Two-Dimensional Flows - Core Definitions

Two-dimensional dynamical systems extend one-dimensional dynamics to planar flows, where trajectories trace out curves in the plane. These systems are described by differential equations that govern continuous-time evolution, revealing rich geometric structures including fixed points, periodic orbits, and separatrices.

DefinitionAutonomous Planar System

An autonomous planar system is a pair of first-order differential equations:

$\frac{dx}{dt} = f(x, y), \quad \frac{dy}{dt} = g(x, y)$

where $f, g: \mathbb{R}^2 \to \mathbb{R}$ are continuous functions. Equivalently, in vector notation:

$\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x})$

where $\mathbf{x} = (x, y)$ and $\mathbf{F} = (f, g)$ is the vector field. The flow $\phi_t(\mathbf{x}_0)$ gives the position at time $t$ of a trajectory starting from $\mathbf{x}_0$ at time $t=0$ .

The geometry of planar flows is visualized through phase portraits, which display trajectories as curves in the $(x, y)$ -plane. The vector field assigns a direction to each point, and trajectories follow these directions, never crossing except at fixed points where the vector field vanishes.

DefinitionFixed Points and Linear Stability

A point $\mathbf{x}^* = (x^*, y^*)$ is a fixed point (or equilibrium) if $\mathbf{F}(\mathbf{x}^*) = \mathbf{0}$ . The linearization at $\mathbf{x}^*$ is the matrix:

$J = \begin{pmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{pmatrix}\bigg|_{\mathbf{x}^*}$

The eigenvalues $\lambda_1, \lambda_2$ of $J$ determine the local behavior near $\mathbf{x}^*$ :

Sink (stable node): both $\text{Re}(\lambda_i) < 0$
Source (unstable node): both $\text{Re}(\lambda_i) > 0$
Saddle: one $\text{Re}(\lambda_i) > 0$ , one $\text{Re}(\lambda_i) < 0$
Center or spiral: $\text{Re}(\lambda_1) = \text{Re}(\lambda_2)$

The classification of fixed points by eigenvalues reveals the local flow geometry. Sinks and sources represent attracting and repelling equilibria, while saddles have stable and unstable manifolds that separate regions of different dynamical behavior. Centers and spirals correspond to oscillatory dynamics, either purely periodic (center) or spiraling inward/outward (spiral).

DefinitionLimit Cycles

A limit cycle is an isolated closed trajectory in the phase plane. It is:

stable (attracting) if nearby trajectories spiral toward it as $t \to \infty$
unstable (repelling) if nearby trajectories spiral away from it
semi-stable if trajectories on one side approach it while those on the other side diverge

Limit cycles represent self-sustained oscillations and are fundamental objects in applied dynamical systems.

ExampleHarmonic Oscillator

The undamped harmonic oscillator is described by:

$\ddot{x} + \omega^2 x = 0$

Converting to a first-order system with $y = \dot{x}$ :

$\dot{x} = y, \quad \dot{y} = -\omega^2 x$

The origin is a center with eigenvalues $\pm i\omega$ . Trajectories are ellipses $\omega^2 x^2 + y^2 = E$ (constant energy), representing periodic oscillations. This system is conservative; every orbit is closed, and there are no isolated limit cycles.

Remark

The distinction between centers and limit cycles is subtle but crucial. A center is surrounded by infinitely many closed orbits forming a continuous family, while a limit cycle is an isolated periodic orbit. Centers typically arise in conservative systems (with a conserved quantity like energy), whereas limit cycles appear in dissipative systems where energy is both added and removed, creating self-sustained oscillations.

Understanding these fundamental objects—fixed points, their stability types, and limit cycles—provides the foundation for analyzing planar flows. The Poincare-Bendixson theorem, which we will encounter later, characterizes the long-term behavior of trajectories in terms of these basic structures, showing that planar flows, while richer than one-dimensional systems, still have constrained asymptotic dynamics compared to higher-dimensional chaos.