Question

(a) Find the eigenvalues and eigenvectors. (b) Classify the critical point \((0,0)\) as to type and determine whether it is stable, asymptotically stable, or unstable. (c) Sketch several trajectories in the phase plane and also sketch some typical graphs of \(x_{1}\) versus \(t .\) (d) Use a computer to plot accurately the curves requested in part (c). \(\frac{d \mathbf{x}}{d t}=\left(\begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right) \mathbf{x}\)

Short answer

Question: Analyze the behavior of the linear system of differential equations with the matrix A = \(\begin{pmatrix}-1 & 0 \\ 0 & -1 \end{pmatrix}\). Find the eigenvalues and eigenvectors, classify the critical point (0,0), sketch several trajectories in the phase plane, and plot the curves accurately.

Step-by-step solution

Find the eigenvalues and eigenvectors

To find the eigenvalues, we need to solve the characteristic equation \(|A-\lambda I| = 0\). In the case of our matrix \(A = \begin{pmatrix}-1 & 0 \\ 0 & -1 \end{pmatrix}\), the characteristic equation is: \(|A-\lambda I| = \left|\begin{array}{cc}{-1-\lambda} & {0} \\\ {0} & {-1-\lambda}\end{array}\right| = (-1-\lambda)^2.\) Solving for \(\lambda\) gives us eigenvalue \(\lambda = -1\) with multiplicity 2. Now, we need to find the eigenvectors associated with this eigenvalue. In order to do this, we will solve the equation \((A-\lambda I) \mathbf{x} = 0\): \((A-\lambda I) \mathbf{x} = \left(\begin{array}{cc}{0} & {0} \\\ {0} & {0}\end{array}\right) \mathbf{x} = 0.\) Any non-trivial solutions to this equation will be eigenvectors of \(A\). Using the equation \(x_{1}=\alpha x_{2}\) for some constant \(\alpha\), we can find the eigenline spanned by eigenvectors: \(\mathbf{x} = \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} = x_{2} \begin{pmatrix} \alpha \\ 1 \end{pmatrix}\), for any constant \(\alpha\). So, there is an infinite number of linearly dependent eigenvectors associated with the eigenvalue \(\lambda = -1\).

Classify the critical point \((0,0)\) and determine its stability

Since our eigenvalue is real and negative \(\lambda = -1\), the critical point \((0,0)\) is a stable node. Furthermore, in this case, it is asymptotically stable, as the solutions converge towards \((0,0)\) exponentially.

Sketch several trajectories in the phase plane and some typical graphs of \(x_1\) versus \(t\).

As we have an asymptotically stable node, all trajectories will converge to \((0,0)\). The eigenvectors form an infinite family of lines in the phase plane that approaches the origin. Some trajectories can be represented by multiplying the eigenvectors by an exponential term \(e^{\lambda t}\), where \(\lambda = -1\). For example, for the eigenvector \(\begin{pmatrix} 1 \\ 1 \end{pmatrix}\), the trajectory will be given by: \(x(t) = e^{-1t} \begin{pmatrix} 1 \\ 1 \end{pmatrix}\) As \(t\) increases, the exponential term vanishes, so the trajectories will approach \((0,0)\) exponentially. Typical graphs of \(x_1\) versus \(t\) will be of the form \(x_{1}(t) = c_{1} e^{-t}\), where \(c_1\) is an arbitrary constant. These graphs are exponentials decaying to zero, reflecting the asymptotic stability of the node.

Use a computer to plot accurately the curves requested in part (c).

In this step, you will need software like MATLAB, Python, or any similar software that can display plots of differential equations and functions. Since I can't use external software here, I encourage you to try plotting the trajectories and functions mentioned in step c in your preferred plotting software, taking into account the properties discussed in the previous steps. The results should visually match the conclusions drawn in the previous steps about the critical point's stability, the trajectories' shape in the phase plane, and the decay of the exponential function \(x_1(t)\) with time.

Key concepts

Critical Point Stability

Understanding how a critical point behaves is essential in analyzing the stability in dynamical systems. A **critical point** is where the system experiences no change, often analyzed as \((0,0)\) in linear systems. The stability of this point can tell us if small disturbances will disappear or grow over time. When we find eigenvalues that are both real and negative, like in our matrix example where \(\lambda = -1\), it indicates that this point in the phase plane is a stable node.

Stable nodes imply that solutions nearby will return to the critical point. In our exercise, where the eigenvalues suggest asymptotic stability, any trajectory will eventually fall back to the point \((0,0)\). This stability type is particularly significant because it means that, as time progresses, any slight deviation from the critical point diminishes, making the system predictable and resilient to small perturbations.

Phase Plane Trajectories

The phase plane offers a visual way to understand the system's behavior over time. Here, we plot all possible states of the system, as determined by the variables. In linear systems such as our example, where the matrix has the form \(\begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix}\), **phase plane trajectories** illustrate how solutions evolve.

These trajectories are particularly telling: with real and negative eigenvalues \(\lambda = -1\) here, we see trajectories moving directly towards the critical point. You can think of these as a family of curves or lines all pointing to a single spot, representing how solutions contract or converge over time.

• Each eigenvector gives its direction of "flow" within the plane.
• The trajectories form a harmonious pattern, symmetrically heading towards the origin.

Visualizing this can often clarify complex concepts, providing an intuitive understanding of stability and movement within the system.

Exponential Decay

Exponential decay refers to how solutions in dynamical systems change over time, typically decreasing to zero. With our system, solutions converge towards the origin, flowing along the negative direction of their eigenvectors.

When solutions behave like \(x(t) = e^{-1t} \cdot \mathbf{x}_0\), where \(\mathbf{x}_0\) represents the initial condition, we see **exponential decay** in action. This mathematical relationship highlights how quickly variables reduce over time. The rate is determined by the negative eigenvalues; here, \(\lambda = -1\).

• As \(t\) increases, \(e^{-t}\) becomes very small.
• Solutions diminish exponentially, indicating fast stabilization.
• In practice, it showcases how a system might respond or stabilize over time bonds.

Exponential decay is a fundamental characteristic in physics, engineering, and even biology, where systems return to a stable state after disturbances.

Asymptotic Stability

**Asymptotic stability** is a stronger form of stability relevant in systems where not only do solutions stay bounded, but they also return to a specific state over time. Our linear system provides a clear example of this scenario.

Here, the presence of real and negative eigenvalues ensures any solution trajectory that starts near the critical point \((0,0)\) will eventually meet it as \(t\) goes to infinity. This is what we call **asymptotic stability**.

• If perturbed, the system naturally recovers, with disturbances smoothing out over time.
• This behavior contrasts with just "stable" where solutions remain close but don't necessarily return to a point.

Asymptotic stability is attractive because it gives us confidence in the predictability of a system's future state, suggesting robust behavior and reliability across practical applications.