ODE systems and dynamic stability

The following code uses this method to obtain an approximate value for the integral of (4.82) in good agreement with that obtained from dbiquad. 

We now resume our discussion of IVPs, starting with a system of hnear first-order ODEs X = Ax, 

Definition A steady state of a dynamic system is one in which the time derivatives of each state variable are zero. A steady state Xs is stable, if following every infinitesimal pertmbation away from ats, the system returns to Xg. A steady state Xs is unstable, if any infinitesimal perturbation causes the system to move away from Xg. A steady state for which a perturbation neither grows nor decays with time is said to be neutrally stable. Stabihty is a property of the particular steady state and not of the differential equation. 

We now determine the stability of Xg = 0 for x = Ax. Let us assume that A is diago-nalizable, so that any v e 91 can be written as the linear combination 

To determine the stability of Xg = 0, we compute the response, starting at xM = e, and examine whether x(t) returns to Xg = 0 or it diverges. That is, if the system is stable, we must have 

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For larger systems such as those typically fotmd in complex chemical problems, non-dimensionalisation may be impractical, and hence, numerical perturbation methods are generally used to investigate system dynamics and to explore timescale separation. By studying the evolution of a small disturbance or perturbation to the nonlinear system, it is possible to reduce the problem to a locally linear one. The resulting set of linear equations is easier to solve, and information can be obtained about the local timescales and stability of the nonlinear system. Several books on mathematics and physics (see e.g. Pontryagin 1962) discuss the linear stability analysis of the stationary states of a dynamical system. In this case, the dynamical system, described by an ODE, is in stationary state, i.e. the values of its variables are constant in time. If the stationary concentrations are perturbed, one of the possible results is that the stationary state is asymptotically stable, which means that the perturbed system always returns to the stationary state. Another possible outcome is that the stationary point is unstable. In this case, it is possible that the system returns to the stationary state after perturbation towards some special directions but may permanently deviate after a perturbation to other directions. A full discussion of stationary state analysis in chemical systems is given in Scott (1990). 

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