Determination of dynamic modes in a two-mode hereditary dynamo system

This study presents a computational analysis of a hereditary dynamical system modeling a two-mode hydromagnetic dynamo with memory. We conduct a numerical investigation of dynamic regimes emerging under variation of the system's control parameters. The hereditary dynamical system is described by a set of integro-differential equations.
Lyapunov exponent analysis serves as a principal method for examining dynamic regimes. To implement this approach, the integro-differential system was reduced to a system of ordinary differential equations. The paper provides a description of the corresponding class of kernels and the reduction result.
As an alternative approach, we employ the 0-1 test for chaos detection. A comparative analysis between the 0-1 test and Lyapunov exponents for a particular case demonstrates their qualitative agreement. Subsequent investigations are primarily utilized the 0-1 test for analyzing the integro-differential system's dynamic regimes.
Notably, this method only discriminates between regular (periodic and asymptotically stationary) and chaotic regimes. For finer classification of regular regimes, we propose an auxiliary method based on analyzing the autocorrelation function characteristics of the solution's time series. Empirical results show that computing the autocorrelation function's expected value effectively distinguishes periodic/quasi-periodic regimes from asymptotically stationary ones.
Both instantaneous and delayed hereditary feedback cases are examined. Simulation results reveal that the model reproduces various dynamic regimes characteristic of actual cosmic dynamo systems.