1. The Pillar of Autonomy
We focus primarily on autonomous systems. A system with the property that $F$ and $G$ in equations (1) do not depend on the independent variable $t$ is said to be autonomous. This independence allows us to interpret trajectories as permanent paths in a fixed phase plane.
For any autonomous system $\mathbf{x}' = \mathbf{f}(\mathbf{x})$, there exists a unique solution satisfying $\mathbf{x}(t_0) = \mathbf{x}_0$. In the phase plane, this ensures that trajectories never cross; the path is determined entirely by the current state, not the time at which you arrived there.
2. Linear Benchmarks vs. Nonlinear Realities
In linear systems $\mathbf{x}' = \mathbf{Ax}$, the origin is typically the lone equilibrium, governed by the determinant $q = a_{11}a_{22} - a_{12}a_{21}$ and the trace. However, nonlinear systems are defined by their critical points—locations where the right-hand side is zero. A major Pitfall is that there may be several, or many, critical points that are competing for influence on the trajectories.
Example: The Nonlinear Pendulum
Unlike the linear spring-mass where the period is constant, the period $T$ of a nonlinear pendulum depends on its amplitude, expressed via the elliptic integral:
$$T = 4\sqrt{\frac{L}{g}} \int_{0}^{\pi/2} \frac{d\phi}{\sqrt{1 - k^2 \sin^2 \phi}}$$
3. Stability and Liapunov's Vision
To analyze these points without solving the equations, we use Liapunov Functions. Let $V$ be defined on some domain $D$ containing the origin. Then $V$ is said to be positive definite on $D$ if $V(0, 0) = 0$ and $V(x, y) > 0$ for all other points in $D$.
As we scale to 3D, we encounter the Lorenz matrix:
$$\begin{pmatrix} u \\ v \\ w \end{pmatrix}' = \begin{pmatrix} -10 & 10 & 0 \\ 1 & -1 & -\sqrt{\frac{8}{3}(r-1)} \\ \sqrt{\frac{8}{3}(r-1)} & \sqrt{\frac{8}{3}(r-1)} & -\frac{8}{3} \end{pmatrix} \begin{pmatrix} u \\ v \\ w \end{pmatrix}$$