Question

Verify that \((0,0)\) is a critical point, show that the system is almost linear, and discuss the type and stability of the critical point \((0,0)\) by examining the corresponding linear system. $$ d x / d t=(1+x) \sin y, \quad d y / d t=1-x-\cos y $$

Short answer

Answer: The critical point (0,0) is a center and its stability is neutral.

Step-by-step solution

Verify the critical point (0, 0)

To find the critical points, set both rates of change equal to 0 and solve for x and y: $$ \begin{aligned} (1+x) \sin y &= 0 \\ 1-x-\cos y &= 0 \end{aligned} $$ In this case, it's easy to see that \((0,0)\) is a critical point, as \(\sin 0 = 0\) and \(1 - 0 - \cos 0 = 1 - 1 = 0\).

Find the Jacobian matrix of the system

The Jacobian matrix of the system of equations is a matrix of partial derivatives, which can be written as: $$ J(x,y) = \begin{bmatrix} \frac{\partial {(1 + x)\sin y}}{\partial x} & \frac{\partial {(1 + x)\sin y}}{\partial y} \\ \frac{\partial {(1 - x - \cos y)}}{\partial x} & \frac{\partial {(1 - x - \cos y)}}{\partial y} \end{bmatrix} $$ Compute the partial derivatives: $$ J(x,y) = \begin{bmatrix} \sin y & (1+x)\cos y \\ -1 & \sin y \end{bmatrix} $$

Evaluate the Jacobian matrix at the critical point(0, 0)

Substitute the critical point \((0,0)\) into the Jacobian matrix: $$ J(0,0) = \begin{bmatrix} \sin 0 & (1+0)\cos 0 \\ -1 & \sin 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} $$

Find the eigenvalues of the Jacobian matrix at the critical point

Find the eigenvalues of the evaluated Jacobian matrix by first calculating its characteristic polynomial: $$\begin{aligned} \det(J(0,0)-\lambda I) &= \det \begin{bmatrix} -\lambda & 1 \\ -1 & -\lambda \end{bmatrix} = \lambda^2 + 1 \end{aligned} $$ Since the characteristic polynomial is \(\lambda^2 + 1 = 0\), this equation has complex eigenvalues with nonzero imaginary parts, \(\lambda = \pm i\). Since the real parts of the eigenvalues are zero, the critical point \((0,0)\) is a center, which is an almost-linear system.

Determine the type and stability of the critical point

As the eigenvalues are purely imaginary, the critical point \((0,0)\) corresponds to a center. In general, a center is considered neutrally stable, as it is not attracting or repelling. Hence, the type of the critical point \((0,0)\) is a center and the stability of the critical point is neutral.

Key concepts

Jacobian Matrix Differential Equations

The Jacobian matrix is a powerful tool used to analyze the behavior of differential equations near critical points. It consists of first-order partial derivatives of each function in a system with respect to all variables.

In the context of the given system of equations, calculating the Jacobian matrix helps us linearize the system near the critical point \( (0,0) \). When we find this matrix at a critical point, it represents the best linear approximation to the system's dynamics near that point. In our case, the Jacobian matrix evaluated at the critical point \( (0,0) \) turns out to be a rotation matrix. This implies that the system exhibits rotational behavior around the critical point.

Understanding the Jacobian is crucial because it directly influences the determination of the critical point's stability - a key factor in predicting the system's long-term behavior.

Eigenvalues of a Matrix

Eigenvalues are a fundamental concept when assessing the stability of a system. They are values that, when a certain matrix is multiplied by a vector, the vector is only scaled and not rotated. To find eigenvalues, we set up and solve the characteristic equation of the matrix.

In the provided exercise, the eigenvalues of the Jacobian matrix are found by calculating the roots of the characteristic polynomial \( \lambda^2 + 1 \). The roots are complex numbers \( \lambda = \pm i \), suggesting oscillatory motion without growth or decay since the real part of the eigenvalues is zero. These complex eigenvalues are keys to determining that our system's behavior near the critical point is that of a neutral center.

Knowing the eigenvalues helps us understand whether perturbations to the system will damp out, persist undamped, or grow exponentially, which speaks directly to system stability.

Neutral Stability Center

A neutral stability center is a particular type of critical point in a dynamical system. It is characterized by undamped periodic motion. In simpler terms, any small push or pull on the system will lead to it moving in a loop without spiraling inwards or outwards.

In our exercise, the term 'neutral stability' is derived from the purely imaginary eigenvalues of the Jacobian matrix at the critical point. This type of critical point neither attracts nearby points (like a spiral) nor repels them. Instead, trajectories are confined to close curves around the critical point, indicating a pattern of constant amplitude oscillations. A good real-world analogy would be a marble freely rotating inside a smooth bowl; it continues circling around the bottom without settling or escaping.

Navigating through these concepts can be tough, but unpacking the role of the Jacobian matrix, eigenvalues, and the type of critical point can provide a significant leap in understanding the stability of systems governed by differential equations.