  Estimating the Domain of Attraction (DA) of non-polynomial systems is a
challenging problem. Taylor expansion is widely adopted for transforming a
nonlinear analytic function into a polynomial function, but the performance of
Taylor expansion is not always satisfactory. This paper provides solvable ways
for estimating the DA via Chebyshev approximation. Firstly, for Chebyshev
approximation without the remainder, higher order derivatives of Lyapunov
functions are used for estimating the DA, and the largest estimate is obtained
by solving a generalized eigenvalue problem. Moreover, for Chebyshev
approximation with the remainder, an uncertain polynomial system is
reformulated, and a condition is proposed for ensuring the convergence to the
largest estimate with a selected Lyapunov function. Numerical examples
demonstrate that both accuracy and efficiency are improved compared to Taylor
approximation.
