In the context of dynamical systems, the stability of fixed points (also called equilibrium points) is determined by the eigenvalues of the system's Jacobian matrix at those points. Specifically, fixed points can be classified as either spirals or nodes based on their eigenvalues, and this classification affects their stability.

1. Stable Spirals (or Spiral Sinks):

   • A fixed point is a stable spiral if the eigenvalues of the Jacobian have complex conjugate pairs with negative real parts. This causes the trajectories to spiral in toward the fixed point.
   • Stability: Stable, since the trajectories approach the equilibrium point over time in a spiral fashion.

2. Stable Nodes:

   • A fixed point is a stable node if both eigenvalues are real and negative. This causes the trajectories to move directly toward the fixed point without spiraling.
   • Stability: Stable, since the trajectories approach the equilibrium point directly, without oscillation.

In short, both stable spirals and stable nodes are considered stable, but they differ in how trajectories approach the equilibrium. Spirals exhibit oscillatory motion while nodes do not.

Would you like further details on how these are determined mathematically?

Here are some related questions:

1. What are the differences between a spiral source and a spiral sink?
2. How are the eigenvalues of a Jacobian matrix calculated for stability analysis?
3. What is the significance of complex eigenvalues in dynamical systems?
4. How do the trajectories differ between stable nodes and unstable nodes?
5. What is a saddle point in the context of dynamical systems?

Tip: In a phase plane, the stability of a fixed point can be visualized by plotting the trajectories of nearby points.