Question

Each exercise lists the general solution of a linear system of the form $$ \begin{aligned} &x^{\prime}=a_{11} x+a_{12} y \\ &y^{\prime}=a_{21} x+a_{22} y \end{aligned} $$ where \(a_{11} a_{22}-a_{12} a_{21} \neq 0\). Determine whether the equilibrium point \(\mathbf{y}_{e}=\mathbf{0}\) is asymptotically stable, stable but not asymptotically stable, or unstable. $$ \begin{aligned} &x=c_{1} e^{-2 t} \cos 2 t+c_{2} e^{-2 t} \sin 2 t \\ &y=-c_{1} e^{-2 t} \sin 2 t+c_{2} e^{-2 t} \cos 2 t \end{aligned} $$

Short answer

Answer: The equilibrium point \(y_0 = (0, 0)\) is asymptotically stable.

Step-by-step solution

Examine the decay of the exponent in the solution

Consider the exponential term \(e^{-2t}\) in both components of the general solution. As \(t\) approaches infinity, we see that \(e^{-2t}\) approaches zero. Specifically, we have \(\lim_{t \to \infty} e^{-2t} = 0.\) This indicates that the amplitude (magnitude) of both components is decaying, which has implications for the stability of the equilibrium.

Investigate the behavior of the trigonometric terms

Now let's investigate the trigonometric terms \(\cos 2t\) and \(\sin 2t\) in the general solution. These terms oscillate between \(-1\) and \(1\) as \(t\) increases. Specifically, their amplitude does not grow or decay, which also has implications for the stability of the equilibrium.

Combine the exponential and trigonometric behavior

Since the exponent \(e^{-2t}\) is decaying and the trigonometric terms are oscillating, their combined behavior shows that the solutions to the system have oscillation with a decaying amplitude. This means that as time goes by, the oscillations will become smaller and smaller, and eventually, the solutions will approach the equilibrium \(\mathbf{y_0} = \mathbf{0}\).

Determine the stability

As we've shown in step 3, the general solution approaches the equilibrium \(\mathbf{y_0} = \mathbf{0}\) as time goes by. Because the amplitude is decaying, this means that the equilibrium is not just stable but asymptotically stable. This is because the oscillation with decaying amplitude implies that the system trajectories will approach and remain arbitrarily close to the equilibrium point as \(t \rightarrow \infty\). In conclusion, the equilibrium point \(\mathbf{y_e} = \mathbf{0}\) is asymptotically stable.

Key concepts

Asymptotic Stability

Asymptotic stability is a key concept in the analysis of differential equations, especially when examining equilibrium points. An equilibrium point is asymptotically stable if solutions that start near this point not only remain close as time progresses but also eventually converge to the equilibrium point itself as time approaches infinity.

When considering linear systems, such as the ones given in the exercise, a crucial factor to determining asymptotic stability is the imaginary part of the system's eigenvalues being zero, while the real part is negative. This combination results in solutions where amplitudes decay over time.

In our exercise, the term \(e^{-2t}\) directly signifies decay in the exponential sense, leading to shrinking oscillations over time. When a system is asymptotically stable, you can expect that any small perturbation will gradually dissipate, causing the system to return to its equilibrium, much like a ball returning to a resting position at the bottom of a bowl after being lightly nudged. This concept is pivotal in engineering and mathematics as it ensures predictability and safety in systems behavior.

Linear Systems of Differential Equations

Linear systems of differential equations involve multiple interdependent equations. These systems arise when modeling a variety of phenomena, from population dynamics to electrical circuits. In the context of our exercise, we're analyzing a two-dimensional linear system.

The general form given involves two equations:

• \(x^{\prime}=a_{11}x + a_{12}y\)
• \(y^{\prime}=a_{21}x + a_{22}y\)

Here, \(a_{11}, a_{12}, a_{21},\) and \(a_{22}\) are constants that determine the interactions between the variables \(x\) and \(y\).

In terms of solution, these equations can often be decoupled or combined into matrix form to find a general solution. The importance of the determinant condition \(a_{11}a_{22} - a_{12}a_{21} eq 0\) ensures that the system is nontrivial, meaning it doesn't collapse to a zero-dimension solution. Therefore, the system possesses unique behaviors worthy of analysis, like stability or instability, which can be classified by interpreting the solution's structure.

Equilibrium Points

Equilibrium points are specific solutions of a system of differential equations where the system does not change over time. In mathematical terms, these are points where all derivatives are zero, indicating that the system is at rest.

In our linear system example, the equilibrium point \(\mathbf{y_{e} = 0}\) serves as the reference point for assessing stability. Analyzing whether a system returns to or moves away from its equilibrium after a small disturbance helps us understand the system’s inherent stability properties.

Regarding the exercise, the equilibrium point \(\mathbf{y_{e} = 0}\) is determined by setting the rates of change \(x'\) and \(y'\) to zero, leading to a simultaneous solution of given linear equations. The solution's behavior near these points provides insight into whether the equilibrium is stable, unstable, or asymptotically stable. Essentially, equilibrium points act as anchors or positions around which the dynamics of the system occur. Understanding these points allows for comprehensive stability analysis, which informs us about both immediate reactions to changes and long-term behavior.