📂Dynamics

Hyperbolicity of Fixed Point in Dynamics

Definition ¹

Let’s consider a space $\mathbb{R}^{p}$ and a smooth function $f , g : \mathbb{R}^{p} \to \mathbb{R}^{p}$, such that the dynamical system can be represented as follows by a vector field or a map. $$ \dot{x} = f(x) \\ x \mapsto g(x) $$ We denote the eigenvalues obtained from the Jacobian matrix at a fixed point $\overline{x}$ as $\lambda_{1} , \cdots , \lambda_{p}$.

1. Depending on whether it is a vector field $f$ or a map $g$, the multipliers $n_{-}, n_{0}, n_{+}$ are defined as follows. $$ f: \begin{align*} n_{-} =& \operatorname{card} \left\{ k : \operatorname{Re} \lambda_{k} < 0 \right\} \\ n_{0} =& \operatorname{card} \left\{ k : \operatorname{Re} \lambda_{k} = 0 \right\} \\ n_{+} =& \operatorname{card} \left\{ k : \operatorname{Re} \lambda_{k} > 0 \right\} \end{align*} $$ $$ g: \begin{align*} n_{-} =& \operatorname{card} \left\{ k : \left| \lambda_{k} \right| < 1 \right\} \\ n_{0} =& \operatorname{card} \left\{ k : \left| \lambda_{k} \right| = 1 \right\} \\ n_{+} =& \operatorname{card} \left\{ k : \left| \lambda_{k} \right| > 1 \right\} \end{align*} $$ Here, $\operatorname{Re}$ denotes the real part of the complex number, $\left| \cdot \right|$ denotes the modulus of the complex number, and $\operatorname{card}$ denotes the cardinality of a set.
2. If $n_{0} = 0$, the fixed point $\overline{x}$ is said to be hyperbolic.
3. If $n_{-} n_{+} \ne 0$, the hyperbolic fixed point is called a hyperbolic saddle.

Explanation

In continuous systems, note to compare the real part of $\lambda_{k}$ with $0$, and in discrete systems, compare the magnitude of $\lambda_{k}$ with $1$.

For a visual understanding, the typical phase portrait of a hyperbolic fixed point in 2D is as follows.

As seen above, the hyperbolic fixed point can often be inferred to possess certain properties just by observing the complex eigenvalues. In many cases, it is considered either a ‘comparatively easy point to understand’ or an ‘unremarkable and generic point’.

• The fact that the eigenvalues are complex implies there is a rotation in the surrounding trajectories.
• If all eigenvalues lie to the left of the complex plane, it implies that nearby points are approaching the fixed point.
• If all eigenvalues lie to the right of the complex plane, it implies that nearby points are moving away from the fixed point.
• If all eigenvalues do not lie solely on either the left or right side of the complex plane, it means that there are at least one stable and unstable manifold, thereby forming a saddle.

As an example not covered above, if the real part is $0$, in other words, if all $\lambda_{k}$ are purely imaginary, the nearby points neither approach nor move away, and the fixed point is called elliptic and is referred to as a center.

Related Topics

• Hyperbolicity of Fixed Points
• Hyperbolicity of Limit Cycles