154k views
5 votes
Provide an example for a linear differential equation x′=Ax with infinitely many fixed points and draw its phase portrait. Hint: What if A has a non-trivial kernel?

User Shangxinbo
by
8.7k points

1 Answer

4 votes

Final answer:

A linear differential equation with infinitely many fixed points occurs when the matrix A has a non-trivial kernel, for example, when A is a zero matrix. In this case, the corresponding differential equation's solution doesn't change over time, implying that each initial condition is a fixed point. The phase portrait consists of flat, horizontal lines for each initial condition, indicating no change over time.

Step-by-step explanation:

When considering a linear differential equation of the form x'=Ax, if the matrix A has a non-trivial kernel, that means there are non-zero vectors belonging to the kernel. In such a case, there are infinitely many fixed points, as each point in the kernel is a fixed point. This occurs when the matrix A is not invertible, or equivalently, when its determinant is zero.

As an example, consider the matrix A being a zero matrix of any size. The corresponding differential equation is x'=0, which implies that the derivative of x with respect to time is zero. This means that any initial condition for x will remain constant for all time, making each initial condition a fixed point. The phase portrait for this system would consist of a flat, horizontal line at each initial condition, indicating no change over time.

Example Phase Portrait for a Zero Matrix A:


  • For a two-dimensional system, draw a set of axes with x and y representing two linearly independent vectors in the plane.

  • Mark several points for different initial conditions in the plane.

  • Draw horizontal lines passing through each initial condition, which represent the trajectories of the solutions over time.

Because we chose a zero matrix for A, all points in the space are fixed points, and their trajectories are simply points in the phase space with no movement. The resulting phase portrait is a grid of fixed points with no trajectories connecting them, illustrating the idea of an indefinite number of fixed points.

User Gene Vincent
by
8.1k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.