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MathidiaMathematics Academy
Dynamical systems
Lesson 34 of 75
Challenge

Bifurcations

Objective. Identify qualitative parameter-driven changes.

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  2. 2. Worked example
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Learn: the key idea

A bifurcation is a qualitative change in the number or stability of fixed points (or other invariant structures) of a dynamical system as a parameter is varied continuously. In a saddle-node bifurcation, two fixed points of opposite stability collide and annihilate as the parameter crosses a critical value; in a pitchfork bifurcation, a single fixed point splits into three as stability transfers between them. The normal form ẋ = μ − x² illustrates a saddle-node: for μ > 0 there are two fixed points at ±√μ, for μ = 0 they merge at x = 0, and for μ < 0 none exist. Example: as μ increases through 0, the system's long-term behaviour changes discontinuously, even though the equation varies smoothly.

Worked example

Identify the bifurcation occurring in the family ẋ = μ − x² as μ passes through zero, and find the fixed points for μ = 4.

4
Illustrated card showing 4
  1. 1Fixed points satisfy μ − x² = 0, so x = ±√μ, which exist only when μ ≥ 0.
  2. 2For μ < 0 there are no real fixed points, for μ = 0 there is exactly one (x = 0), and for μ > 0 there are two: this is a saddle-node bifurcation at μ = 0.
  3. 3For μ = 4, solve x² = 4 to get x = ±2.
  4. 4Checking stability, f'(x) = −2x gives f'(2) = −4 < 0 (stable) and f'(−2) = 4 > 0 (unstable).

Answer: Saddle-node bifurcation at μ = 0; fixed points x = 2 (stable) and x = −2 (unstable) at μ = 4

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Common mistake. Students often assume a bifurcation requires the equation itself to change discontinuously; in fact the vector field varies smoothly with the parameter, and it is only the qualitative structure of fixed points that changes abruptly.

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