Homoclinic Points

Homoclinic Points are points of intersection between the stable and unstable manifolds of a hyperbolic fixed point (or periodic orbit) in a dynamical system.

Consider the Non Linear Map of the equation:

$$f(x,y)=\left(\frac{x}{2},2y-7x^2\right)$$

The Stable Manifold $S$ is a parabola tangent to the x-axis at 0; the Unstable Manifold $U$ is the y-axis.

Jacobian Matrix Matrix for this space:

$$\mathcal{D}\vec{f}=\left(\begin{array}{cc}\frac{1}{2}& 0\\ -14x& 2\end{array}\right)$$

Eigenvalues for this are $\frac{1}{2},2$, with corresponding eigenvectors $\left[\begin{array}{c}1\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\end{array}\right]$, tangent to $s$ and $u$

In this example, $S$ and $U$ don’t intersect (except at the origin). But what happens when they do?

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Poincaré discovered during his study of the Three Body Problem that the Stable Manifold and Unstable Manifolds of a Saddle fixed point can intersect each other ( We note that $S$ and $U$ can not intersect themselves )

Assume $\vec{p}$ is Fixed or periodic and is a saddle. Also assume ${\vec{h}}_0\ne \vec{p}$ is a point of intersection of $S$ and $U$, in other words:

$${\vec{h}}_0\in S(\vec{p})and{\vec{h}}_0\in U(\vec{p})$$

We then say ${\vec{h}}_0$ is a homoclinic point, and if ${\vec{h}}_0\in S(\vec{p})$, then so is ${\vec{h}}_1$ ( where ${\vec{h}}_1=\vec{f}({\vec{h}}_0)$, same for ${\vec{h}}_2$ and so on ). Also ${\vec{h}}_{-1}=f^{-1}({\vec{h}}_0)$, and ${\vec{h}}_{-2}=f^{-2}({\vec{h}}_0)$ are in $S(\vec{p})\dots$ in fact the entire forward and backward orbit of ${\vec{h}}_0$ is in $S$. And by the same reasoning

To understand the geometry of how $S$ and $U$ can intersect, we need tools for analyzing how linear maps deform space:

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Let $N$ be the unit disc in ${\mathbb{R}}^m$ and let $A$ be an $m\times m$ matrix. Let $s_1^2\ge s_2^2\ge \dots s_m^2\ge 0$ and ${\hat{u}}_1,{\hat{u}}_2,\dots ,{\hat{u}}_m$ be the eigenvalues and unit eigenvectors of the $m\times m$ matrix $A{A}^T$.

1. ${\hat{u}}_i$ are mutually orthogonal unit vectors
2. the axes of the ellipse $AN$ are $s_i{\hat{u}}_i$ for $1\le i\le m$.

Somestimes we call $s_i$ the singular values of $A$, indexed in descending order ( think Singular Value Decomposition ).

Example

$$A=\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ 0& 2\end{array}\right]{\lambda }_1=\frac{1}{2},{\lambda }_2=2$$

To see how $A$ maps the unit disc, we look at $A{A}^T$:

$$\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ 0& 2\end{array}\right]\left[\begin{array}{cc}\frac{1}{2}& 0\\ \frac{1}{2}& 2\end{array}\right]=\left[\begin{array}{cc}\frac{1}{2}& 1\\ 1& 4\end{array}\right]$$

which has eigenvalues $s_1^2\approx 4.3$ and $s_2^2\approx 0.23$, $s_1\approx 2.1$ and $s_2\approx 0.48$.

and corresponding eigenvectors:

$${\hat{u}}_1\approx \left[\begin{array}{c}0.26\\ 0.97\end{array}\right]and{\hat{u}}_2\approx \underset{\text{or negation of this}}{\underbrace{\left[\begin{array}{c}-0.97\\ 0.26\end{array}\right]}}$$

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