Periodic solutions and flip bifurcation in a linear impulsive system

In this paper, the dynamical behaviour of a linear impulsive systemis discussed both theoretically and numerically. The existence andthe stability of period-one solution are discussed by using adiscrete map. The conditions of existence for flip bifurcation arederived by using the centre manifold theorem and bifurcationtheorem. The bifurcation analysis shows that chaotic solutionsappear via a cascade of period-doubling in some interval ofparameters. Moreover, the periodic solutions, the bifurcationdiagram, and the chaotic attractor, which show their consistencewith the theoretical analyses, are given in an example.