Stability

Ordinary differential equations: Asymptotics and stability

Stability

Frequent behaviour of a solution of a differential equation is that it approaches a constant value in the course of time and that the constant value itself is a particular solution, namely an equilibrium. We already had a short look at the behaviour of sigmoidal solutions of the logistic differential equation. Let us now explore the following differential equation of limited exponential growth:

$\frac{\dd y}{\dd t}=2-y$ In the slope field two solution curves are shown and they illustrate that a solution of this differential equation always approaches $y=2$ for large values of $t$ . $y(t)=2$ itself is a solution of the differential equation; it is called the equilibrium solution (shortly, equilibrium), fixed point solution, or steady state.

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The general case of limited exponential growth is not difficult either. Consider the differential equation

$\frac{\dd y}{\dd t}=r\cdot (a-y)$ for positive constants $r$ and $a$ . There exists an equilibrium and we can find it by solving the equation $\frac{\dd y}{\dd t}=0$ . The solution of the linear equation $r\cdot (a-y)=0$ is easy to find and is $y=a$ .

Of course you can also release the requirements of positivity of $r$ and $a$ .

Stability and asymptotics of limited exponential growth Consider the differential equation

$\frac{\dd y}{\dd t}=r\cdot (a-y)$ for constants $r\neq 0$ and $a$ . The solution

$y=a$ is an equilibrium.
If $r>0$ , then any solution of the differential equation always approaches $y=a$ for large values of $t$ . In other words, we have attracting equilibrium. We also refer to this as a stable equilibrium or attractor.
If $r<0$ , then any solution of the differential equation different from the equilibrium increasingly moves away from $y=a$ for large values of $t$ . In other words, we have a repelling equilibrium. We also refer to this as an unstable equilibrium or repeller.

Example of a semi-stable equilibrium Besides a stable or unstable equilibrium, often referred to as the non-hyperbolic equilibria, there is a third option, namely, a semi-stable equilibrium, also referred to as a half-stable equilibrium. In this case, the behaviour of a solution depends on where the solution is located with respect to the equilibrium. The previously discussed ODE with the blow-up phenomenon

$\frac{\dd y}{\dd t}=y^2$ illustrates this.

There is an equilibrium $y=0$ . The general solution is a rational function of the form

$y(t)=\frac{1}{c-t}$ for some constant $c$ . But in fact, this formula describes two solutions: for $t<c$ we have a solution that runs away from the equilibrium over time and for $t>c$ we have a solution that approaches the equilibrium as time passes. In other words: a solution curve with positive function values moves away from the equilibrium in the course of time and a solution curve with negative function values approaches the equilibrium in the course of time. The equilibrium is from one side a repeller and from the other side an attractor.