Chaos, Bifurcations, Sensitive Dependence
Ch 12 — Part II — Higher-Order & Nonlinear
Bifurcations are qualitative changes in the solution structure as a parameter varies. Chaos describes deterministic but unpredictable behavior in nonlinear systems.
Saddle-node bifurcation
In \(y' = \mu + y^2\), for \(\mu<0\) there are two equilibria (one stable, one unstable); at \(\mu=0\) they collide; for \(\mu>0\) none exist. The number of equilibria changes as \(\mu\) passes through 0.
Pitchfork bifurcation
\(y' = \mu y - y^3\): one equilibrium \(y=0\) for \(\mu \le 0\), three (\(0\) and \(\pm\sqrt\mu\)) for \(\mu > 0\). The origin switches stability.
Sensitive dependence
Systems like the Lorenz equations exhibit trajectories that diverge exponentially from nearby initial conditions (\(\|\delta(t)\| \sim e^{\Lambda t}\) where \(\Lambda\) is a Lyapunov exponent). Small measurement errors in ICs render long-term prediction impossible even though the system is deterministic.
Practice Problems
Hint
Solution
If \(\mu > 0\), equilibria \(\pm\sqrt\mu\) (one stable, one unstable). At \(\mu=0\), single double root. \(\mu<0\): no real equilibria. Saddle-node at \(\mu=0\).
Answer: Saddle-node at \(\mu = 0.\)
Hint
Solution
For \(\mu<0\): only \(y=0\), stable. For \(\mu>0\): \(y=0\) unstable, \(\pm\sqrt\mu\) stable. This is a supercritical pitchfork.
Answer: Supercritical pitchfork at \(\mu=0.\)
Hint
Solution
Small uncertainties in initial conditions grow exponentially, so after time \(\sim (1/\Lambda)\log(1/\delta_0)\) the initial uncertainty fills the available state space.
Answer: Lyapunov exponent bounds predictability.