the first terminal of the resistor R18. The second terminal of the resistor R18 is connected to pin "6" of the voltage comparator U9. Pins 1, 3 and 5 of the voltage comparator U6 are grounded.
The system used in the present invention is as follows:
wherein a and b are non-negative parameters, beta1 cos(w1t)、β2 cos(w2t) is a slow variable, w1,2(w1,2<<0) Small enough that there is a step difference between the excitation frequency and the system natural frequency, resulting in two time scale coupling effects. Then, a fast-slow characteristic may occur, generating a cluster oscillation in the system. Without additional parameter beta2 cos(w2t),w1<<1, the system has only one slow parameter excitation:
when the slow excitation is performed, as shown in FIGS. 2(a) and (b), the slow excitation is performed1 cos(w1t) periodically passing through the supercritical bifurcation value gammaPBWhen 0, a hysteresis behavior can be produced: if the motion trajectory starts from the left, it will track E0For a period of time, when gamma is beta1 cos(w1t) passing through the branching point of the supercritical branch, E0Losing stability. Then E from instability can be observed0Sudden changes to other stable attractors, then supercritical Hopf bifurcation points appear, and the motion trail oscillates with considerable amplitude towards a non-zero equilibrium branch E±The limit rings of (a) are close. This phenomenon continues until a point of separation from the host occurs. Then, the track will followVector field of limit cycle and stable non-zero equilibrium point E occurs-Or E+. As γ is gradually decreased, the motion trajectory will return to the origin, which is a period. This phenomenon is called "delayed PB/subphopf/co-sink connection".
At beta2 cos(w2t), the oscillation of the system exhibits more complex dynamics than without β1cos(w1t) are completely different, the delayed supercritical branch bifurcation behavior is more complex and has more interesting kinetic properties. This is the 1 st mode of operation. As shown in fig. 3(a) - (c), each tufting cycle consists of two phases, i.e. one is a quiescent state and the other is a spike state. In fig. 3, delayed branch bifurcation may generate a phenomenon similar to fig. 2(a), but the motion trajectories in fig. 3(a) - (c) show different characteristics from those in fig. 2. In fig. 3, a distinct oscillation can be observed, the frequency of which varies with β2 cos(w2t) increases with increasing frequency. Take FIG. 3(a) as an example, where w1＝0.005,w2We consider g (t) cos (0.005t) as a slow variable, and then the corresponding fast subsystem can be written as:
as shown in fig. 4(a), γ is a bifurcation parameter. The fast subsystem consists of two non-zero balanced branches: e+And E-The oscillation phenomenon and four extreme points occur. Obviously, each hair cluster period consists of a large amplitude oscillation frequency wLAnd the frequency w of the oscillating of the hair in clustersbComposition, FIG. 3(c) shows wL＝w2In the case of (a), w2＝nw1，β1>β2。
As shown in FIG. 5(b), at β1 cos(w1t) during one period, due to the presence of beta2 cos(w2t) an additional force, g (t) ═ β1 cos(w1t)+β2 cos(w2t) will be at β1 cos(w1t) fluctuates around. g (t) is oneIncrease or decrease the alternating process, resulting in E±Becomes tortuous.
As shown in FIG. 5(d), along E-The trajectory of the branch becomes meandering. E-The motion curve of (a) contains two parts, one starting from C to H and the other starting from H to the branch point of PB.
As shown in FIG. 5(c), the system frequency is equal to β2 cos(w2Frequency of t), i.e. wL＝w2. In addition, FIG. 3 also shows the frequency of tufting and β1 cos(w1t) are equal in frequency.
FIG. 6 is a schematic diagram of the 2 nd mode of operation, as compared to the 1 st mode of operation (β)1>β2) The amplitude of the 2 nd operation mode satisfies beta1<β2. (a) - (c) denotes beta1＝0.2，β21.6. In this mode, the number of states of the quiescent state and the spike state is larger in each clustering period.
Due to beta1<β2G (t) inherits beta2 cos(w2t) of the first and, at the same time, w2＝nw1Is constant, which means that a beta is excited1 cos(w1t) in one period, two beta are excited2 cos(w2t) will repeat n times, crossing gamma in the forward direction PB0 and γsup HB1.0n times, so that g (t) also passes through gamma in the forward direction PB0 and γsupHB± 1.0n times, as shown in fig. 6(d) - (f). Whenever g (t) passes in the forward direction by γ PB0 and γsup HBA simple "subphopf/homing connection" burst oscillation induced by the delayed branch point occurs at ± 1.0.
Fig. 7(a) - (c) correspond to the fast-slow analysis diagrams of fig. 6 for the 2 nd operation mode (a) - (c), respectively. (d) - (f) fast subsystem
Where n is 2,3 and 4, the bifurcation diagram. γ ═ cos (0.005t) is a slow variable. When n is 2,3 and 4, 2,3 and 4 independent branches induced by delayed branch point are observed "SupHopf/Homojunctional "clustering oscillation phenomenon.
FIG. 8 shows the 2 nd mode of operation of the system, where (a) denotes w1＝0.005，w20.025 and (b) represents w1＝0.005，w20.05. At beta2 cos(w2t), 5 and 10 independent "subphopf/homed connection" cluster oscillations induced by delayed branch points were observed in each cycle, respectively.
Selecting circuit parameters by utilizing Matlab numerical simulation:
the sine wave voltage source V1 is +/-1V, f is 0.8Hz, V2 is +/-0.3V, and f is 0.8 Hz;
the capacitance C1 ═ C2 ═ C3 ═ 10 nF;
the resistance R1-R2-50 k omega,
R3＝R4＝R7＝R8＝R11＝R12＝R13＝R14＝R15＝R16＝100kΩ，
R5＝R6＝R9＝R10＝R17＝R18＝10kΩ；
when the parameter value a is 2, b is 1, beta1＝1.5，w1At 0.005, a fast-slow analysis of the "delayed PB/supHB/homozygote junction" clustering was obtained as shown in fig. 2.
Selecting circuit parameters by utilizing Matlab numerical simulation:
the sine wave voltage source V1 is +/-1V, f is 0.8Hz, V2 is +/-0.3V, and f is 0.8 Hz;
the capacitance C1 ═ C2 ═ C3 ═ 10 nF;
the resistance R1-R2-50 k omega,
R3＝R4＝R7＝R8＝R11＝R12＝R13＝R14＝R15＝R16＝100kΩ，
R5＝R6＝R9＝R10＝R17＝R18＝10kΩ；
when the parameter value a is 2, b is 1, beta1＝1.5，w1＝0.005,w2When the value is 0.025, the 1 st operation mode is obtained as shown in fig. 3.
Selecting circuit parameters by utilizing Matlab numerical simulation:
the sine wave voltage source V1 is +/-1V, f is 0.8Hz, V2 is +/-0.3V, and f is 0.8 Hz;
the capacitance C1 ═ C2 ═ C3 ═ 10 nF;
the resistance R1-R2-50 k omega,
R3＝R4＝R7＝R8＝R11＝R12＝R13＝R14＝R15＝R16＝100kΩ，
R5＝R6＝R9＝R10＝R17＝R18＝10kΩ；
when the parameter value a is 2, b is 1, beta1＝0.2,β2＝1.6,w1＝0.005,w2When the value is 0.025, the 2 nd operation mode is obtained as shown in fig. 6.
The multi-frequency slow excitation Lorenz derivative system capable of generating the novel complex clustering phenomenon is simple in circuit structure, and compared with a clustering oscillation circuit in a general three-dimensional continuous system, the multi-frequency slow excitation Lorenz derivative system capable of generating the novel complex clustering phenomenon is more complex in dynamic behavior.
The above embodiments are merely some preferred examples of the present invention, and are not intended to limit the present invention, and other variations and modifications may be made by those skilled in the art based on the present invention. Therefore, any simple modification made on the basis of the present invention still falls within the protection scope of the technical solution of the present invention.
Claims (4)
1. A multi-frequency slow excitation Lorenz derivative system capable of generating a novel complex clustering phenomenon is characterized in that: the voltage source circuit comprises input sine wave voltage sources V1 and V2, capacitors C1, C2 and C3, multipliers A1 and A4, resistors R1 and R18, and voltage comparators U1 and U9, wherein the second end of the resistor R1 is connected with the second end of the resistor 2, the first end of the resistor R3 and the "-" end of a voltage comparator U1, the "6" pin of the voltage comparator U1 is connected with the second end of the resistor R3 and the first end of the resistor R4, the "1", "3" and "5" pins of the voltage comparator U1 are grounded, the "-" end of the voltage comparator U2 is connected with the second end of the resistor R4 and the first end of the capacitor C1, the second end of the capacitor C1 is connected with the "6" pin of the voltage comparator U2 and the first end of the resistor R95, and the "1" 3 "pin" and "ground" pin of the voltage comparator U2 are grounded, the "-" terminal of the voltage comparator U3 is connected to the second terminal of the resistor R5 and the first terminal of the resistor R6, the second terminal of the resistor R6 is connected to the "6" pin of the voltage comparator U3, the "1", "3" and "5" pins of the voltage comparator U3 are grounded, the first terminal of the sine wave voltage source V1 is connected to the "a" terminal of the multiplier A3, the second terminal of the sine wave voltage source V1 is grounded, the first terminal of the sine wave voltage source V2 is connected to the "B" terminal of the multiplier A3, the second terminal of the sine wave voltage source V2 is grounded, the "C" terminal of the multiplier A3 is floating, the output terminal is connected to the "X" terminal of the multiplier a2, the output terminal of the multiplier a1 is connected to the first terminal of the resistor R7, the output terminal of the multiplier a2 is connected to the first terminal of the resistor R12, the second terminal of the resistor R7 is connected to the second terminal of the resistor R12 and the second terminal of the resistor R11, A "6" pin of the voltage comparator U4 is connected to the second end of the resistor R11 and the first end of the resistor R8, a "1", "3" and "5" pin of the voltage comparator U4 is grounded, a "1" end of the voltage comparator U5 is connected to the second end of the resistor R8 and the first end of the capacitor C2, a second end of the capacitor C2 is connected to the "6" pin of the voltage comparator U5 and the first end of the resistor R9, a "1", "3" and "5" pin of the voltage comparator U5 is grounded, a "minus" end of the voltage comparator U6 is connected to the second end of the resistor R38 and the first end of the resistor R10, a second end of the resistor R10 is connected to the "6" pin of the voltage comparator U6, and a "1" end "and" 3 "pin of the voltage comparator U6, The pin "5" is grounded, the output terminal of the multiplier a4 is connected to the first terminal of the resistor R13, the second terminal of the resistor R13 is connected to the second terminal of the resistor R14, the first terminal of the resistor R15 and the "-" terminal of the voltage comparator U7, the "6" terminal of the voltage comparator U7 is connected to the second terminal of the resistor R15 and the first terminal of the resistor R6, the "1", "3" and "5" pins of the voltage comparator U7 are grounded, the "-" terminal of the voltage comparator U8 is connected to the second terminal of the resistor R16 and the first terminal of the capacitor C3, the second terminal of the capacitor C3 is connected to the "6" terminal of the voltage comparator U8 and the first terminal of the resistor R17, the "1", "3" and "5" terminals of the voltage comparator U8 are grounded, the "-" terminal of the voltage comparator U9 is connected to the second terminal of the resistor R17 and the second terminal of the resistor 18, the second end of the resistor R18 is connected with the pin "6" of the voltage comparator U9, and the pins "1", "3" and "5" of the voltage comparator U6 are grounded.
2. The multi-frequency slow excitation Lorenz derivative system capable of generating novel complex clustering phenomena as claimed in claim 1, wherein: the system is described by the following equation:
wherein a and b are non-negative parameters, beta1cos(w1t)、β2cos(w2t) is a slow variable, w1,2(w1,2<<0) Small enough that there is a step difference between the excitation frequency and the system natural frequency, resulting in two time scale coupling effects.
3. The multi-frequency slow excitation Lorenz derivative system capable of generating new complex clustering phenomena as claimed in claim 2, wherein: circuit parameter fixing:
the sine wave voltage source V1 is +/-1V, f is 0.8Hz, V2 is +/-0.3V, and f is 0.8 Hz;
the capacitance C1 ═ C2 ═ C3 ═ 10 nF;
the resistance R1-R2-50 k omega,
R3＝R4＝R7＝R8＝R11＝R12＝R13＝R14＝R15＝R16＝100kΩ，
R5＝R6＝R9＝R10＝R17＝R18＝10kΩ；
when the parameter value a is 2, b is 1, beta1＝1.5，w1At 0.005, a "delayed PB/supHB/homozygote connection" clustering pattern was obtained.
4. The multi-frequency slow excitation Lorenz derivative system capable of generating new complex clustering phenomena as claimed in claim 2, wherein: circuit parameter fixing:
the sine wave voltage source V1 is +/-1V, f is 0.8Hz, V2 is +/-0.3V, and f is 0.8 Hz;
the capacitance C1 ═ C2 ═ C3 ═ 10 nF;
the resistance R1-R2-50 k omega,
R3＝R4＝R7＝R8＝R11＝R12＝R13＝R14＝R15＝R16＝100kΩ，
R5＝R6＝R9＝R10＝R17＝R18＝10kΩ；
when the parameter value a is 2, b is 1, beta1＝1.5,w1＝0.005,w2When the value is 0.025, the 1 st working mode is obtained;
when the parameter value a is 2, b is 1, beta1＝0.2,β2＝1.6,w1＝0.005,w2When 0.025, the 2 nd operation mode is obtained.
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