Stability Solutions of Differential Equations

In mathematics, a stability solution of equations refers to a situation in which a minor disturbance in a system does not have a significant impact on the system. A function f(x) is said to be stable in terms of the solution of a differential equation if every other solution starts fairly close to it. When x = 0, it stays similar to it for subsequent x values. As x increases, the difference between the solutions reaches zero, and the solution is said to be asymptotically stable. A solution is said to be unstable if it lacks any of these properties.

In this article, we will discuss the stability of equilibrium solutions and conditions for asymptotically stable differential equations. 

Stability of Equilibrium Solutions

The behaviour of integral curves near an equilibrium solution is used to classify its stability – they represent the graphs of specific solutions satisfying initial conditions whose initial values,y₀, vary only slightly from the equilibrium value.

The equilibrium solution is said to be stable, or asymptotically stable if the nearby integral curves all converge towards an equilibrium solution as t increases. The long-term behaviour of such a solution is unaffected by minor (or often large) changes in its initial state.

When the integral curves near an equilibrium solution all diverge away from it as t increases, the equilibrium solution is said to be unstable. Such a solution is highly sensitive to even minor changes in its initial state, as shown by the previous example, where the smallest change in initial value results in completely different behaviours (in both long- and short-term)

As a result, the solution y = 0 is an unstable equilibrium solution in the logistic equation example, whereas y = K is an (asymptotically) stable equilibrium solution.

Plotting y'=f(y)versus y is an alternative graphical form. This graph is easier to draw, but it contains the same amount of information as the direction field. It's close to calculus's First Derivative Test for local extrema. Any interval where f(y) > 0 will rise, as indicated by a rightward arrow (they are separated by equilibrium solutions / critical points, which are the graph's horizontal intercepts). (As we know y is the horizontal axis in this map, and its coordinates increase from left to right, from (- to +). Similarly, on any interval where f(y) < 0, y is decreasing. 

This fact will be indicated by a leftward arrow. To summarise: f(y)>0, y increases, resulting in a rightward arrow; f(y)₀, y decreases, resulting in a leftward arrow. The following is how the outcome can be interpreted: If y=c is an equilibrium solution with f(y)=0

1. The equilibrium solution y=c is unstable if f(y)₀ is to the left of c and f(y)>0 is to the right of c. (The arrows on both sides are visually moving away from c.)

2. The equilibrium solution y=c is asymptotically stable if f(y)>0 on the left of c and f(y)₀ on the right of c. (The arrows on both sides are visually heading towards c.)

A leftward arrow indicates that y is decreasing as t increases. It refers to the direction field's downward-sloping arrows. A rightward arrow indicates that y grows as t grows. On the direction field, it corresponds to upward-sloping arrows.

Asymptotically Stable Differential Equations

Stable solution of differential equation, for example, the solution \[y = ce^{-x}\] of the equation \[y = -y\]is asymptotically stable, because the difference of any two solutions \[c_{1}e^{-x}\] and \[c_{2}e^{-x}\] is \[(c_{1} - c_{2})e^{-x}\]which always approaches zero as x increases.

Stable Semistable Unstable Differential Equations

If there are other solutions on both sides of an equilibrium equation, it is said to be asymptotically stable. If there are other solutions that approach this equilibrium solution on one side of the equilibrium solution, and other solutions diverge from this equilibrium solution on the other side of the equilibrium solution, the equilibrium solution is said to be Semi-Stable. If other solutions diverge from this equilibrium solution on both sides of the equilibrium solution, it is said to be unstable.

If \[\frac{dy}{dx} = f(t,y)\]is a differential equation, then the equilibrium solutions can be obtained by setting \[\frac{dy}{dx} = 0\]. For example, if \[\frac{dy}{dx} = y(y+2)\] then the equilibrium solutions can be obtained by solving y(y+2)=0 for y. We hence see that  y=0 and y=-2 are the equilibrium solution.

The following image is the slope field of the differential equation \[\frac{dy}{dx} = (y-1)^{2} (y-2) (y-3)\] which has three equilibrium solutions y=1,y=2 and y=3.

(Image will be uploaded soon)

The equilibrium solution y=1 is green and is a semi-stable equilibrium solution.

The equilibrium solution y=2  is yellow and is an asymptotically stable equilibrium solution.

The equilibrium solution y=3 is a red colour, indicating that it is an unstable equilibrium solution.