Control devices for projection reflections in the phase space of strange attractors of dynamic chaos generators

Abstract

One of the ways to experimentally test the proposed mathematical models that exhibit the properties of deterministic chaos is to implement them using «analog computers». Depending on the dimensionality of the system, three or more signals are obtained, the spectrum of which can be analyzed. However, the most obvious evidence of chaotic behavior is strange attractors. A two-dimensional version of attractors is obtained by applying two signals to an oscilloscope that is switched to the X-Y mode. In this way, three projections are obtained, although the minimum dimensionality of the system implies that the object is three-dimensional. There are digital ways of displaying strange attractors in 3D using special consoles; they are connected to the system under study and to a personal computer. The greater the accuracy of such a set-top box, the higher its cost. However, it is possible to implement the display of a strange attractor on the oscilloscope screen in pseudo-3D using simple mathematical operations. With this approach, no information is lost during signal processing, and the cost of the device is lower. The structure of the object of study can be compared to a mathematical simulation by rotating it in phase space immediately after connecting three signals to the console, without additional programs. The basic mathematical operations are realized with the help of operational amplifiers, inverters, analog multipliers, and sin/cos potentiometer analogs. The article is devoted to a number of devices-attachments to the oscilloscope that make it possible to rotate strange attractors in pseudo 3D along two or three axes. The presented works contain device schematics and the necessary information for independent implementation. These studies demonstrate the sequence of development of the idea, the gradual departure from the analog sin/cos potentiometer to its digital counterparts, and the expansion of the rotation range from 90 to 360 degrees. The possibility of drawing a sectional plane along one of the axes and obtaining a Poincaré section is controlled. The main structural elements of the devices are defined, and the operation of some of them is briefly described. For a better understanding of the operation of such devices, images illustrating rotations in phase space are shown. A certain number of images were converted from black and white to color and further processed. The prospective development of such devices has been determined.