ABSTRACT
In linear system theory, if the variables which characterize the behavior of a system are either constant or periodic functions of time, the system is said to be in steady state. In a stable linear system, the steady state can be seen as a limit behavior, approached either as the actual time t tends to +∞ or, alternatively, as the initial time t0 tends to −∞ (the two viewpoints being in fact equivalent). For a general nonlinear dynamical system, concepts yielding to a notion of steady-state repose on certain fundamental ideas dating back to the works of H. Poincaré and G.D. Birkhoff.∗ In particular, a fundamental role is played by the concept of ω-limit set of a given point, which is defined as follows. Consider an autonomous ordinary differential equation
x˙ = f (x), (47.1) in which x ∈ Rn. Suppose that f (x) is locally Lipschitz. Then, it is well-known that, for any x0 ∈ Rn, the solution of Equation 47.1 with initial condition x(0) = x0, denoted in what follows by x(t, x0), exists on some open interval of the point t = 0 and is unique.
