Analytical and Numerical Study of Bifurcation and Chaos for Nonlinear Systems

Abstract

This research work is an effort which deals with the analytical and numerical study of bifurcation, chaos and control of chaos through synchronization for nonlinear systems. This work will be a comprehensive, though not exhaustive, investigation to find the equilibrium points and their stability, identification and classification of bifurcation point, synchronization and chaos in integer as well as fractional order (FO) dynamical systems. Here we studied bifurcation and chaos via analytical and numerical techniques with specific applications to FO modified stretch-twist-fold (MSTF) flow, Nuclear Spin Generator (NSG) system and an attitude system of Quad−rotor Unmanned Aerial Vehicle (QUAV). A newly introduced 3D nonlin ear system, derived from Sprott B, C, Van der Schrier-Mass and Munmuangsaen Srisuchinwong chaotic systems, also investigated for FO analysis and existence of zero-Hopf bifurcation. During investigation of fractional MSTF, it revealed that the system displayed dynamical properties and behaved chaotically with effective dimension P (sum of fractional orders) less than 2. Adams-Bashforth-Moulton (ABM) method is utilized to analyze chaotic behavior. Synchronization of FO MSTF flow via active control at different order also achieved. NSG system examined for existence of Si’lnikov type of chaos. The research deals ˆ with the analysis of heteroclinic Si’lnikov-type orbits within an NSG chaotic sys- ˆ tem. Method of undetermined coefficient applied for analytical analysis of hete roclinic orbits. As an outcome, the Si’lnikov criteria assures that the NSG system ˆ has a Smale horseshoe type chaos. The retardational effect on a new attitude system of quad−rotor unmanned aerial xvii vehicle (QUAV) is examined. Significance of the situation where remote controller of attitude system cannot prevent flipping and consequently, shortens the life of QUAV is provided. Moreover, bifurcations aroused due to this effect in attitude system are analyzed with the help of normal form and averaging theory respec tively. A new 3D autonomous system derived from Sprott B, C, Van der Schrier-Mass and Munmuangsaen Srisuchinwong chaotic systems also examined for existence of zero Hopf bifurcation. Novelty of the study is investigation of integer and FO synchronization of derived system with famous Lorenz model by active control method under the same parametric values and initial conditions. By taking an example we proved that a couple of integer order chaotic dynamical system is synchronized if and only if its FO is synchronized, for same parametric values and initial conditions. We also compared three different numerical techniques for synchronization. This research work also gives the answer of question that is it sufficient to investigate synchronization of an integer order chaotic system when its FO also exists? This investigation contributes to minimize the control cost for a class of dynamical systems when such control is achieved through synchroniza tion. Numerical simulations, where necessary, are also provided to authenticate the analytical results.