Definitions of hyperbolic fixed points for a flow and a map

dynamical systemsfixed points-

The definition of a hyperbolic fixed point for a map is

Let $p\in M$ be a fixed point of f. Then p is a $\textbf{hyperbolic fixed point}$ if none of the eigenvales of $DF_{p}$ are on the unit circle.

The definition for a flow is

Let $\varphi_{t}$ be a flow generated by a vector field X. Let p be a fixed point of X. Then p is a $\textbf{hyperbolic fixed point}$ if none of the real parts of the eigenvales of $DX_{p}$ has zero real part.

Questions:

  1. My understanding of the differences between flow and map is that the former is defined for continuous-time dynamical systems, while the latter is defined for discrete-time systems. Is this the only distinction?

  2. Are the two definitions consistent?

I have seen this question, but it seems to only address the linear case?

Best Answer

  1. Not sure if I understand this question, but yes I think so. Flows are used to define continuous-time dynamical systems and maps are used to define discrete-time dynamical system.
  2. Yes, these definitions are consistent. One common way to get a discrete-time dynamical system from a continuous-time one is to look at the flow for a fixed time $\tau$. Specifically, $\phi_\tau:\mathbb R^n \to \mathbb R^n$ is a diffeomorphism and so we can define a discrete-time dynamical system by applying the map $\phi_\tau$. The definitions are consistent in the sense that if $p$ is a hyperbolic fixed point of the continuous-time system then it is a hyperbolic fixed point for the discrete-time system. This can be shown be noting that if $p$ is a fixed point, then $[D\phi_\tau] (p) = \exp(A\tau)$ where $A$ is the matrix from linearizing around $p$. Generally, results about hyperbolic fixed points in discrete time have analogous results for continuous time and vice versa.
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