Editor's Note: This article mainly comes from the introduction; Chaotic circuit; EWB simulation analysis; Hardware circuit debugging; The conclusion is discussed. It mainly includes: the performance of nonlinear system is complex and changeable, chaos is the random motion of nonlinear dynamic system which is sensitive to initial conditions under certain parameter conditions, circuit theory analysis shows that chaos phenomenon is also very common in nonlinear circuits, and forced systems above the second order have at least one nonlinear device, construct a nonlinear resistance circuit, and use EWB(ElectronicsWorkbench). Computer simulation analysis of the circuit in Figure 3 shows that the capacitor voltage and inductor current in the circuit have irregular oscillation similar to noise, and a straight line can be observed on the oscilloscope screen. Using this circuit, we can also observe that periodic window and chaos exist not only in the circuit, but also in the circuit. For more information, see.

1 Introduction

The performance of nonlinear systems is complex and changeable. For a long time, people have fully studied the equilibrium state and periodic oscillation state in nonlinear circuits, and obtained many useful results. It was not until an important simulation result appeared more than 40 years ago that the research in nonlinear field entered a new era. 1963, E.N.Lorenz, a famous meteorologist at the Massachusetts Institute of Technology, discovered an abnormal situation while studying a meteorological model. Lorenz has repeatedly experimented on the computer for a long time, and the results are all the same, which is different from the classical understanding. Its characteristic is that the response always appears random oscillation, and the state trajectory never moves repeatedly in a region, which is later called chaos [1][2].

Chaos is a kind of random motion which is sensitive to initial conditions and produced by nonlinear dynamic system under certain parameter conditions. The fundamental reason of chaotic motion is the nonlinearity of motion equation; Chaotic motion is random in nature and very sensitive to initial values. If the initial values of the two actions are slightly different, there will be a great unpredictable deviation between the two actions after a long time. Chaos is a common phenomenon in nature and a unique complex state of nonlinear systems.

2 chaotic circuit

2. Theoretical analysis of1circuit

Chaos is also common in nonlinear circuits, and the circuit presents chaos. In principle, two situations should be considered [3][4]:

(1) mandatory system above second order; An autonomous system of three or more orders;

(2) at least one nonlinear device.

The third-order autonomous circuit shown in figure 1 consists of four linear components (two capacitors, one inductor and one linear resistor) and one nonlinear resistor.

2.2 the structure of nonlinear resistance circuit

The nonlinear resistor can be made into a negative impedance circuit through an operational amplifier. When it is greater than a certain voltage value, the operational amplifier begins to saturate. By connecting two such operational amplifiers in parallel, a nonlinear resistor with a volt-ampere curve as shown in Figure 2 can be obtained, and the complete circuit is shown in Figure 3.

3EWB simulation analysis

The circuit of Figure 3 is simulated and analyzed by EWB(ElectronicsWorkbench) software. Here we take C 1 = 0.3474UF, C2 = 0.0 155UF, L1=11.0534MH, R1=1. R5 = 3.0811kΩ, R6 =18.596kΩ, r7 = 21.7kΩ, which are substituted into the piecewise linear characteristic equation of nonlinear resistance. By changing different values of W 1, different state trajectories can be obtained. The state track at w 1 = 1. 14kω is shown in Figure 4, and the voltage time domain waveforms at C2 and C 1 are shown in Figures 5 and 6 respectively.

The results show that the capacitance voltage and inductance current in the circuit oscillate irregularly like noise, which is a bounded steady-state process, and the trajectory on the state plane never passes through repeatedly according to some inherent law. This butterfly-shaped figure is called a chaotic attractor. Chaotic attractor, also known as strange attractor, is unique in chaotic motion, which has a complex structure of stretching, folding and stretching, so that the system keeps diverging exponentially in a limited space, that is, all movements outside the attractor are close to the attractor, corresponding to a stable direction; However, all the motions reaching the interior of the attractor are mutually exclusive and correspond to the unstable direction.

In computer simulation analysis, if the initial state is changed, its response will change greatly, because chaotic motion is very sensitive to the initial state.

4 hardware circuit debugging

A printed circuit board is manufactured according to the circuit in fig. 3. Considering the nominal values of component parameters, in the actual circuit, C 1 = 0.33UF, C2 = 0.0 15UF, L1= 65438+1100mh, R1= 5./kloc. Connect the output signals S2-OUT and S 1-OUT to the CH 1 and CH2 probes of the oscilloscope respectively, and select the X-Y mode as the working mode. Adjust W 1 to the minimum, and a straight line can be observed on the oscilloscope screen. Adjust W 1, and the straight line becomes an ellipse. When it reaches a certain position, increase the magnification of the oscilloscope and fine-tune W 1 reversely, and you can see that the curve begins to change with double periods, and the curve increases from one period to two periods, from two periods to four periods, ... until a series of endless circular curves. If we continue to fine-tune W 1, the single attractor suddenly becomes a double attractor, and only the circular curve can be seen filling and jumping between the two outward vortex attractors. This is the chaotic attractor, which is characterized by global stability and local instability. Fine-tune W 1 to make it around 1. 1kω, and the actual characteristics observed by the oscilloscope are very close to the results of computer analysis.

Using this circuit, we can also observe the periodic window. If we carefully adjust W 1, the original chaotic attractor suddenly appears a three-period image. If we continue to fine-tune W 1, the chaotic attractor will appear again. This phenomenon is called periodic window.

The above results show that chaotic oscillation with this characteristic in nonlinear circuits has profound theoretical value and has changed many traditional understandings. Classical theory is mainly based on linearity, symmetry, reversibility, orderliness and stability, and produces very regular results. Modern theory is characterized by nonlinearity, asymmetry, irreversibility, disorder and instability, and has evolved a very peculiar motion mechanism, and chaos is such a typical representative.

5 concluding remarks

Chaos exists not only in circuits, but also in earthquake, meteorology, machinery, chemistry, control, physiology and other fields. The research and application of chaos has formed a new science, involving mathematics, physics, biology, chemistry, astronomy, economics and engineering technology, and has had a far-reaching impact on the development of these disciplines. Chaos contains a wide range of physical contents, and the study of these contents needs more in-depth mathematical theories, such as differential dynamics theory, topology, fractal geometry and so on. At present, the research focus of chaos has turned to chaos, quantum and spatiotemporal chaos, and synchronization and control of chaos in multidimensional dynamic systems.

refer to

[1] e.n. Lorenz The essence of chaos [M]. Beijing: Meteorological Press, 1997.

[2] james gleick. Chaos opens a new science [M]. Shanghai: Shanghai Translation Publishing House, 1990.

Jinfeng Gao. Nonlinear circuit and chaos [M]. Beijing Science Press 2005.

Wang Xingyuan. Chaos in complex nonlinear systems [M]. Beijing: Electronic Industry Press, 2003.

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