The article suggests using modern instruments of dynamic analysis, i.e. the theories of phase, cointegration, and bifurcation analysis and the catastrophe theory to improve the methodology to study the dynamic pattern of the development indices of complex hierarchical systems (CHS) and their relationship. The article elaborates the main directions for creating research models, which would describe the interaction between the development indices of CHS, grounded on estimating and analyzing pre-crisis, crisis and post-crisis phenomena in hierarchical social and economic territorial systems. A conceptual framework algorithm is designed to model the dynamic pattern of the CHS development using modern economic and mathematical instruments to study the dynamics of time-series data and assess the relationship of CHS indices. Complex models have been implemented to monitor the key CHS development indices based on the phase and cointegration analysis of the relationship between the following processes: investment and GDP; GDP and industrial production dynamics; GDP dynamics and import volumes dynamics; wages dynamics and industrial production dynamics; migration and natural population growth. As part of the implementation of a comprehensive model for monitoring key indices of CHS development based on bifurcation analysis and the catastrophe theory, the supercritical Hopf bifurcation is built in the relationship model of imports and GDP; surfaces of the functions of Kaldor’s model and a three-dimensional Kaldor’s model are constructed. The suggested complex toolkit for research models of the CHS development instability gives us the opportunity to draw conclusions about the reasons and factors of the occurrence of endogenous (self-generating) fluctuations and bifurcations; about the probability of catastrophes and crises arising in complex hierarchical economic systems. The solution of problems caused by the CHS development instability on the basis of complex application of phase, cointegration and bifurcation analysis will allow us to predict crisis situations in advance and to offer methods of their prevention, to find complex ways out of crisis situations.
Continuation and Bifurcation Analysis in Helicopter Aeroelastic Stability Problems
