  Investigation of the critical levels and catastrophes in the complex systems
of different nature is useful and perspective. Mathematical modeling and
analysis is presented for revealing and investigation of the phenomena and
critical levels in a development of complex systems for various natures
associated with diverse complicated factors, in particular with shifted
arguments of the system. Intensive research in this direction and developed
techniques may optimize management of the complex systems in
financial-economic, natural and other fields. Construction of adequate
mathematical models for development of complex systems, critical modes and
their effective control are important tasks for a wide range of contemporary
issues as shown in paper on examples. Critical levels in development of
economic, banking, technical, political and other systems are necessary to
determine and anticipate, to manage their system requirements and provide
stable development, without being hit in a critical situations, leading to
growing oscillations of the system settings.
