THE METHOD OF COMPLETE BIFURCATION ANALYSIS FOR PREDICTING THE BEHAVIOR OF COMPLEX SYSTEMS

Abstract

Numerous studies of nonlinear dynamical systems have revealed a wide variety of dynamic behaviors. Nonlinear effects that occur when the initial conditions and parameters change can appear in different and sometimes unexpected ways. This work focuses on studying how complex dynamics arise in nonlinear systems. A bilinear oscillator is used as an example to show how the behavior of the system changes as its parameters vary. The study is based on a systems approach and uses the method of complete bifurcation groups. To investigate nonlinear effects, analog simulation or the direct scanning method is usually employed. However, these methods are somewhat limited in their ability to predict many dynamic characteristics, including bifurcations that lead to unstable, rare, and chaotic solutions. The paper is devoted to the parametrical analysis of periodic and chaotic oscillations in non-linear dynamic systems. The description of the regular approach to construction of bifurcation diagrams is offered to your attention. This approach allows building complete bifurcation diagrams constructing into consideration regimes the analysis of which is inaccessible by traditional methods. For example, regimes with a very small area of existence are identified as rare attractors. Such approach is used in a method of complete bifurcation groups and realized in software SPRING that developing in the Riga Technical University under guidance of Professor Mikhail Zakrzhevsky. The description of this approach and results of its application are presented in this report. The object of the research is a dynamical system whose behavior is described by the second order ordinary differential equation or a system of such equations. The dynamical system state at any time moment is characterized by the values of its phase coordinates and behavior – by variation of the phase coordinates in time. The behavior of a determinate dynamical system is uniquely defined by its initial state and the fixed parameters. The initial state is given by the values of the phase coordinates at the start. In the system behavior we can distinguish a transient process, which ends by a stationary state or regime. The stationary regime can be periodic with a period equal to the mapping period, subharmonic with a period multiple to the mapping period, almost periodic and chaotic. This method is effective for scientific research because it allows a systematic study of the main scenarios of birth and conditions for the formation of chaotic and rare attractors, which depend on parameters defined by engineering systems. Applications of this method include the simulation and analysis of nonlinear dynamic systems, the prediction of technical disasters, and the development of new vibration technologies and other technical systems.