parameter a > 0. If the system is perturbed to the value its subsequent sequence of values is If a < 1, these numbers get closer and closer to 0 regardless of the starting value while if a > 1 the numbers get larger and larger without bound. But if a = 1, the numbers do neither of these: instead, all future values of x equal the value Thus the case a = 1 exhibits marginal stability. System response A marginally stable system is one that, if given an impulse of finite magnitude as input, will not "blow up" and give an unbounded output, but neither will the output return to zero. A bounded offset or oscillations in the output will persist indefinitely, and so there will in general be no final steady-state output. If a continuous system is given an input at a frequency equal to the frequency of a pole with zero real part, the system's output will increase indefinitely (this is known as pure resonance). This explains why for a system to be BIBO stable, the real parts of the poles have to be strictly negative (and not just non-positive). A continuous system having imaginary poles, i.e. having zero real part in the pole(s), will produce sustained oscillations in the output. For example, an undamped second-order system such as the suspension system in an automobile (a mass–spring–damper system), from which the damper has been removed and spring is ideal, i.e. no friction is there, will in theory oscillate forever once disturbed. Another example is a frictionless pendulum. A system with a pole at the origin is also marginally stable but in this case there will be no oscillation in the response as the imaginary part is also zero (jw = 0 means w = 0 rad/sec). An example of such a system is a mass on a surface with friction. When a sidewards impulse is applied, the mass will move and never returns to zero. The mass will come to rest due to friction however, and the sidewards movement will remain bounded. Since the locations of the marginal poles must be exactly on the imaginary axis or unit circle (for continuous time and discrete time systems respectively) for a system to be marginally stable, this situation is unlikely to occur in practice unless marginal stability is an inherent theoretical feature of the system. Stochastic dynamics Marginal stability