14-10-2010, 12:28 PM
Chaos report.pdf (Size: 1.11 MB / Downloads: 119)
Tanuj Gupta
Ashutosh Thakur
Sujil V.
Bachelor of technology in “ELECTRICAL & ELECTRONICS ENGINEERING”
ANALYSIS OF CHAOTIC SYSTEMS
ABSTRACT
Analysis and control of dynamical systems have been explored since long by various researchers as it is vital to design systems to meet out desired behaviour successfully. Chaotic systems belong to a special class of systems which has drawn tremendous amount of interest because of its applications in various fields of science and engineering. These systems are different than conventional nonlinear systems in the sense that these are highly sensitive to initial conditions and parameter variations. These systems exhibit quite complicated behaviour named chaos when variations are introduced to initial conditions or parameters. Such systems can attain chaotic behaviour in many different ways. In this report, we are going to look at the period-doubling approach to chaos. It is indeed a remarkable fact that all systems which make a transition from order to disorder by the period-doubling route, irrespective of the exact functional forms that represent these systems, display common properties. It turns out that given certain qualitative constraints on the system considered, thus allowing it to take a certain route to complexity, we can make statements on the quantitative properties displayed too. Specifically, we look at one-dimensional maps, functions which map the interval (0, 1) to itself here. The concept of universality guarantees that it is sufficient to study the most basic of non-linear maps, the Logistic Map to arrive at broad features for all maps with the same qualitative behaviour as the Logistic Map. We therefore proceed to look at the Logistic map and Henon map(2nd order) itself, in some detail, to analyze how the inherent period-doubling nature of the route to aperiodicity leads to the emergence of scaling coefficients.
Linear and Non-Linear System
System
A system can be defined as the group of interacting or interdependent entities forming an integrated whole. A system can be represented by a group of mathematical equations. The behavior of the defining equations of the system defines the type of system whether linear or non-linear. Most natural systems are non-linear in nature but can be studied by making linear approximations.
A dynamical system is described in terms of differential equations or difference equations that describe its behavior for a short period of time. To determine the behavior for longer periods it is necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers.
One of the major classifications of systems is on the basis of linearity.
• Linear systems
• Non-linear systems
Linear systems
A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.
Linear systems satisfy the properties of superposition and homogeneity.
Homogeneity
As we increase the strength of the input signal, say we double it then the output then we can predict that the output will also double.
Additivity
If x1 gives an output y1 and x2 gives an output y2 then on taking the combined effect of both we get the output as sum of individual inputs.