Shrimps as a fundamental object of nonlinear dynamics. Overview, properties and quasi-periodic generalization

Mathematics and Computers in Simulation. 2027. Vol. 2. P. 574–592.

Kuznetsov A. P., Sedova Y. V., Sataev I. R., Stankevich N.

Universal bifurcation structures on the parameter plane of nonlinear systems, known as shrimps, are discussed. An overview of numerous examples from various fields of science is given. Shrimps are typical for both maps of different dimensions and flows. The varieties of shrimp are discussed and relevant examples are given. The similarity property on the parameter plane is illustrated, which generates smaller and smaller shrimps, as well as the effect of noise on shrimps. Gener alizations to the case of quasi-periodic shrimps for the maps are presented. The modification of the picture with a variation of an additional parameter is discussed. In this case, the destruction of shrimps and quasi-periodic analogues of Arnold tongues occurs with the emergence of complex quasi-periodic structures, a resonant web and chaos. Examples of quasi-periodic structures and shrimps in chaos for flow systems are presented for three types of radio-physical generators.

Keywords: chaos, Lyapunov exponents, quasi-periodicity, shrimp structures

Publication based on the results of: Машинное обучение и нелинейная динамика: пересечение, взаимодействие и синтез (2027)