Question

Describe and graph trajectories of the given system. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rr} -7 & 1 \\ 3 & -5 \end{array}\right] \mathbf{y} $$

Short answer

Answer: The eigenvalues of the matrix A are -4 and -8. 2. What are the eigenvectors corresponding to these eigenvalues? Answer: The corresponding eigenvectors are \(\mathbf{v}_1 = \left[\begin{array}{c} 1\\ 3 \end{array}\right]\) for eigenvalue -4, and \(\mathbf{v}_2 = \left[\begin{array}{c} 1\\ -1 \end{array}\right]\) for eigenvalue -8. 3. What type of trajectories does the system have based on the eigenvalues found? Answer: Since both eigenvalues are negative, the system has stable trajectories, meaning all trajectories approach the origin as time increases. 4. What do the eigenvectors represent on the phase portrait of the given system? Answer: The eigenvectors serve as the axes, or directions, of the stable system, on which the trajectories approach as time increases.

Step-by-step solution

Find the eigenvalues of the matrix

First, let's find the eigenvalues of the matrix. To do this, we need to solve the characteristic equation given by: \(|A - \lambda I| = 0\), where A is the matrix, \(\lambda\) are the eigenvalues, and I is the identity matrix. For the given matrix A: $$A = \left[\begin{array}{rr} -7 & 1 \\ 3 & -5 \end{array}\right]$$ The characteristic equation is: $$|\left[\begin{array}{cc} -7-\lambda & 1 \\ 3 & -5 - \lambda \end{array}\right]| = (\lambda + 7)(\lambda + 5) - (1)(3) = \lambda^2 + 12\lambda + 32 = 0$$ The eigenvalues are the roots of this quadratic equation.

Solve the Quadratic Equation

Use the quadratic formula to find the eigenvalues: $$\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-12 \pm \sqrt{12^2 - 4(1)(32)}}{2(1)}$$ After solving, we get two eigenvalues: $$\lambda_1 = -4 \quad \text{and} \quad \lambda_2 = -8$$

Find the Eigenvectors

Now that we have the eigenvalues, we need to find their corresponding eigenvectors. We plug each eigenvalue into \((A-\lambda I)\mathbf{v}=\mathbf{0}\), where \(\mathbf{v}\) is the eigenvector and \(\mathbf{0}\) is the zero vector. For \(\lambda_1 = -4\), we have: $$\left[\begin{array}{cc} -3 & 1 \\ 3 & -1 \end{array}\right]\mathbf{v}=\mathbf{0}$$ Row reduce to echelon form, and we get: $$\left[\begin{array}{cc} 1 & -\frac{1}{3} \\ 0 & 0 \end{array}\right]\mathbf{v}=\mathbf{0}$$ So the first eigenvector is: $$\mathbf{v}_1 = \left[\begin{array}{c} 1\\ 3 \end{array}\right]$$ For \(\lambda_2 = -8\), we have: $$\left[\begin{array}{cc} 1 & 1 \\ 3 & 3 \end{array}\right]\mathbf{v}=\mathbf{0}$$ Row reduce to echelon form, and we get: $$\left[\begin{array}{cc} 1 & 1 \\ 0 & 0 \end{array}\right]\mathbf{v}=\mathbf{0}$$ So the second eigenvector is: $$\mathbf{v}_2 = \left[\begin{array}{c} 1\\ -1 \end{array}\right]$$

Sketch the Phase Portrait

With the eigenvalues and eigenvectors, we can now sketch the phase portrait for the given system. Since both eigenvalues are negative, the system is stable (all trajectories approach the origin as time increases). The eigenvectors serve as the axes, or directions, of this stable system. 1. On a Cartesian Plane, draw the origin (0, 0) and label it. 2. Draw the eigenvector \(\mathbf{v}_1\) as an arrow from the origin pointing to (1, 3) and label it as \(\lambda_1 = -4\). 3. Draw the eigenvector \(\mathbf{v}_2\) as an arrow from the origin pointing to (1, -1) and label it as \(\lambda_2 = -8\). 4. Now, sketch the trajectories approaching these eigenvectors (arrows) as time increases. Make sure these trajectories go towards the origin (indicating that they are stable). The resulting graph will be the phase portrait of the given system, showing the trajectories of the system as time progresses.

Key concepts

Eigenvalues and Eigenvectors

When studying the behavior of a system of differential equations, eigenvalues and their associated eigenvectors reveal critical information about the system's long-term behavior. Let's consider a two-dimensional system of linear differential equations, which can typically be represented in matrix form as
\[ \mathbf{y}^{\prime} = A \mathbf{y}, \] where \( A \) is a square matrix. The eigenvalues of \( A \) are the solutions to the characteristic equation
\[ \det(A - \lambda I) = 0, \] where \( I \) denotes the identity matrix and \( \lambda \) represents the eigenvalues.

These eigenvalues can tell us whether the system's solutions grow, decay, or oscillate over time. For instance, negative real parts indicate a decaying behavior and stability in the context of equilibrium.

To find the eigenvectors, we solve
\[ (A - \lambda I)\mathbf{v} = \mathbf{0}. \] The eigenvector \( \mathbf{v} \) associated with each eigenvalue provides the direction in which the system evolves over time. Conceptually, these directions will become the axes of the phase portrait, allowing us to visualize the trajectories of the system in the phase plane. This exercise incorporated solving for and understanding the connections between these eigenvalues and eigenvectors to gain insights into the system's dynamics.

Differential Equations System

In the context of the given problem, the system of differential equations takes the form of a first-order linear system, where the derivative of \( \mathbf{y} \) is given as a product of a constant matrix and the variable vector \( \mathbf{y} \). Such systems are compact and incredibly informative. Through matrix operations, they encapsulate the rates of change for multiple interacting variables within the system.

The solution approach typically requires finding the eigenvalues and eigenvectors of the matrix, which directly relate to the system's characteristic behaviors such as stability, oscillation, or divergence. The system given in the exercise is autonomous, meaning that the matrix \( A \) does not change over time, allowing the use of linear algebra techniques to analyze the system categorically.

The key to understanding these systems lies in the interplay between their algebraic structure and geometric behavior, where solutions can be visualized by trajectories in the phase plane. The dynamical properties of these systems provide fundamental insights in various fields including physics, engineering, and applied mathematics.

Characteristic Equation

The characteristic equation is a central concept in solving systems of differential equations as it provides the means to identify the eigenvalues of the matrix. For a matrix \( A \), the characteristic equation is defined as
\[ \det(A - \lambda I) = 0. \] In the exercise, determining the characteristic equation allows us to find the crucial values that reveal the flow and stability of the system.

Specifically, the determinant results in a polynomial where the eigenvalues \( \lambda \) are the roots. These roots are found using algebraic methods such as factoring or applying the quadratic formula, as was done in the solution. The eigenvalues are not just numerical placeholders; they have profound implications on the system's behavior. They are indispensable in mapping out the solution space and understanding the equilibrium's nature—whether it's stable, unstable, or a saddle point. Each eigenvalue corresponds to an eigenvector, creating a line along which the system evolves, a concept that we use in the next step, trajectories graphing.

Trajectories Graphing

Graphing trajectories is a powerful visual tool for understanding the qualitative behavior of a system of differential equations. Trajectories represent possible paths that the system's state can follow over time in the phase plane. To graph these trajectories, we use the eigenvalues and eigenvectors as a starting point.

Stable and Unstable Manifolds

The trajectories bend towards stable eigenvectors and away from unstable ones. They are straight lines only if they lie directly on top of the eigenvectors.

When sketching the phase portrait, we draw these trajectories approaching or diverging from the equilibrium point (the origin in this case), which represent the long-term behavior of the system.

Interpretation of the Sketch

In our problem, the negative eigenvalues indicate that trajectories will approach the origin, showing a stable system. The eigenvectors provide direction, and drawing trajectories curving towards these lines gives us a visual sketch of the system's stability landscape. Each trajectory corresponds to a different initial condition, mapping out how the system could potentially evolve from various starting states.

This graphical approach to understanding the dynamic behavior of systems is part of the broader field of phase plane analysis and contributes significant insight into diverse scientific and engineering disciplines.