Complex dynamics in a linear impulsive system

The dynamical behavior of a linear impulsive system is discussed by means of both theoretical and numerical ways. This paper investigates the existence and stability of the equilibrium and period-one solution, the discontinuous jumps of eigenvalues, and the conditions for system possessing infinite period-two, period-three, and period-six solutions. By using discrete maps, the conditions of existence for Neimark–Sacker bifurcation are derived. In particular, chaotic behavior in the sense of Marotto’s definition of chaos is rigorously proven. Moreover, some detailed numerical results of the phase portraits, the periodic solutions, the bifurcation diagram, and the chaotic attractors, which are illustrated by some interesting examples, are in good agreement with the theoretical analysis.