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MathidiaMathematics Academy
Dynamical systems
Lesson 31 of 75
Introduction

Flows and phase portraits

Objective. Analyse autonomous systems geometrically.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Flows and phase portraits describe how solutions of a system of autonomous ordinary differential equations move through state space, visualised as trajectories tangent to the vector field defined by the system. For ẋ = f(x,y), ẏ = g(x,y), each point in the plane has an associated velocity vector, and sketching these vectors reveals the qualitative behaviour of all solutions without solving explicitly. Key features to identify are fixed points, where f = g = 0, and the direction and curvature of nearby trajectories, which together build the portrait. Example: the system ẋ = y, ẏ = −x has circular trajectories around the origin, since x² + y² is conserved along the flow.

Worked example

Describe the phase portrait of the flow ẋ = y, ẏ = −x near the origin.

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  1. 1Find fixed points: setting ẋ = y = 0 and ẏ = −x = 0 gives the single fixed point (0,0).
  2. 2Check a conserved quantity: d/dt(x² + y²) = 2xẋ + 2yẏ = 2xy − 2yx = 0, so x² + y² is constant along trajectories.
  3. 3Since x² + y² = r² is fixed on each trajectory, orbits lie on circles centred at the origin.
  4. 4The direction of motion follows ẋ = y, so on the positive y-axis x increases, giving anticlockwise circulation.

Answer: Concentric circles around the origin, traversed anticlockwise

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Common mistake. Students often sketch trajectories crossing each other; for an autonomous system, distinct trajectories can never cross, since the vector field is single-valued at each point and would make the future path ambiguous.

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