A methodology for estimating the region of attraction for autonomous nonlinear systems is developed. The methodology is based on a proof that the region of attraction can be estimated accurately by the zero sublevel set of an implicit function which is the viscosity solution of a time-dependent Hamilton–Jacobi equation. The methodology starts with a given initial domain and yields a sequence of region of attraction estimates by tracking the evolution of the implicit function. The resulting sequence is contained in and converges to the exact region of attraction. While alternative iterative methods for estimating the region of attraction have been proposed, the methodology proposed in this paper can compute the region of attraction to achieve any desired accuracy in a dimensionally independent and efficient way. An implementation of the proposed methodology has been developed in the Matlab environment. The correctness and efficiency of the methodology are verified through a few examples.

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The flight envelope plays an important role in flight safety. The concept of posing the flight envelope as a region of attraction is explored further, and it is investigated whether the stable manifold for the region of attraction computation is an efficient method for determining envelope. The stable manifold describes the flight dynamic envelope of an aircraft in an explicit representation, which means that the computation needs to be done only on the envelope, not the entire state space. In this paper, the stable manifold is computed by using a fast method which reduces the computation to solving a system of partial differential equation. Then, the stable manifold grows in the way of advancing front mesh generation framework. The stable manifold is then applied to the envelope determination of a nonlinear F-16 model. The result is compared to the results obtained with the level set method, demonstrating that the stable manifold provides a feasible and accurate result to the dynamic envelope. The proposed method is then used to investigate the effect of actuator failure on the flight safety. The proposed method can also be used as a safety assessing tool during the design phase of an aircraft.

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This paper is concerned about the dynamic behavior for the following high order nonlinear difference equation