Abstract

TOPOLOGICAL ENTROPY OF THE LINEARIZED LORENZ EQUATIONS

C. Udeogu1 and A. Ibrahim2

---

In [3], Lorenz noted that the stability of a solution x(t),y(t),z(t), of his famous systems of differential equations may be formally investigated by considering the behaviour of small superposed perturbations x_0 (t),? y?_0 (t),? z?_0 (t), where such perturbations are temporarily governed by some linearized equations. In this paper, topological entropy is used as a tool to explain the state of the idealized convection governed by the Lorenz differential equations. We use the method in [1] to compute the topological entropy of this linearized equations (equation 2) at three different steady state solutions of convection given in [3] by: x=y=z=0,x=y=??(b(r-1),) z=r-1, sticking with Lorenz? values for the constants r, ?, and b. We see that the result (value) obtained after each of the computations is positive, corroborating a result by Jiagang Yang ( [4], Theorem A). Keywords: Topological Entropy, Linearized Lorenz Equations, Convection, Eigen value.

PDF