A New Method of Time Domain Solution and Stability Analysis for Periodic Orbits of Piecewise Linear Dynamic Systems

This paper proposes a new method to get time solutions of periodic orbits and to determine their stability for piecewise linear dynamic systems. The state space of piecewise linear dynamic system is cut into some linear subspaces by several switching surfaces. By solving together all the equations of periodic orbit in these subspaces with MATLAB, the coordinates of periodic orbit on each switching surface and the running time on each subspace are obtained, from which the time expressions in sections of periodic orbit can be derived. Based on these expressions, the Poincare mapping equation and the Jacobian matrix of periodic orbits can be deduced. According to the eigenvalues of the Jacobian matrix, the stability of the periodic orbit can be determined. Using 3rd-order and 4th-order Chaus circuits as examples, the time expressions of many periodic orbits are obtained and their stability is determined respectively by the new method. The results are exact the same as that of digital simulations, which shows the new method is correct and practical.