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Steady states
Steady states [equilibria, fixed points] for the differ- ential equation of the form
x′(t) = f (x) are those values of x that satisfy f (x) = 0.
Question of interest: what is the stability of such steady states? If x is perturbed from its steady state value x∗, does it return to x∗^ or move away from x∗?
Stability analysis
For equations of the form x′(t) = f (x), there are two approaches to determine the stability of fixed points:
- Graphical stability analysis
- Linear stability analysis
Graphical stability analysis for the cooling
problem with equation x′(t) = k(21 − x)
Boardwork...
Graphical stability analysis for the general
problem with equation x′(t) = f (x)
Boardwork...
Graphical stability analysis: observations
We note the following relationship between the sta- bility of a fixed point x∗^ of the differential equation x′(t) = f (x) and the slope of f (x):
- If f ′(x∗) > 0, then x∗^ is unstable
- If f ′(x∗) < 0, then x∗^ is stable
Linear stability analysis
We are considering
dx dt
= f (x) (1)
with steady state x∗, that is, f (x∗) = 0.
Introduce a small perturbation y from x∗, that is, let
x = x∗^ + y (2)
Substitute (2) into (1), and expand the right-hand- side with a Taylor series to get:
d(x∗^ + y) dt
= f (x∗^ + y) dy dt
= f (x∗) + f ′(x∗)y + O(y^2 )
Since x∗^ is a fixed point, we can replace f (x∗) on the right hand side by 0. If, in addition, we can safely neglect all the terms in the Taylor series that have been collected in the term O(y^2 ), then we are left with the following equation for the perturbation:
dy dt
= f ′(x∗)y.
We recognize that f ′(x∗) is some constant, λ say. The equation for the perturbation thus is the linear equa- tion dy dt
= λy, which we studied previously (the world’s simplest dif- ferential equation).
The solution for this last differential equation is y(t) = y 0 eλt.
- If λ < 0, then y(t) → 0 as t → ∞.
- If λ > 0, then y(t) → ±∞ as t → ∞.
That is, the perturbation dies out if λ = f ′(x∗) < 0, and grows if λ = f ′(x∗) > 0. In the special case that λ = f ′(x∗) = 0, the terms collected in the term O(y^2 ) become important, and other techniques of analysis are required. The theorem presented earlier follows.
