The usual classification of fixed points as used in linear stability analysis is based on planar systems (un-/stable node, un-/stable spiral point, saddle). I need to extend this classification to a 4D phase-space (arising from 4 coupled ODEs).

Are there any "canonical" termini or ways to describe the 4D cases?

For example: the eigenvalues are complex conjugates "pairwise":

  • $a+bi$
  • $a-bi$
  • $-c+di$
  • $-c-di$

for all parameters positive (i.e. combination of stable and unstable spiral point character).


1 Answer 1


In short: In general you'll have direct products of the 2-D cases. That's why they're so heavily studied, because they also provide the basis for higher dimensions.

Your example is a focus-focus.

While a hyperbolic equilibrium in 2-D can be a node, a saddle, or a focus; in 3-D it can be a node or a saddle or the products focus-node or saddle-focus. Note that in 3-D you cannot have, say, the product "saddle-saddle" because each saddle has two eigenvalues (positive and negative real parts), thus two saddles have 4 eigenvalues and can only exist in 4-D.

The reasons why this is so are:

  1. Hartman–Grobman theorem tells us that, for a smooth, hyperbolic system (i.e., C$^1$ with no eigenvalue with Re$\;= 0\,$), we can use its linearization to assess the system's behavior in the vicinity of a fixed point;

  2. linear systems with the same numbers of eigenvalues with Re$\;< 0$ and Re$\;> 0$ and no Re$\;= 0$ are topologically equivalent; and

  3. the linearized system space is a direct sum of the subspaces determined by its eigenvectors, and the dynamics in these subspaces are independent.

The vector subspaces of point 3 correspond to the invariant manifolds of the original (non-linearized) system. Also, from Kuznetsov:

bifurcations occur in "essentially" the same way for generic n-dimensional systems [as in 1-D and 2-D systems, due to] invariant manifolds on which the system exhibits the corresponding bifurcations, while the behavior off the manifolds is somehow "trivial", for example, the manifolds may be exponentially attractive.

If the system is not hyperbolic, other equilibria are possible, such as center, center-saddle, center-focus-focus, etc. They play an important role in Hamiltonian systems, and their classification is well covered in the Bolsinov and Fomenko's Integrable Hamiltonian Systems: Geometry, Topology, Classification.

Additional references:
* stability of equilibria for $n$-dimensional nonlinear systems of differential equations: examples
* Scholarpedia - Equilibrium

  • 1
    $\begingroup$ Wonderful answer. $\endgroup$ Commented Feb 15, 2018 at 3:46

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