Question

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

Short answer

In the given 2x2 matrix representing a linear system of first-order differential equations, the eigenvalues are λ1 = 1√2 and λ2 = -1√2, with their corresponding eigenvectors v1 = (1, 1 - 1√2) and v2 = (1, 1 + 1√2). Since both eigenvalues have different signs, the system has a saddle point. Trajectories will move away from the origin in the direction of eigenvector v1 and approach the origin in the direction of eigenvector v2. Graphing the trajectories involves plotting the eigenvectors and the saddle point at the origin (0, 0).

Step-by-step solution

Find the eigenvalues of the matrix

To find the eigenvalues of the matrix, we must solve the characteristic equation, which is given by the determinant of the matrix minus the λ times the identity matrix: $$ \text{det}\begin{pmatrix} 1-\lambda & 1 \\ 1 & -1-\lambda \end{pmatrix}= (1-\lambda)(-1-\lambda) - 1 \cdot 1 $$ Now calculate the determinant: $$ \lambda^2 - 2 = 0 $$ Using the quadratic formula, we get the eigenvalues: $$ \lambda_1 = 1 \sqrt{2}, \lambda_2 = -1 \sqrt{2} $$

Find the eigenvectors corresponding to the eigenvalues

To find each eigenvector, plug the eigenvalues back into the equation \((A - \lambda I)v = 0\). Start with \(\lambda_1 = 1 \sqrt{2}\): $$ \begin{pmatrix} 1 - 1 \sqrt{2} & 1 \\ 1 & -1 - 1 \sqrt{2} \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$ We obtain the following system of equations: $$ (1 - 1 \sqrt{2})x_1 + x_2 = 0 \\ x_1 + (-1 - 1 \sqrt{2})x_2 = 0 $$ Simplify the system of equations and find the eigenvector corresponding to \(\lambda_1\): $$ \begin{pmatrix} 1 \\ 1 - 1\sqrt{2} \end{pmatrix} $$ Now, find the eigenvector for \(\lambda_2 = -1\sqrt{2}\) similarly: $$ \begin{pmatrix} 1 \\ 1 + 1\sqrt{2} \end{pmatrix} $$

Analyze the eigenvectors and eigenvalues

The eigenvalues \(\lambda_1 = 1\sqrt{2}\) and \(\lambda_2 = -1\sqrt{2}\) determine the behavior of the system. Since both eigenvalues are real and have different signs, the system has a saddle point, which means trajectories will either approach the origin along one eigenvector or move away from the origin along the other eigenvector. The eigenvectors are given by: $$ v_1 = \begin{pmatrix} 1 \\ 1 - 1\sqrt{2} \end{pmatrix}, \quad v_2 = \begin{pmatrix} 1 \\ 1 + 1\sqrt{2} \end{pmatrix} $$ These eigenvectors give us the direction and shape of the trajectories.

Graph the trajectories

To graph the trajectories, start by plotting the eigenvectors \(v_1\) and \(v_2\). Since the eigenvalue associated with \(v_1\) is positive (\(1\sqrt{2}\)), trajectories will move away from the origin in the direction of \(v_1\). Similarly, as the eigenvalue associated with \(v_2\) is negative (-\(1\sqrt{2}\)), trajectories will approach the origin in the direction of \(v_2\). Graph the saddle point at the origin (0, 0), trajectories moving away from the origin in the direction of eigenvector \(v_1\), and trajectories approaching the origin in the direction of eigenvector \(v_2\).

Key concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are foundational concepts in linear algebra and are particularly critical in analyzing the behavior of dynamical systems described by differential equations. An eigenvalue, typically denoted by \( \lambda \), represents the scalar value for which there exists a non-zero vector (the eigenvector), which does not change its direction when a linear transformation is applied to it.

This transformation is associated with a matrix, often representing the coefficients of a system of linear differential equations. To find an eigenvalue, we look for scalars \( \lambda \) such that the equation \( (A - \lambda I)v = 0 \) holds true, where \( A \) is the linear transformation matrix, \( I \) is the identity matrix, and \( v \) is the potential eigenvector.

The eigenvectors then correspond to specific directions in the space which are invariant under the transformation \( A \). These vectors affirm the tendency of trajectories in a dynamic system to orient along certain lines as they evolve over time. In the example given, the eigenvectors provide the direction along which the system either converges to or diverges from the equilibrium.

Characteristic Equation

The characteristic equation is the tool we use to determine the eigenvalues of a matrix. It is formed by setting the determinant of \( A - \lambda I \) equal to zero, where \( A \) is our matrix and \( I \) is the identity matrix. This equation is a polynomial where the roots correspond to the eigenvalues of \( A \).

For instance, with the matrix presented in the exercise, after computing the determinant, we obtain the characteristic equation \( \lambda^2 - 2 = 0 \), which upon solving gives us the eigenvalues of the system. In practice, finding the characteristic equation is the cornerstone in solving systems of differential equations since it leapfrogs us to discovering the eigenvalues, which are imperative for understanding system behavior.

Saddle Point

A saddle point in the context of differential equations is a type of equilibrium point characterized by having eigenvalues with mixed signs—this is indicative of an unstable equilibrium. In the phase plane, a saddle point is depicted as the point where trajectories cross before diverging along one eigenvector or converging along another.

From the eigenvalues obtained, \( \lambda_1 = \sqrt{2} \) and \( \lambda_2 = -\sqrt{2} \), we can interpret that the system's equilibrium at the origin is a saddle point. Trajectories are attracted along the direction of the eigenvector associated with the negative eigenvalue and repelled along the direction corresponding to the positive eigenvalue. A saddle point often signifies a delicate balance, with behavior drastically changing depending on initial conditions.

Phase Plane Analysis

In phase plane analysis, we examine the behavior of a two-variable dynamical system by graphing its trajectories in a plane where each axis represents one of the variables. It enables us to visualize how the system evolves over time and identify patterns such as fixed points, limit cycles, and other kinds of attractors.

In the given system, phase plane analysis entails plotting the trajectories dictated by the eigenvectors, taking into account the influence of their respective eigenvalues. We find that because we have a saddle point, the graph will showcase trajectories emanating from or spiraling into the point where the eigenvectors intersect, giving us an intuitive grasp of the potential long-term behavior of the system. Overall, the phase plane is an insightful tool that simplifies complex dynamical behavior into a comprehensible visual format.