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:
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- For a two-dimensional system, draw a set of axes with x and y representing two linearly independent vectors in the plane.
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- Mark several points for different initial conditions in the plane.
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- 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.