📂Dynamics

Multidimensional Nonlinear Maps

Definition of Nonlinear Maps

1. If the map $\mathbf{f} : \mathbb{R}^{m} \to \mathbb{R}^{m}$ is not linear, it is said to be nonlinear.

Buildup ¹

It is hard to prove that a map is linear, but easy to show it is nonlinear; linear problems are easy, but nonlinear problems are challenging. Almost everything in this universe is nonlinear, and because nonlinearity is difficult, humans start by trying to convert nonlinearities into linearities.

In the figure above, although the given curve is clearly curved, if you look at a very small range, it appears almost straight. If we ignore some errors, within a sufficiently small range, the nonlinearity is almost the same as linearity. This method can be applied to linear maps, thereby viewing nonlinear maps as virtually linear maps.

Taylor approximation shows that $f(x+h) \approx f(x) + h f '(x)$ and around $x$, at a neighborhood $B(x; h)$, $f(x+h)$ is expressed as a linear equation with respect to $h$. To extend this to multivariable functions, the introduction of the Jacobian is necessary. It is because of such ideas that the development of analysis was absolutely essential.

Given a small vector $\mathbf{h}$ such that $\mathbf{f}$ has linearity within the neighborhood $B \left( \mathbf{p} ; | \mathbf{h} | \right)$ of the fixed point $\mathbf{p}$ of $\mathbf{f}$: $$ \mathbf{f} ( \mathbf{p} + \mathbf{h} ) \approx \mathbf{f} ( \mathbf{p} ) + D \mathbf{f} ( \mathbb{ p} ) \cdot \mathbf{h} \\ \implies \mathbf{f} ( \mathbf{p} + \mathbf{h} ) - \mathbf{f} ( \mathbf{p} )\approx D \mathbf{f} ( \mathbb{ p} ) \cdot \mathbf{h} $$ If $\mathbf{f}$ has linearity, then $\displaystyle \mathbf{f} ( \mathbf{p} + \mathbf{h} ) \approx \mathbf{f} ( \mathbf{p} ) + \mathbf{f} ( \mathbf{h} )$, meaning: $$ \mathbf{f} ( \mathbf{h} ) = D \mathbf{f} ( \mathbb{ p} ) \cdot \mathbf{h} $$ In other words, $\mathbf{f}$ is a linear map $T_{D \mathbf{f} ( \mathbb{ p} )} : \mathbb{R}^{m} \to \mathbb{R}^{m}$ that sends $\mathbf{h} \in \mathbb{R}^{m}$ in the neighborhood of $\mathbf{p}$ to $D \mathbf{f} ( \mathbb{ p} ) \cdot \mathbf{h} \in \mathbb{R}^{m}$.

Thus, just like we discussed a dynamic system using the eigenvalue of matrix $A$ corresponding to the linear map $T_{A}$, for the fixed point $\mathbf{p}$ of the nonlinear map $\mathbf{f}$, we can continue the same discussion with $$ D \mathbf{f} ( \mathbb{ p} ) = \begin{bmatrix} {{\partial f_{1} } \over {\partial x_{1} }} ( \mathbb{ p} ) & \cdots & {{\partial f_{1} } \over {\partial x_{n} }} ( \mathbb{ p} ) \\ \vdots & \ddots & \vdots \\ {{\partial f_{m} } \over {\partial x_{1} }} ( \mathbb{ p} ) & \cdots & {{\partial f_{m} } \over {\partial x_{n} }} ( \mathbb{ p} ) \end{bmatrix} $$

Definition of Hyperbolic and Saddles in Nonlinear Maps

Let’s call the eigenvalues of $D \mathbf{f} ( \mathbf{p})$ as $\lambda_{1} , \cdots , \lambda_{m}$. 2. If $| \lambda_{1} | \ne 1, \cdots , | \lambda_{m} | \ne 1$, then $\mathbf{p}$ is said to be hyperbolic. 3. If for hyperbolic $\mathbf{p}$, there exists at least one $i,j$ satisfying $\begin{cases} | \lambda_{i} | >1 \\ | \lambda_{j} | <1 \end{cases}$, then $\mathbf{p}$ is said to be a saddle.

Criteria when Not a Saddle

When $\mathbf{p}$ is not a saddle, the following theorems can be used to judge whether it is a sink or source.

• [1]: If $| \lambda_{1} | < 1, \cdots , | \lambda_{m} | < 1$, then $\mathbf{p}$ is a sink.
• [2]: If $| \lambda_{1} | > 1, \cdots , | \lambda_{m} | > 1$, then $\mathbf{p}$ is a source.