Question

Prove that (a) If all eigenvalues of \(A\) have negative real part, every solution of \(Y^{\prime}=A Y\) approaches zero as \(t\) tends to infinity. (b) If some eigenvalue of \(A\) has positive real part, \(Y^{\prime}=A Y\) has an unbounded solution for all \(t \geqslant 0\). (c) If all eigenvalues of \(A\) have negative and zero real parts, \(Y^{\prime}=A Y\) has a bounded solution for all \(t \geqslant 0\).

Short answer

Answer: If all eigenvalues of A have negative real parts, every solution of \(Y' = AY\) approaches zero as \(t\) tends to infinity. If some eigenvalues have positive real parts, the system has an unbounded solution for all \(t \geq 0\). If all eigenvalues have negative or zero real parts, the system has a bounded solution for all \(t \geq 0\).

Step-by-step solution

Case (a): All eigenvalues have negative real parts

For this case, let's first recall that the general solution for a system of linear differential equations is given by \(Y(t) = c_1 e^{\lambda_1 t} v_1 + c_2 e^{\lambda_2 t} v_2 + \cdots + c_n e^{\lambda_n t} v_n\) where \(\lambda_i\) are the eigenvalues of matrix A and \(v_i\) are the corresponding eigenvectors. If all eigenvalues have negative real parts, then we can write \(\lambda_i = -\alpha_i + i\beta_i\), where \(\alpha_i > 0\). The general solution becomes: \(Y(t) = \sum_{i=1}^n c_i e^{(-\alpha_i + i\beta_i) t}v_i\). As \(t\) tends to infinity, \(e^{-\alpha_i t}\) tends to zero for each term, since all the eigenvalues have negative real parts. Consequently, every solution of \(Y' = AY\) approaches zero as \(t\) tends to infinity.

Case (b): Some eigenvalues have positive real parts

For this case, suppose \(A\) has an eigenvalue \(\lambda_j\) with a positive real part: \(\lambda_j = \alpha_j + i\beta_j\), where \(\alpha_j > 0\). The corresponding term in the general solution will be \(Y_j(t) = c_j e^{(\alpha_j + i\beta_j) t}v_j\). As \(t\) tends to infinity, the exponential term \(e^{\alpha_j t}\) will increase without bound since \(\alpha_j > 0\). This means that \(Y_j(t)\) is unbounded, and since it is part of the general solution \(Y(t)\), it implies that \(Y' = AY\) has an unbounded solution for all \(t \geq 0\).

Case (c): All eigenvalues have negative or zero real parts

For this case, all eigenvalues have real parts that are either negative or zero. Given the general solution \(Y(t) = c_1 e^{\lambda_1 t} v_1 + c_2 e^{\lambda_2 t} v_2 + \cdots + c_n e^{\lambda_n t} v_n\), we can label the eigenvalues as follows: - For negative real parts: \(\lambda_i = -\alpha_i + i\beta_i\), where \(\alpha_i > 0\). - For zero real parts: \(\lambda_i = i\beta_i\). For the terms with negative real parts, the exponential function \(e^{-\alpha_i t}\) tends to zero as \(t\) tends to infinity, making those terms bounded. For the terms with zero real parts, the exponential function becomes \(e^{i\beta_i t}\). Applying Euler's formula, \(e^{i\beta_i t} = \cos(\beta_i t) + i\sin(\beta_i t)\). Both the cosine and sine functions are bounded between -1 and 1, so the terms with zero real parts are also bounded. Since all terms in the general solution are bounded, the system \(Y' = AY\) has a bounded solution for all \(t \geq 0\).

Key concepts

Eigenvalues

In linear algebra, eigenvalues are a fundamental concept. When dealing with matrices, eigenvalues help us understand a lot about the matrix's properties. Given a square matrix \(A\), an eigenvalue \(\lambda\) is a special number such that when it is subtracted from the matrix \(AI\) (where \(I\) is the identity matrix), the matrix becomes non-invertible. Basically, if there exists a vector \(v\) (not equal to the zero vector) such that \(Av = \lambda v\), then \(\lambda\) is an eigenvalue of \(A\). Each eigenvalue is associated with an eigenvector \(v\).

• To find eigenvalues, solve the characteristic equation, \(\det(A - \lambda I) = 0\).
• Eigenvalues can be real or complex numbers.
• They provide insights into the behavior of matrix equations.

Eigenvalues with negative real parts suggest that related systems of equations will "die out" or stabilize over time, as solutions approach zero. Conversely, positive real parts imply solutions will grow unbounded, and zero real parts indicate cyclical behavior.

Linear Differential Equations

Linear differential equations play an essential role in understanding dynamic systems. These are equations where the function and its derivatives appear linearly. A standard form for such an equation is \(Y^{\prime} = AY\), where \(A\) is a matrix and \(Y\) is a vector function.

The solutions to linear differential equations can be found by diagonalizing the matrix \(A\). This often leads us to find the eigenvalues and eigenvectors of \(A\):

• The eigenvalues tell us how each component of the vector \(Y\) behaves over time.
• Eigenvectors provide us the directions in which these behaviors occur.

The sum of terms involving exponential functions usually describes the solution, each exponentially influenced by the respective eigenvalues. If after evaluating, these exponentials shrink in magnitude over time, it hints at stability. If some grow infinitely, it reflects instability or unbounded behavior.

Bounded Solutions

What makes a solution bounded? In simple terms, a solution to a differential equation is characterized as bounded if it doesn't grow without limit. Specifically, it remains within a fixed range for all time \(t\geq 0\).

Boundedness in the context of differential equations, particularly those expressed as \(Y^{\prime} = AY\), is tied closely to the eigenvalues of \(A\):

• If all eigenvalues have negative or zero real parts, the solution will be bounded. The negative drives the terms to zero, while zero corresponds to bounded oscillations.
• If any eigenvalue has a positive real part, that corresponding term must grow without bound, thus leading to an unbounded overall solution.

Understanding the condition for bounded solutions helps us predict if a system will stabilize (stay within limits) or behave erratically with time. It's crucial in fields where predicting long-term behavior accurately matters, such as control systems and ecology.