Problem 22 - Advanced Engineering Mathematics

Question

Letting \(y=x^{\prime}\) we obtain the plane autonomous system $$\begin{array}{l} x^{\prime}=y \\ y^{\prime}=-8 x+6 x^{3}-x^{5} \end{array}$$ Solving \(x^{5}-6 x^{3}+8 x=x\left(x^{2}-4\right)\left(x^{2}-2\right)=0\) we see that critical points \(\operatorname{are}(0,0),(0,-2),(0,2),(0,-\sqrt{2}),\) and \((0, \sqrt{2}) .\) The Jacobian matrix is $$\mathbf{g}^{\prime}(\mathbf{X})=\left(\begin{array}{cc} 0 & 1 \\ -8+18 x^{2}-5 x^{4} & 0 \end{array}\right)$$ and we see that \(\operatorname{det}\left(\mathbf{g}^{\prime}(\mathbf{X})\right)=5 x^{4}-18 x^{2}+8\) and the trace of \(\mathbf{g}^{\prime}(\mathbf{X})\) is 0. since det \(\left(g^{\prime}((\pm \sqrt{2}, 0))\right)=-8<0,(\pm \sqrt{2}, 0)\) are saddle points. For the other critical points the determinant is positive and linearization discloses no information. The graph of the phase plane suggests that (0,0) and (±2,0) are centers.

Step-by-Step Solution

Answer

(\pm \sqrt{2}, 0) are saddle points; (0,0) and (±2,0) are centers.

Step 1: Identify the Autonomous System

The given autonomous system is:\[\begin{array}{l}x' = y \y' = -8x + 6x^3 - x^5\end{array}\]This is a system of differential equations where we have expressed the second derivative of \(x\) in terms of \(x\) and \(y\). The goal is to find the critical points and analyze their stability using the Jacobian matrix.

Step 2: Find Critical Points

Critical points occur where both derivatives are equal to zero:1. Set \(x' = y = 0\)2. Set \(y' = -8x + 6x^3 - x^5 = 0\)Solving the second equation, we factorize:\[x(-8 + 6x^2 - x^4) = 0\]This gives us critical points: \((0, 0), (0, -2), (0, 2), (0, -\sqrt{2}), (0, \sqrt{2})\).

Step 3: Compute the Jacobian Matrix

The Jacobian matrix \( \mathbf{g}^{\prime}(\mathbf{X}) \) is given by:\[\mathbf{g}^{\prime}( extbf{X}) = \begin{pmatrix}0 & 1 \-8 + 18x^2 - 5x^4 & 0\end{pmatrix}\]This matrix helps us determine the type and stability of the critical points by evaluating the determinant and trace.

Step 4: Evaluate Determinant and Trace

The determinant of the Jacobian matrix is:\[\operatorname{det}(\mathbf{g}^{\prime}(\mathbf{X})) = 5x^4 - 18x^2 + 8\]And the trace is:\[\operatorname{trace}(\mathbf{g}^{\prime}(\mathbf{X})) = 0\]The sign of the determinant, along with the trace, provides insight into the nature of the critical points.

Step 5: Analyze Stability of Critical Points

For positive determinant values, the linearization method is inconclusive with 0 trace. For \((\pm \sqrt{2}, 0)\), the determinant is negative (det = -8), indicating these are saddle points. The phase diagram suggests that (0,0) and (±2,0) behave as centers based on their positive determinant values and 0 trace.

Key Concepts

Critical Points

Critical points play a fundamental role in the analysis of autonomous systems. They are the solutions where the derivatives of the system are zero, indicating a potential equilibrium state. In the given system:

• We find critical points by setting the equations for the derivatives of both variables, \( x' = y \) and \( y' = -8x + 6x^3 - x^5 = 0 \), equal to zero.
• By solving these equations, we identify the critical points: \((0, 0)\), \((0, -2)\), \((0, 2)\), \((0, -\sqrt{2})\), and \((0, \sqrt{2})\).

These points indicate where the system may reach a steady state, making them crucial for further stability analysis. By understanding critical points, we can explore the behavior of the system near these equilibrium states.

Jacobian Matrix

The Jacobian matrix is a mathematical tool that represents the linear approximation of a system of differential equations at a given point. For the autonomous system described, the Jacobian matrix is:

• \[ \mathbf{g}^{\prime}(\mathbf{X}) = \begin{pmatrix}0 & 1 \-8 + 18x^2 - 5x^4 & 0\end{pmatrix}\]
• This matrix is constructed from the partial derivatives of the system's equations.
• It helps in determining the nature of the critical points by evaluating the determinant and trace of the matrix.

The use of the Jacobian in stability analysis is crucial because it provides a way to assess how small perturbations around the critical points affect the system's behavior.

Stability Analysis

Stability analysis uses the Jacobian matrix to determine the stability of critical points. The process involves examining the determinant and trace of the Jacobian at these points:

• The determinant \( \operatorname{det}(\mathbf{g}^{\prime}(\mathbf{X})) = 5x^4 - 18x^2 + 8 \) is key to assessing the behavior near each critical point.
• A positive determinant typically indicates potential stability, while a negative determinant suggests instability.
• For the given system, the negative determinant at \((\pm \sqrt{2}, 0)\) signifies saddle points, which are unstable.
• Critical points with positive determinant and zero trace are centers, suggesting neutral stability in its immediate vicinity.

Stability analysis reveals which equilibria are stable and predicts the system's response to small disturbances.

Phase Plane Analysis

Phase plane analysis is a graphical method to visualize the trajectories of a system of differential equations. By plotting the system's vector field and analyzing the critical points on a phase plane diagram, we gain insights into the system behavior over time:

• The phase plane consists of all possible states of the system plotted against each other (here, \(x\) vs. \(y\)).
• Trajectories in the phase plane reveal the paths the system can take.
• Centers are seen as closed loop trajectories, which is typical of oscillatory behavior around \((0,0)\) and \((\pm 2,0)\) in this system.
• Saddle points are illustrated by trajectories that diverge, consistent with instability found in stability analysis.

Phase plane analysis complements algebraic stability results, providing a comprehensive view of the system dynamics and how states evolve over time.