WHAT IS CHAOS? Sajid Iqbal 1 , Suhail Aftab Qureshi 2 and Muhammad Shafiq 3 1 Faculty of Engineering, University of Central Punjab, Lahore. 2 Department of Electrical Engineering, UET, Lahore. 3 College of Engineering, Islamia University, Bahwalpur. sajid.iqbal@engg.ucp.edu.pk ABSTRACT: - Chaos occurs widely in nonlinear dynamical systems. It is an aperiodic behavior in a deterministic system that shows sensitive dependence on initial conditions. Chaotic system exhibits apparently random and unpredictable behavior. In a deterministic system starting from an exactly known initial condition we can repeat the sequence of outcomes as many times as we feel like whereas in a random system the sequence of the outcomes cannot be repeated. This new science grounds understanding of order in apparent randomness and simplicity in apparent complexity. Index terms: - Bifurcation, chaos, nonlinear dynamics, strange attractor. I. INTRODUCTION Chaos is often a catchy name for nonlinear dynamics. This term is used to explain the apparently complex behavior of so called simple, linear and well behaved systems. Chaotic behavior looks erratic and almost random like the behavior of system strongly influenced by external random noise. The mathematical definition of chaos is unpredictable long time behavior arising in a deterministic dynamical system because of sensitivity to initial conditions (popularly referred to as the butterfly effect). Chaos theory is defined as the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems [1-2,6]. The chaos occurs in very simple systems that are almost free of noise. Actually, these systems are fundamentally “deterministic”; i.e., the precise knowledge of initial conditions of the system principally allows us to predict future behavior of that system. Consequently chaos may be described as a bounded, aperiodic, and noisy like oscillation: a deterministic system appears to behave randomly even though there is no random input. In unstable nonlinear systems a variety of strange effects are observed including subharmonics, quasiperiodic oscillation, intermittency and chaotic behavior [1- 2]. A. Properties of Chaotic System The chaotic dynamical systems must have the following characteristics: 1. They must be sensitive to initial conditions. 2. Their periodic orbits must be dense. 3. They must be topologically mixing. “Sensitivity to initial conditions” means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Therefore, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior. “Topologically mixing” means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region [1-2]. B. Importance of Chaos Theory The scientists and engineers are absorbed by chaos theory due to two reasons: 1. Chaos theory has provided theoretical and experimental tools to categorise and understand complex behavior that had confused earlier theories. 2. Chaos is universal; it shows up in mechanical oscillators, electric circuits, chemical reactions, optical systems, nerve cells, lasers etc. The chaotic behavior shows incredible qualitative and quantitative universal properties. These universal properties are independent of system details. This universality means that what can be learnt by studying the chaotic behavior of a mechanical oscillator can be applied immediately to understand the chaotic behavior of other systems [3]. C. Nonlinear Dynamics The word chaos is a catchphrase used to describe a particularly complex behavior of dynamical systems. Chaos is actually only one kind of behavior exhibited by these systems. 1