Nonlinear dynamical systems, fixed points, and bifurcations

Equipment

• “Parameter μ” slider (−100…+100)
• “Initial condition x₀” slider
• Vector-field plot of $\dot{x}$ versus $x$ with evolution arrows
• Plot of the solution x(t) integrated by Euler's method

Procedure

1. Find the equilibria on the vector field

   Set μ > 0 and watch the curve $\dot{x} = \mu - x^2$ cross the horizontal axis at two points: $x^* = \pm\sqrt{\mu}$. Increase μ and watch the pair move apart; lower μ toward 0 and see them approach each other.

2. Probe stability with initial conditions

   Keep μ > 0 and drag “Initial condition x₀” across the upper equilibrium, then watch x(t). Trajectories starting from $x_0 > -\sqrt{\mu}$ converge to $+\sqrt{\mu}$ (stable); those starting below drift to $-\infty$ — confirming the upper fixed point is unstable, consistent with the sign of $f'(x) = -2x$.

3. Cross the saddle-node bifurcation

   Lower μ through 0: the two equilibria merge at $x^* = 0$ and annihilate — a saddle-node bifurcation. For μ < 0 every trajectory escapes; as a check, record the μ value where the pair appears and compare with the algebraic prediction $\mu_c = 0$.

Simulation

Experiment history

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