Dynamical Systems

Many engineering and natural systems are dynamical systems. For example a pendulum is a dynamical system. l mg 2 Figure 1. Simple pendulum. State The state of the dynamical system specifies it conditions. For a pendulum in the absence of external excitation shown in the figure, the angle and the angular velocity uniquely define the state of the dynamical system. Plots of the state variables against one another are referred to as the phase space representation. Every point in the phase space identifies a unique state of the system. For the pendulum, a plot of θ versus & θ is the phase space representation. The equation of motion of a dynamical system is given by a set of differential equations. That is) f(x x t , = & (1) where x is the state and t is time. The dynamical system is linear if the governing equation is linear. For the pendulum shown in Figure 1, the equation of motion is given as ⎩ ⎨ ⎧ θ ω − ω ζω − = ω ω = θ sin 2 2 o o & & where l g 2 o = ω (2) and the dynamical system is nonlinear. For small amplitude oscillation, and the equation of motion becomes , sin θ ≈ θ ⎩ ⎨ ⎧ θ ω − ω ζω − = ω ω = θ 2 o o 2 & & (3) The dynamical system is now linear. In Equations (2) and (3) o ω is the natural frequency and ζ is the damping coefficient. A system is said to autonomous if time does not appear explicitly in the equation of motion. The equation of motion of nonautonomous systems, however, explicitly