Question

Study the stability of the equilibrium pasitions for the following equations: 1) \(\dot{x}=0\); 3) \(\left\\{\begin{array}{c}\dot{x}_{1}=x_{2}, \\\ \dot{x}_{2}=-x_{1}\end{array}\right.\) 4) \(\left\\{\begin{array}{l}\dot{x}_{1}=x_{1} ; \\ \dot{x}_{2}=-x_{2} ;\end{array}\right.\) 5) \(\left\\{\begin{array}{c}\dot{x}_{1}=x_{2}, \\ \dot{x}_{2}=-\sin x_{1}\end{array}\right.\).

Short answer

Answer: The equilibrium position for Equation 4 is unstable, as it is a saddle point.

Step-by-step solution

(Find the Equilibrium Positions)

Since \(\dot{x}=0\), there are no dynamics in this system, and the equilibrium positions occur for any \(x\). #Step 2: Determine the Stability of Equilibrium Positions#

(Stability of Equilibrium Positions)

There are no dynamics and no changes in the system, making any x-value an equilibrium position with stability. --- #Equation 3# 3) \(\left\\{\begin{array}{c}\dot{x}_{1}=x_{2}, \\\ \dot{x}_{2}=-x_{1}\end{array}\right.\) #Step 1: Identify the Equilibrium Positions#

(Find the Equilibrium Positions)

The equilibrium positions occur when \(\dot{x}_{1} = 0\) and \(\dot{x}_{2} = 0\). This is the case when \(x_1 = 0\) and \(x_2 = 0\). #Step 2: Determine the Stability of Equilibrium Positions#

(Stability of Equilibrium Positions)

The Jacobian matrix is: \(J=\begin{bmatrix}0 & 1 \\\ -1 & 0\end{bmatrix}\) The eigenvalues of the Jacobian matrix are \(\lambda = \pm i\). Since the real parts of the eigenvalues are zero, this equilibrium position is a center, making it neutrally stable. --- #Equation 4# 4) \(\left\\{\begin{array}{l}\dot{x}_{1}=x_{1} ; \\\ \dot{x}_{2}=-x_{2};\end{array}\right.\) #Step 1: Identify the Equilibrium Positions#

(Find the Equilibrium Positions)

The equilibrium positions occur when \(\dot{x}_{1} = 0\) and \(\dot{x}_{2} = 0\). This is the case when \(x_1 = 0\) and \(x_2 = 0\). #Step 2: Determine the Stability of Equilibrium Positions#

(Stability of Equilibrium Positions)

The Jacobian matrix is: \(J=\begin{bmatrix}1 & 0 \\\ 0 & -1\end{bmatrix}\) The eigenvalues of the Jacobian matrix are \(\lambda_1 = 1\) and \(\lambda_2 = -1\). Since the eigenvalues have different signs, this equilibrium position is a saddle point, which implies instability. --- #Equation 5# 5) \(\left\\{\begin{array}{c}\dot{x}_{1}=x_{2}, \\\ \dot{x}_{2}=-\sin{x_{1}}\end{array}\right.\). #Step 1: Identify the Equilibrium Positions#

(Find the Equilibrium Positions)

The equilibrium positions occur when \(\dot{x}_{1} = 0\) and \(\dot{x}_{2} = 0\). This is the case when \(x_1 = n\pi\) (for integer values of n) and \(x_2 = 0\). #Step 2: Determine the Stability of Equilibrium Positions#

(Stability of Equilibrium Positions)

To study the stability of those equilibrium positions, we linearize the system around each equilibrium point by computing the Jacobian matrix: \(J(x_1, x_2) = \begin{bmatrix}\frac{\partial \dot{x}_1}{\partial x_1} & \frac{\partial \dot{x}_1}{\partial x_2} \\\ \frac{\partial \dot{x}_2}{\partial x_1} & \frac{\partial \dot{x}_2}{\partial x_2}\end{bmatrix} = \begin{bmatrix}0 & 1 \\\ -\cos{(x_1)} & 0\end{bmatrix}\) For \(x_1 = n\pi\), the Jacobian matrix is given by: \(J(n\pi, 0) = \begin{bmatrix}0 & 1 \\\ (-1)^n & 0\end{bmatrix}\) The eigenvalues of \(J(n\pi, 0)\) are \(\lambda = \pm\sqrt{(-1)^n}\). The equilibrium points for even values of n (when \(-1)^n = 1\)) are stable because the eigenvalues are purely imaginary with zero real parts, leading to centers. For odd values of n (when \(-1)^n = -1\)), the eigenvalues are real and have opposite signs, making the equilibrium points unstable saddle points.

Key concepts

Ordinary Differential Equations

Ordinary differential equations (ODEs) are fundamental in the study of mathematics, physics, and engineering. These equations relate a function with its derivatives, expressing the change in a system. They provide a description of the dynamics of a system, where the rate of change of a variable is a function of the variable itself or other variables. For example, the simple equation \( \dot{x} = 0 \) is an ODE representing a system with no change over time—meaning that for any value of \( x \) the system is in equilibrium.

In real-world situations, ODEs can get quite complex, with equations such as \( \dot{x}_1 = x_2 \) and \( \dot{x}_2 = -\sin{x_1} \) from the original exercise, which represent nonlinear dynamics. These equations are often solved to determine the state of a system at a future time given an initial condition, and to analyze stability of its equilibrium positions. Understanding the solutions to these ODEs and their behavior is crucial for designing stable systems in fields like control engineering and dynamical systems analysis.

Jacobian Matrix

The Jacobian matrix is a powerful tool in analyzing the behavior of multivariable systems, often used in the context of ODEs. It consists of first-order partial derivatives of a vector-valued function with respect to another vector. Essentially, when evaluating the stability of a system of equations, the Jacobian provides information on how small changes in the state variables affect the rates of change.

For instance, in the case of \( \dot{x}_1 = x_1 \) and \( \dot{x}_2 = -x_2 \), the Jacobian matrix (\( J = \begin{bmatrix}1 & 0 \ 0 & -1\end{bmatrix} \)) helps us to see how small perturbations near the equilibrium point will evolve. A positive entry on the diagonal tells us that a deviation in that direction grows with time, indicating instability, while a negative entry implies that the system will return to equilibrium, suggesting stability. The Jacobian is thus essential in determining the nature of equilibrium points and predicting system behavior near these points.

Eigenvalues of a Matrix

Eigenvalues are a set of scalars associated with a linear system of equations, which provide significant insights into the matrix's properties and the system it describes. In the stability analysis of equilibrium points, the eigenvalues of the Jacobian matrix are of particular importance. They tell us how a system responds to small disturbances around an equilibrium.

For example, if the eigenvalues of the Jacobian matrix are all negative real numbers, the equilibrium point is stable; any small perturbation will decay over time, and the system will return to its equilibrium state. Conversely, if any eigenvalue has a positive real part, it indicates that the system may respond to small disturbances by moving away from the equilibrium point—suggesting instability, as in the case with \( \lambda_1 = 1 \) for the system \( \dot{x}_1 = x_1 \).

In the case of pure imaginary eigenvalues, such as \( \lambda = \pm i \) found in one step of the solution, the system doesn't move towards or away from the equilibrium, but orbits it in a periodic motion, hence, it is neutrally stable. This nuanced understanding of the eigenvalues is crucial for predicting how a system might behave over time and is extensively used in the analysis and design of stable systems in various engineering fields.