BIFURCATION IN DYNAMICAL SYSTEMS. DETERMINISTIC CHAOS. QUANTUM CHAOS

Transformation of caustic structures of the catastrophe type during the propagation of electromagnetic waves in cold plasma

A. S. Kryukovskii, D. V. Rastyagaev

Russian New University, Moscow, Russia

Abstract: The purpose of the work is to investigate the possibilities of formation of centers of topological caustic singularities of the catastrophe type during probing of a unimodal plasma layer by electromagnetic waves. The centers of caustic singularities correspond to the focusing regions of electromagnetic fields of different orders. Therefore, their study is an urgent task. Methods. The article develops a method for calculating the position of the centers of singularities in an extended parameter space, which in addition to coordinates include the height of the plasma layer, the angle of the ray exit, the ratio of the plasma frequency to the operating frequency, and cubicity. The Hamilton-Lukin bicharacteristic method is used to calculate the ray trajectories. Results. Mathematical modeling is performed using the example of a flat-layered plasma layer with a cubic dependence of the electron concentration on the height. Explicit expressions for the eikonal derivatives up to the eighth order inclusive are obtained, which makes it possible to determine the centers of the main cuspoid catastrophes. Graphs are constructed for the dependences of coordinates, the height of the trajectory reflection from the plasma layer, the distance from the radiation source to the plasma layer, the ray exit angle, and the ratio of the plasma frequency to the operating frequency on cubicity for a butterfly-type catastrophe. It is shown that the height of the trajectory reflection from the plasma layer, the distance from the radiation source to the plasma layer, and the height of the singularity reach maximum values for a parabolic layer. It is established that a butterfly-type singularity occurs even when not only the function itself describing the electron concentration is continuous, but also its derivative. Conclusion. The developed approach allows us to find the centers of not only the “butterfly” type catastrophe, but also the centers of other topological cuspoid singularities: “cusp”, “swallow tail”, “wigwam”, “star”, which is of great practical importance in studying the propagation of radio waves in ionospheric plasma.  

Keywords: caustic structures, singularities, radio wave propagation, bicharacteristic system, wave catastrophes.

UDC: 537.87

Received: 24.03.2025
Revised: 30.09.2025
Accepted: 23.05.2025

DOI: 10.18500/0869-6632-003182