Spectral Analysis and Numerical Ranges in Banach Space and Measuring Chaos of Lorenz System

Abstract

This paper represents the basic concepts spectrum analysis and numerical range dynamical system in Banach space. Li-Yorke chaos in Banach spaces is defined and their related theorems, definitions, lemmas, propositions, and corollaries are redefined, and some are proofed. Spectral and numerical analysis of chaotic dynamical systems are established through Lyapunov exponents, equilibrium etc. Stability analysis of dynamical system in Banach space is proofed that chaos has existed in chaotic systems. We have analysed the Lorenz system in Banach space with different equilibrium points and its stability are discussed. Measuring Chaos (Lorenz system) in Banach space are verified through basin of attraction, Poincare sections, strange attractors bifurcation analysis at different equilibrium points, and dissipativity natures and existence of the bounded attractor etc.