Theoretical and applied results obtained in the theory of diffraction and propagation of electromagnetic waves with the use of methods of wave catastrophes are reviewed. The basic concepts of the theory are considered: classification of wave catastrophes, methods for constructing uniform asymptotics applied to describe fields in such regions, and analysis of the field structure. A general description of classes of special functions involved in uniform asymptotic decompositions of wave fields, properties of these functions, and methods of their calculation are presented. Applied problems that are solved efficiently by means of methods of wave catastrophes are discussed.

A numerical simulation of single-hop and double-hop propagation of frequency-modulated signals in anisotropic ionospheric plasma is performed on the basis of a numerical solution to a space–time Hamil

Methods of mathematical simulation of propagation of short radio waves in the Earth’s ionosphere are considered. Simulation is performed with allowance for new models of the Earth’s magnetic field, the electron concentration in the ionosphere, and experimental data obtained by means of the radio tomography method. Results of application of extended bicharacteristic system methods, the Maslov canonic operator method, and the catastrophe theory as the mathematical apparatus developed for the considered problems are described.

A mathematical apparatus for solving differential equations of special type by the methods of main, edge, and corner catastrophes is developed. The fundamentals of the wave catastrophe theory are cons

Propagation of short waves along a short path is modeled numerically using the bicharacteristic method with the IRI-2012 global model of ionospheric electron density, which was corrected in accordance with the experimental data, and the model of the Earth magnetic field (World Magnetic Model). It is demonstrated that periodic ionospheric disturbances may produce vertical structures in amplitude maps. Ray structures typical of the case under study and numerically calculated amplitude maps are obtained. The calculated data are qualitatively compared to experimental results.