Question

Using Theorem \(9.7 .2,\) show that the linear autonomous system $$ d x / d t=a_{11} x+a_{12} y, \quad d y / d t=a_{21} x+a_{22} y $$ does not have a periodic solution (other than \(x=0, y=0\) ) if \(a_{11}+a_{22} \neq 0\)

Short answer

Question: Prove that the linear autonomous system, given by: $$ \frac{dx}{dt} = a_{11}x + a_{12}y, \quad \frac{dy}{dt} = a_{21}x + a_{22}y, $$ has no periodic solution other than \(x=0, y=0\) if \(a_{11} + a_{22} \neq 0\) using Theorem 9.7.2. Answer: Since \(a_{11} + a_{22} \neq 0\), it implies that \(Tr(A) \neq 0\). The system does not satisfy both conditions of Theorem 9.7.2, which means it does not have periodic solutions other than the equilibrium point \(x=0, y=0\).

Step-by-step solution

Write down Theorem 9.7.2

Theorem 9.7.2 states that: For a system of the form $$ \dot{x} = Ax, $$ where \(A\) is a constant matrix, the solutions are either equilibrium points or periodic orbits if and only if \(Tr(A) = 0\) and \(det(A) > 0\). Here, \(Tr(A)\) is the trace of matrix A, which is the sum of its diagonal elements, and \(det(A)\) is the determinant of matrix A.

Compute the trace and determinant

In our given system, the matrix \(A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\), so we have: $$ Tr(A) = a_{11} + a_{22}, $$ and $$ det(A) = a_{11}a_{22} - a_{12}a_{21}. $$

Apply the conditions from Theorem 9.7.2

According to Theorem 9.7.2, the system has periodic solutions if and only if \(Tr(A) = 0\) and \(det(A) > 0\). Our task is to show that the system does not have periodic solutions other than \(x=0, y=0\) if \(a_{11} + a_{22} \neq 0\). Since we are given that \(a_{11} + a_{22} \neq 0\), it implies that \(Tr(A) \neq 0\). Therefore, the system does not satisfy both conditions of Theorem 9.7.2, which means it does not have periodic solutions other than the equilibrium point \(x=0, y=0\).

Key concepts

Periodic Solution in Autonomous Systems

In the study of differential equations, particularly in the field of dynamical systems, a periodic solution refers to a solution that repeats itself after a fixed period. That is, for the solution \( x(t), y(t) \) to a system of equations, it is periodic if there exists a positive number \( T \) such that \( x(t + T) = x(t) \) and \( y(t + T) = y(t) \) for all values of \( t \) within the domain of the solution.

To reveal whether a linear autonomous system exhibits periodic behavior, we look to specific theorems such as the one mentioned in the exercise. In our case, Theorem 9.7.2 offers insight into the conditions necessary for periodic solutions. It states that for a constant matrix \( A \) governing the system, periodic orbits are possible if and only if the trace of \( A \) is zero and its determinant is positive. This means that the tragectory in the phase plane is bound to form a closed loop, thus representing a periodic nature of solutions. Applying this to the given autonomous system, we conclude that the nonzero trace excludes the possibility of periodic solutions (apart from the trivial equilibrium point), ensuring that students grasp the pivotal role of matrix properties in determining system behavior.

Trace of a Matrix

When analyzing linear systems, understanding the trace of a matrix is crucial. The trace, denoted \( Tr(A) \) for a matrix \( A \), is the sum of its diagonal entries. For a 2x2 matrix \( A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \) as in our exercise, the trace is simply \( a_{11} + a_{22} \).

What does the trace tell us? In the context of linear autonomous systems, the trace can determine stability and behavior of solutions. If the trace is zero, and when combined with a positive determinant, it suggests the possibility of periodic solutions. However, a nonzero trace, specifically if it is also positive, can be an indicator of diverging solutions – solutions that grow indefinitely over time. Therefore, through calculating the trace, students gain valuable insights into the behavior of solutions without having to solve the entire system.

Determinant of a Matrix

The determinant of a matrix, represented as \( det(A) \) for a matrix \( A \), is a scalar value that provides a lot of information about the matrix. In our 2x2 matrix \( A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \) scenario, the determinant is \( a_{11}a_{22} - a_{12}a_{21} \).

What role does the determinant play in autonomous systems? Apart from being involved in matrix invertibility, the determinant is critical for understanding system dynamics. A positive determinant implies that the system preserves the orientation of vectors, while a negative determinant implies a flip in orientation. Additionally, as per Theorem 9.7.2, a positive determinant (along with a zero trace) is essential for the existence of periodic solutions. Therefore, the determinant is not merely a calculation—it is a window into the qualitative properties of the system itself, reinforcing the nuances of matrix algebra that students must master in analyzing dynamical systems.