[Math] stability of equilibria for $n$-dimensional nonlinear systems of differential equations: examples

dynamical systemsordinary differential equationsstability-in-odes

I'm currently self-studying dynamical systems. I'm trying to summarize what can be said about the stability of equilibrium points for an $n$-dimensional non-linear system of differential equations:

$\frac{d}{dt}\bf{x} = \bf{f}(\bf{x})$

Let's say I already have found the equilibria $\bf{x}^{*}$ (for which $\bf{f}(\bf{x}^{*}) = \bf{0}$). Can someone point me to a reference which will help me classify the stability types of these equilibria (whether they are saddle points, centres, sources, sinks, etc.).

Specifically, I want to know what can be said by using linearizations (ie. with Jacobians) – I realize that what can be said using just this method is probably quite limited.

I know that the Hartman-Grobman theorem is helpful. The way I understand it, this theorem says that if the Jacobian of the system evaluated at $\bf{x}^{*}$ has eigenvalues $\{ \lambda_{j} \}_{j=1}^{n}$ such that $\text{Re}(\lambda_{j})\neq0$ for all $j$, then:

  • if all eigenvalues are negative then it is asymptotically stable (sink)
  • if all eigenvalues are positive then it is unstable (source)
  • if there is a mix of positive and negative eigenvalues, then its a saddle point

Is this correct? Can you say anything else if $\text{Re}(\lambda_{j})=0$ for one or more $j$?

Every reference I come across deals with $two$-dimensional systems pretty heavily, and then omits a detailed discussion on higher-dimensional systems. What I am really interested in is a document which could list me some examples dealing with the stability of systems that have dimension $higher$ than two. Can someone help me out?

Best Answer

First for general n-D case:Full classification (via linearization) in higher dimensions gets complicated. But the general idea is that you may have products of the basic equilibria. For example, in restricted three-body problem, linearization around some fixed points tells us that they are a product of a saddle and two centers (in 6D). This is said to be a rank-1 saddle.

For your second question on whether one can say anything if real part is 0, yes! But you have to move beyond mere linearization. The whole center-manifold theory is build for that. Take a look at books on intro to dynamical systems theory book such as Wiggins or Perko. In addition, Wiggins gives explicit examples of some 3D cases.

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