A method for determining the stability of a class of autonomous nonlinear continuous chaotic dynamical systems

This paper presents a method for analyzing the stability of a nonlinear chaotic system that is substantially different from existing methods. First, the systems of equations are rewritten in terms of equivalence classes of ratios of polynomials of the operator ξ over the real field. Next, conditions are developed using the Banach Fixed Pont Theorem that guarantees that the solution to the system of equations is exponentially stable. Then the concept of an equivalent system is derived. If a system has an equivalent system that satisfied the conditions that guarantee exponentially stability, then the solution to the original system of equations is shown to be exponentially stable. Finally, if an equivalent system of equations that satisfy the conditions that guarantee exponentially stability cannot be found, an approximation to an equivalent system that is exponentially stable in some region of the state space is developed and is used to determine the stability of the system.