Question

In each exercise, locate all equilibrium points for the given autonomous system. Determine whether the equilibrium point or points are asymptotically stable, stable but not asymptotically stable, or unstable. $$ \begin{aligned} &y_{1}^{\prime}=y_{2}-1 \\ &y_{2}^{\prime}=y_{1}+2 \\ &y_{3}^{\prime}=-y_{3}+1 \\ &y_{4}^{\prime}=-y_{4} \end{aligned} $$

Short answer

Question: Determine the stability of the equilibrium point for the given system of equations. $$ \begin{aligned} &y_{1}^{\prime}=y_{2}-1 \\\ &y_{2}^{\prime}=y_{1}+2 \\\ &y_{3}^{\prime}=-y_{3}+1 \\\ &y_{4}^{\prime}=-y_{4} \end{aligned} $$ Answer: The equilibrium point is (-2,1,1,0). The stabilities for each variable at this point are: y1 and y2 are unstable, while y3 and y4 are asymptotically stable.

Step-by-step solution

Locate the equilibrium points

Set the following equations equal to zero and solve for each variable: $$ \begin{aligned} &y_{1}^{\prime}=y_{2}-1=0 \\\ &y_{2}^{\prime}=y_{1}+2=0 \\\ &y_{3}^{\prime}=-y_{3}+1=0 \\\ &y_{4}^{\prime}=-y_{4}=0 \end{aligned} $$ We can find the equilibrium points by solving these equations as follows: $$ \begin{aligned} &y_2=1\\\ &y_1=-2\\\ &y_3=1\\\ &y_4=0 \end{aligned} $$ The equilibrium points are \((y_1,y_2,y_3,y_4)=(-2,1,1,0)\).

Determine the stability of the equilibrium points

To determine the stability of the equilibrium point, let's analyze the behavior of each equation's derivative in the neighborhood of the equilibrium point. For \(y_1\), since \(y_{1}^{\prime}=y_{2}-1\), we have \(y_{1}^{\prime}>0\) if \(y_{2}>1\) and \(y_{1}^{\prime}<0\) if \(y_{2}<1\). Thus, \(y_1\) is unstable around \(y_2 = 1\). For \(y_2\), since \(y_{2}^{\prime}=y_{1}+2\), we have \(y_{2}^{\prime}>0\) if \(y_{1}>-2\) and \(y_{2}^{\prime}<0\) if \(y_{1}<-2\). Thus, \(y_2\) is unstable around \(y_1 = -2\). For \(y_3\), since \(y_{3}^{\prime}=-y_{3}+1\), we have \(y_{3}^{\prime}>0\) if \(y_3<1\) and \(y_{3}^{\prime}<0\) if \(y_3>1\). Thus, \(y_3\) is asymptotically stable around \(y_3 = 1\). For \(y_4\), since \(y_{4}^{\prime}=-y_{4}\), we have \(y_{4}^{\prime}<0\) if \(y_4>0\); \(y_4\) is always decreasing. Thus, \(y_4\) is asymptotically stable around \(y_4 = 0\). The equilibrium point \((-2,1,1,0)\) is classified as follows: - Unstable for \(y_1\) and \(y_2\) - Asymptotically stable for \(y_3\) and \(y_4\)

Key concepts

Asymptotically Stable

When discussing the dynamic behavior of systems, the term asymptotically stable plays a crucial role. An equilibrium point is said to be asymptotically stable if, when the system is slightly deviated from this point, it returns to it as time progresses. In essence, the system's state will not only stay close to the equilibrium after small disturbances but will actually tend to it over time.

In our example, both y_3 and y_4 exhibit this type of stability. For y_3, any initial value less than 1 causes the system to increase towards the equilibrium, and any initial value greater than 1 causes the system to decrease back to the equilibrium. Similarly, for y_4, if it starts above 0, it monotonically decreases towards 0. It's important to note that asymptotic stability implies stability, but not all stable systems are asymptotically stable.

Stability Analysis

Performing a stability analysis requires us to investigate the behavior of a system near its equilibrium points. Through this analysis, we learn whether the system will return to equilibrium (stable), move away from it (unstable), or stay neutral with neither tendency. To conduct a basic stability analysis, we look at the sign changes in the derivatives of the system's functions.

In the given exercise, the analysis shows that the equilibrium points for y_1 and y_2 are unstable since the directional behavior of their derivatives indicates that the system will move away from these points. For y_3 and y_4, the reverse is true, indicating their asymptotic stability. This simple sign-change criterion is often a starting point for more sophisticated stability analyses, including those using Lyapunov methods or Jacobian matrices.

Autonomous System

An autonomous system is a system of differential equations where the rates of change are functions of the system's state only and not of time directly. Such systems have their behavior fully determined by their initial conditions, without any explicit time dependence in the equations governing their evolution.

The system presented in our exercise is autonomous: the derivatives of y_1, y_2, y_3, and y_4 each depend solely on their respective variables and not on time. This property significantly simplifies the analysis of equilibrium points and the study of system dynamics as it allows us to focus on the spatial configuration of the system's states, rather than on their evolution over time.