AN INSIGHT INTO CHAOTIC SYSTEMS

ABSTRACT

This study offers a profound exploration of chaotic systems, employing the Lorenz equation as a fundamental model given as dt/dx = Ϭ(y-x)……………………………………………………..(1) dt/dy = x(𝜌-z)-y…………………………………………………...(2) dy/dz = xy-βz ………………………………………………………(3) The Lorenz system, is a three coupled ordinary differential equations having chaotic solutions for certain parameter values and initial conditions. In particular the equations describes the rate of change of three quantities with respect to time; x is proportional to the rate of convection, y is proportional to the horizontal temperature variation and z is proportional to the vertical temperature variation, where p, q, r are system parameters proportional to certain physical dimension of the layer. Our research integrate the study of chaotic dynamics with function composition and Jacobian matrix, further extending its application to ergodic theory’ This was done by taking the composition of the three coupled ordinary differential equation, by first rewriting the equation in its compact form. Taking the composition of the system f0f, the behavior of the dynamical system described by the composition of the function is heavily influenced by the large value of r. the quadratic and linear transformations in each coordinate are shaped by specific values of p, q, r.  By examining the behavior of the solution at a specific point in the domain, [0,1,0], shows that the system produce the vector [q,-1.0] at this point.