Systems of Autonomous Equations

Definition

A system of autonomous equations is a set of coupled first-order differential equations where the rates of change of multiple state variables depend only on the variables themselves, not on time explicitly. $\frac{dx}{dt} = F(x, y), \quad \frac{dy}{dt} = G(x, y)$

• How to read: “dx/dt equals F of x comma y; dy/dt equals G of x comma y.”
• Meaning: Coupled first-order ODEs where rates depend only on current state variables, not on time explicitly. The rules of motion are fixed.

Why It Matters

Autonomous systems allow us to model the long-term behavior of a system (like a predator-prey relationship) without needing to know the exact time. They reveal the ‘attractors’ and ‘stable points’ that dictate the ultimate fate of a dynamic process.

Core Concepts

• Phase Plane: The $xy$-coordinate system representing the state space of the system.
  • How to read: “x-y plane.”
  • Meaning: The phase plane—each point $(x,y)$ is a possible system state; trajectories trace how the state evolves over time.
• Trajectories: The paths $(x(t), y(t))$ followed by the system over time. Through any non-equilibrium point, exactly one trajectory passes.
• Vector Field: At every point $(x, y)$, the vector $(F, G)$ represents the velocity and direction of the system’s state change.
• Equilibrium Points: Points where $F(x, y) = 0$ and $G(x, y) = 0$ simultaneously. These are the fixed points of the system.

Connected Concepts

• Graphs of Polar Equations
• Polar Equations of Conics
• Substitution Method for Linear Systems and Elimination Method for Linear Systems