Chaos

Chaotic systems have been widely studied by many scientists and engineers from different viewpoints. Chaos and non-linear dynamics are presently active fields of interdisciplinary research, attracting the attention of researchers on account of its wide applicability in the physical world. Chaotic systems are dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly known as the butterfly effect. Since chaos phenomenon in weather models which was first observed by Edward Lorenz in 1963, a large number of chaos phenomena and chaos behaviour have been discovered in social, economical, biological and electrical systems (El-Dessoky, 18 june 2007).

One of the key problems in the study of chaotic systems is the chaos control. The chaos control problem is concerned with stabilization of the chaotic attractor to a periodic orbit or a stable equilibrium point. The pioneering works of Ott et al (1998), are the most important works in this direction. It was promptly followed by application of various engineering control techniques like feedback control (Boris Kramer, august 17 2017), adaptive control (LOZANO, 2014), passive control (QI Dong-lian,2002), backstepping control (Vincent, january 2008), active control (Vincent, june 2008)etc. for achieving this objective.

Pecora and Carroll (1990) introduced a method to synchronize two identical chaotic systems and showed that it was possible for some chaotic systems to be completely synchronized. From then onwards, chaos synchronization has been widely explored in a variety of fields including physical systems, chemical systems, ecological systems, secure communications, etc. (Kenny Headington, 2004). Recently, the concept of synchronization has been extended to the scope, such as generalized synchronization, phase synchronization, lag synchronization, and anti-synchronization (AS), in which the state vectors of synchronized systems have the same absolute values but opposite signs. Therefore, the sum of two signals can converge to zero when AS appears (Singh Piyush Pratap, 2010).

Chaos has developed over time. For example, Ruelle and Takens (2000) proposed a theory for the onset of turbulence in fluids, based on abstract considerations about strange attractors (Sajad Jafari, 2009). In the early 1970‟s, May was working on a model that addressed how insect birth rate varied with food supply. He found that at certain critical values, his equation required twice time to return to its original state, the period having doubled in value. After several period-doubling cycles, his model became unpredictable, rather like actual insect populations tend to be unpredictable. Since May’s discovery with insects, mathematicians have found that this period-doubling is a natural route to chaos for many different systems (El-Dessoky, 18 june 2007).

In recent times, study on chaotic dynamics has entered a new phase. Coupled to the demonstration of chaos in a wide range of situations and its study, many researchers are interested in utilizing the basic knowledge of the theory of chaos either to analyze chaotic experimental time series data or to use the presence of chaos to achieve some practical goals like synchronization and control, which has applications in secure communications, description of financial times series prediction, techniques of neural networks and genetic algorithms(AZZAZ, may 2011). Chaotic dynamics essentially is one of the attributes of nonlinearity of an evolving process. There exists domains of operation in which apart from general nonlinear features one can expect the richness of chaotic behaviour. As a matter of fact every Natural system is a store house of such behaviours and in all probability chaos assumes the operating paradigm in such systems(Maiti, august 2013).

Chaos is one of the subtle behaviours of a nonlinear system. It has its intrinsic academic interest which involve a number of concepts definitions and can be studied from very elementary nonlinear differential or difference equations(Khaki-Sedigh, 6 april 2008). The essential mathematics is embedded in the dynamical systems theory which focuses on the qualitative and quantitative behaviours of first order differential /difference equations. Following the ideas of Poincare (2001) (First one to identify the richness of nonlinear system behaviour) on the qualitative description of system behaviour, some geometrical methods have evolved which provide information without explicitly solving the system (Maiti, august 2013). An interesting and useful fact, rather a dominating factor of a nonlinear system is its capability of producing multiple equilibrium states unlike linear system which sticks on to a unique steady state. How to realize or avoid a particular state invites the role of stability theory (Uğur Erkin Kocamaz, 2006).

In the context of synchronizing two non-identical oscillators, the most prevalent method is to couple the systems suitably, so that they asymptotically follow the same path on the attractor. The master-slave, or drive-response formalism is commonly used for synchronization and anti-synchronization problems involving non identical oscillators. Two chaotic systems (master- slave) are anti-synchronized, when sum of their states will converge to zero asymptotically and amplitude of states will be equal in magnitude but with opposite phase. State with equal magnitude but opposite in phase is also an important phenomenon in case of chaotic system synchronization (Singh Piyush Pratap, 2015).

In this research, Lu (Vaidyanathan1, 2011) and Lorenz (Spassova, 2013) Chaotic system is considered for anti-synchronization and synchronization using adaptive control as well as parameters estimation when system parameters are unknown. Stabilization and convergence of error dynamics is achieved using Lyapunov stability theory(Piyush Pratap Singh ⇑, 1 october 2014; Singh Piyush Pratap, 2015).