Dynamical Systems on the Sphere | 3 Nov. 2019 |
To start, consider a linear dynamical systems , , for which the solution is given by . We can map this solution to the sphere via . The corresponding vector field for is given by The beauty is the following, to find the equilibria, consider . Of course, is not valid since . However, take any eigenvector of , then is still an eigenvector, plus it lives on . If we plug such a scaled (from now on consider all eigenvectors to live on ) into we get , which is rather cool. The eigenvectors of are (at least) the equilibria of . Note, if , then so is , each eigenvector gives rise to two equilibria. We say at least, since if is non-trivial, then for any , we get . Let , then what can be said about the stability of the equilibrium points? Here, we must look at , where the -dimensional system is locally a -dimensional linear system. To do this, we start by computing the standard Jacobian of , which is given by As said before, we do not have a -dimensional system. To that end, we assume with diagonalization (otherwise peform a Gram-Schmidt procedure) and perform a change of basis . Then if we are interested in the qualitative stability properties of (the eigenvector of ), we can simply look at the eigenvalues of the matrix resulting from removing the row and column in , denoted . We do two examples. In both figures are the initial conditions indicated with a dot while the equilibrium points are stars.
Now, a valid question is, why do stable equilibrium points (for on ) become unstable on the sphere? For our example, we see that the full (minus ) line: all s.t. collapses to just two points: . Not just any two points, but two points surrounded by instability, hence the positive definite Jacobian. In a similar fashion it follows that the unstable asymptotes now relate to (finite!) attracting equilibria . The fact that is attracting from all directions while relates to a saddle follows from the corresponding eigenvalues, . From the figure you can take away that is the attracting plane, resulting in being an attracting great arc (since ). We observe that by construction it seems impossible to have 1 stable pole since the equilibrium points come in pairs. You might say, sure, but these vector fields where created by using linear systems. Perhaps there is some wild differential equation which does the trick? The answer is yes, we can however not remove the equilibrium points completely, we cannot comb a hairy ball. |