A student asked me:
"With regards to Problem Sheet 7, question 2 (the first question on the sheet), after sketching the nullclines and determining the direction of flow at the nullclines, as well as the nature of the equilibrium point at (0.5,0.5) - stable equilibrium, how should I attempt to sketch the phase portrait, i.e. the diagram on the right in the answers. "
It may be instructive to go through the steps of how to try sketching a phase portrait:
Locally near equilibria:
1. Find the equilibria. (Depending on the type of the equations, this may be easy or impossible. In most exercises this is easy.)
2. Calculate the linearization of the vector field (ie Jacobian) at these equilibria and deduce , where possible (ie when hyperbolic), what the phase portrait should look like near these equilibria.
More globally:
1. Where relevant or possible, determine a bounded invariant set to which all solutions are attracted (most of our examples have such a region).
2. Try drawing nullclines where the vector field (and thus the tangent to solution curves) is horizonal and vertical. These nullclines may be helpful. In simple examples, often the nullclines can be computed explicitly.
3. Where there are saddles, it may be useful to try sketching where stable and unstable manifolds may end up.
4. Determine possibilities for the \(\omega\)-limit sets, in view of the Poincare-Bendixson theory.
All-in-all, this is what you should be doing when sketching a phase portrait. It often not possible to get all the properties of the flow this way, so that there still may be some unknowns. In particular, it is often hard to rule out the existence of a periodic solution around an attracting or repelling equilibrium point (unless it lies at the border of a forward invariant set, in which case no periodic solutions can encircle it).
If at the exam, you think there is some ambiguity or unknown property of the phase portrait, you should just write that. "Sketching" a phase portrait is precisely that: provide those features which you are certain of and discuss which additional features may or may not be present, on the basis of the theory.
For the specific problem in question: it is completely reasonable to conclude that around the attracting equilibrium (0.5,0.5), the Poincare-bendixson theorem leave open the possibility of an \(\omega\)-limit set that is a periodic solution encircling this equilibrium. The argument I give in the model answer is correct, but not so easily verifiable (so you are not supposed to discover such a subtle property in an exam question, for instance).
I hope this helps.
"With regards to Problem Sheet 7, question 2 (the first question on the sheet), after sketching the nullclines and determining the direction of flow at the nullclines, as well as the nature of the equilibrium point at (0.5,0.5) - stable equilibrium, how should I attempt to sketch the phase portrait, i.e. the diagram on the right in the answers. "
It may be instructive to go through the steps of how to try sketching a phase portrait:
Locally near equilibria:
1. Find the equilibria. (Depending on the type of the equations, this may be easy or impossible. In most exercises this is easy.)
2. Calculate the linearization of the vector field (ie Jacobian) at these equilibria and deduce , where possible (ie when hyperbolic), what the phase portrait should look like near these equilibria.
More globally:
1. Where relevant or possible, determine a bounded invariant set to which all solutions are attracted (most of our examples have such a region).
2. Try drawing nullclines where the vector field (and thus the tangent to solution curves) is horizonal and vertical. These nullclines may be helpful. In simple examples, often the nullclines can be computed explicitly.
3. Where there are saddles, it may be useful to try sketching where stable and unstable manifolds may end up.
4. Determine possibilities for the \(\omega\)-limit sets, in view of the Poincare-Bendixson theory.
All-in-all, this is what you should be doing when sketching a phase portrait. It often not possible to get all the properties of the flow this way, so that there still may be some unknowns. In particular, it is often hard to rule out the existence of a periodic solution around an attracting or repelling equilibrium point (unless it lies at the border of a forward invariant set, in which case no periodic solutions can encircle it).
If at the exam, you think there is some ambiguity or unknown property of the phase portrait, you should just write that. "Sketching" a phase portrait is precisely that: provide those features which you are certain of and discuss which additional features may or may not be present, on the basis of the theory.
For the specific problem in question: it is completely reasonable to conclude that around the attracting equilibrium (0.5,0.5), the Poincare-bendixson theorem leave open the possibility of an \(\omega\)-limit set that is a periodic solution encircling this equilibrium. The argument I give in the model answer is correct, but not so easily verifiable (so you are not supposed to discover such a subtle property in an exam question, for instance).
I hope this helps.
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ReplyDeleteI really like the solution that you have provided. I had a similar question that I needed to solve and I could simply solve by making possible solution of the vector.
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