What is a phase line portrait?
The path travelled by the point in a solution is called a trajectory of the system. A picture of the trajectories is called a phase portrait of the system. In the animated version of this page, you can see the moving points as well as the trajectories.
What is a phase portrait in differential equations?
Similar to a direction field, a phase portrait is a graphical tool to visualize how the solutions of a given system of differential equations would behave in the long run. Just like a direction field, a phase portrait can be a tool to predict the behaviors of a system’s solutions.
What do phase portraits tell us?
Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the state space. This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value.
What is a nodal sink?
Sinks have coefficient matrices whose eigenvalues have negative real part. nodal sink — real unequal eigenvalues, (c) focus sink — real equal eigenvalues; two independent eigenvectors, and. (d)
What do phase portraits tell you?
How do you find the Nullclines of a system?
Alge- braically, we find the x-nullcline by solving f(x, y)=0. points where the vectors are horizontal, going either to the left or to the right. Algebraically, we find the y-nullcline by solving g(x, y)=0. (to the left of the y-axis) move to the right if below the line x + y = 2 and to the left if above it.
When is the phase portrait stable or unstable?
phase portrait is a saddle (which is always unstable). If 0 < D < T 2/4, the eigenvalues are real, distinct, and of the same sign, and the phase portrait is a node, stable if T < 0, unstable if T > 0. If 0 < T 2/4 < D, the eigenvalues are neither real nor purely imaginary, and the phase portrait is a spiral, stable if T < 0, unstable if T > 0.
Is the phase portrait rank 1 or degenerate?
This has rank 1 and the phase portrait is degenerate, as the Mathlet says. are attracted to some point on the line, and the Mathlet labels these orbits (rays) OK. But the line x=y is labeled with 2 arrows that go to infinity in both directions.
How are phase portrait and eigenvalues related?
Sliders allow manipulation of the matrix entries over . By viewing simultaneously the phase portrait and the eigenvalue plot, one can easily and directly associate phase portrait bifurcations with changes in the character of the eigenvalues.