Applied Mathematical Methods

38

Stability of Dynamic Systems

Solution methods of the previous chapter are applied in the present one to develop a methodology to study the dynamic behaviour of practical systems. This analysis would assist us in drawing important conclusions, especially regarding stability, of dynamic systems. In the context, we also study the Lyapunov’s method of analyzing a system on the question of stability.

Second Order Linear Systems

First, we take up a system of two first order linear differential equations

y′₁    =    a₁₁ y₁ + a₁₂ y₂,

y′₂    =    a₂₁ y₁ + a₂₂ y₂;

or,

 

y′ = Ay.                 (38.1)

Such a system may arise from a pair of coupled coordinates of a physical system. Quite often, it arises through the reduction of a single second order ...