Chaos analysis and control in fractional order systems using fractional chebyshev collocation method

In this paper, fractional Chebyshev collocation method is proposed to study Lyapunov exponents (LEs) and chaos in a fractional order system with nonlinearities. For this purpose, the solution of the fractional order system is discretized by N-degree Gauss-Lobatto-Chebyshev (GLC) polynomials where N is an integer number. Then, the discrete orthogonality relationship for the Chebyshev polynomials is used to obtain the fractional Chebyshev differentiation matrix. The differentiation matrix is then used to convert the nonlinear fractional differential equations to a system of nonlinear algebraic equations with the collocation points as the unknowns. The dominant LE (other than the zero LE) that corresponds to the time dimension is then computed by measuring the exponential rate of the trajectory deviations initiated slightly off the attractor point. The proposed technique is implemented to a damped driven pendulum with fractional order damping and the convergence of the dominant LE is studied versus the number of Chebyshev collocation points. The LE analysis is also verified by studying the system time and frequency responses for different values of the bifurcation parameter. Furthermore, the LE obtained by the proposed method for the analogous integer order system is compared with those obtained by the Jacobian technique and Gruwald-Letnikov approximation. Finally a fractional state feedback controller is designed to control the chaotic system to a desired equilibrium or periodic trajectory such that the error dynamics are time invariant or time periodic, respectively. The numerical example studied is the damped driven pendulum with fractional dampers.