ISEGORIA / MATH ENCYCLOPEDIA
Dynamical systems: stability and chaos
Phase portraits, changing equilibria, and sensitive dependence.
Before you begin: Derivatives and elementary differential equations
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. A spiral in phase space
The arrows give the instantaneous direction of motion. Change the growth rate to turn a decaying spiral into a growing one. The trajectory is an exact solution, sampled for display.
Worked example. At a=-0.2 the radius is multiplied by exp(-0.2t).
Watch out. A center at a=0 is stable but not asymptotically stable.
What changes sign at the stability threshold?
The real parts of the eigenvalues a±i.
2. A saddle-node bifurcation
Move the parameter through zero. Two equilibria are born when the graph first touches and then crosses the horizontal axis. Arrows show the sign of the velocity.
Worked example. For μ=1, x=1 is attracting and x=-1 is repelling.
Watch out. At μ=0 the equilibrium attracts from only one side. A zero derivative alone does not classify stability.
How many real equilibria exist when μ is negative?
None, because x² cannot equal a negative number.
3. The logistic map
Compare two orbits whose starting values differ by one millionth. Parameter changes can produce convergence, cycles, or sensitive dependence. The lower panel samples long-run values across parameters.
Worked example. For r=2, most interior initial values approach 1/2.
Watch out. A finite orbit is not a proof of chaos. Periodic windows occur inside the complicated parameter region.
Does r greater than 3 always imply chaos?
No. Stable periodic cycles also occur.