Stability Analysis of Periodic Solutions of a Certain Ultra-hyperbolic Boundary Value Problem

In this paper, stability analysis of periodic solutions of ultra-hyperbolic boundary value problem was investigated using eigenvalue approach. Periodic solutions for the ultra-hyperbolic equation was obtained using the method of separation of variables and the proof of periodicity was conducted for the independent variables. Test for periodicity was conducted for the independent variables and the results show that the system obey periodic solutions. Stability results for the ultra-hyperbolic equation was obtained by converting the system of equations into first order equivalent system and further obtained the matrix for the given system. The result obtained show that the equilibria points were not asymptotically stable. Eight eigenvalues were obtained in which four of them are parameter dependent and the other four eigenvalues are equal to one. Furthermore, numerical simulations were used to demonstrate the behavior of the system for different values of the parameters extending known results in literature. Application of our results can be seen in inverted pendulum where amplitude of oscillation grows exponentially or without bound.