Question

A wide variety of phenomena can occur when an equilibrium point is completely degenerate, i.e., when the Jacobian is the zero matrix. We will look at just one. Consider the system $$ \begin{aligned} &x^{\prime}=x y \\ &y^{\prime}=x^{2}-y^{2} \end{aligned} $$ a) Show that the Jacobian at the origin is the zero matrix. b) Plot the solutions through the six points \((0, \pm 1)\), and \((\pm \sqrt{2}, \pm 1)\). Plot additional solutions of your choice. c) Compare what you see with the behavior of solutions near a saddle point.

Short answer

a) Jacobian at origin is zero matrix. b) Plot solutions from given points. c) Compare behavior to a saddle point, noting differences due to degeneracy.

Step-by-step solution

Define the System and Equilibrium Point

The given system of differential equations is \(x' = x y\) and \(y' = x^2 - y^2\). The equilibrium point given is the origin \((0,0)\). At equilibrium points, the derivatives are zero.

Find the Jacobian Matrix

The Jacobian matrix \(J(x,y)\) of the system is formed by taking partial derivatives of each equation with respect to \(x\) and \(y\). For \(x' = xy\), \(\frac{\partial}{\partial x}(xy) = y\) and \(\frac{\partial}{\partial y}(xy) = x\). For \(y' = x^2 - y^2\), \(\frac{\partial}{\partial x}(x^2 - y^2) = 2x\) and \(\frac{\partial}{\partial y}(x^2 - y^2) = -2y\). Thus, the Jacobian matrix is \(\begin{pmatrix} y & x \ 2x & -2y \end{pmatrix}\).

Evaluate the Jacobian at the Equilibrium Point

Substitute \(x = 0\) and \(y = 0\) into the Jacobian matrix \(\begin{pmatrix} y & x \ 2x & -2y \end{pmatrix}\) to get \(\begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix}\). This confirms the Jacobian at the origin is a zero matrix.

Plot the Solutions Through Given Points

The given points are \((0, \pm 1)\) and \((\pm \sqrt{2}, \pm 1)\). Plot the trajectories originating from these points using a direction field or by numerical simulation of the system. Observe how each trajectory behaves in proximity to these initial points.

Examine Additional Solution Plots

Plot additional solutions using other points near the origin or other interesting areas of the phase plane. This can help reveal more about the nature of the system and its behavior around the origin.

Compare with Saddle Point Behavior

A saddle point in a dynamical system is characterized by trajectories being repelled along one direction and attracted along another. Compare the behavior of solutions near the origin in this degenerate system to that of a typical saddle point. Typically, this will mean observing some hyperbolic-like behavior, but perhaps with unique characteristics due to the degeneracy.

Key concepts

Jacobian Matrix

In differential equations, the Jacobian matrix plays a crucial role, especially when examining the stability of equilibrium points. It provides a linear approximation of a dynamical system near an equilibrium. For a given system of equations, the Jacobian matrix is composed of the first derivatives of each equation with respect to each variable.
For the system given:

• The derivative of \(x' = xy\) with respect to \(x\) results in \(y\)
• The derivative with respect to \(y\) results in \(x\)
• For \(y' = x^2 - y^2\), the partial with respect to \(x\) is \(2x\)
• And with respect to \(y\) is \(-2y\)

This leads to the Jacobian matrix:\[J(x, y) = \begin{pmatrix} y & x \ 2x & -2y \end{pmatrix}\]Evaluating the Jacobian matrix at the origin \((0,0)\) yields a zero matrix, indicating the equilibrium point is completely degenerate.
This implies the standard linear analysis might not provide sufficient information on the system's behavior, so further analysis is needed.

Equilibrium Points

Equilibrium points occur where the system's derivatives equal zero, indicating no change at those points. These are crucial in understanding system dynamics because they represent steady-state solutions where the system neither evolves nor declines.
For the given system:

• The equilibrium point is located at the origin \((0,0)\)
• Substituting \((0,0)\) into the equations \(x' = xy\) and \(y' = x^2 - y^2\) confirms zero derivatives

Equilibrium points can be classified into various types based on their stability and how they influence nearby trajectories. Since the Jacobian at this equilibrium is the zero matrix, it tells us that standard techniques may not completely capture the dynamics in its vicinity, suggesting complex behavior at and around the origin.

Saddle Points

Saddle points are a specific type of equilibrium point characterized by particular stability properties. While some directions in the phase space are attractive, others are repulsive. This mixed stability results in a characteristic 'saddle' shape in the phase plane.
In many systems, saddle points can be identified through the eigenvalues of the Jacobian matrix at the point. A saddle point typically has both negative and positive real parts in its eigenvalues, indicating its dual nature of attracting and repelling.
In the given system:

• Since the Jacobian at \((0,0)\) is a zero matrix, traditional eigenvalue analysis fails
• This suggests that while the point might exhibit saddle-like behavior, it might be inherently more complex due to degeneracy

To understand the behavior around this degenerate saddle point, plotting solutions near \((0,0)\) can be insightful, showing trajectories behaving in hyperbolic or even unexpected manners.

Phase Plane Analysis

Phase plane analysis is a visual method for studying dynamical systems. It involves plotting the trajectories on a plane, providing insights into the system's behavior over time.
For our system of equations,

• Each trajectory plotted through points like \((0, \pm 1)\) and \((\pm \sqrt{2}, \pm 1)\) reveals how solutions evolve
• By observing these trajectories, we can infer tendencies such as attraction, repulsion, or cycling behaviors around equilibrium points

This approach is particularly useful here because of the zero Jacobian matrix at the origin. It visually demonstrates complex phenomena that are not evident from linear analysis alone. Observing the phase plane may reveal non-linear dynamics that are characteristic of degenerate systems, which can seem hyperbolic and saddle-like but with distinctive features unique to such systems. Therefore, phase plane analysis is essential for a detailed understanding of this system's intricacies.