The logistic map $x_{n+1} = \mu x_n(1 - x_n)$ exhibits a rich bifurcation structure as $\mu$ varies from 0 to 4:

1. $\mu \in (0, 1)$: Fixed point $x^* = 0$ is stable
2. $\mu \in (1, 3)$: Transcritical bifurcation at $\mu = 1$; fixed point $x^* = 1 - 1/\mu$ is stable
3. $\mu \in (3, 1+\sqrt{6})$: Period-doubling at $\mu = 3$; stable period-2 orbit emerges
4. $\mu \in (1+\sqrt{6}, \mu_\infty)$: Cascade of period-doublings (4, 8, 16, ...) accumulating at $\mu_\infty \approx 3.569946$
5. $\mu > \mu_\infty$: Chaotic regime with windows of periodic behavior

The Feigenbaum constant $\delta \approx 4.669$ governs the rate of accumulation of bifurcation points.

The forced Duffing oscillator $\ddot{x} + \delta\dot{x} + \alpha x + \beta x^3 = \gamma \cos(\omega t)$ exhibits multiple bifurcations as forcing amplitude $\gamma$ varies:

• For small $\gamma$: periodic response locked to forcing frequency
• Saddle-node bifurcations create hysteresis loops in the amplitude-frequency response
• For $\beta < 0$ (negative cubic stiffness): potential has two wells, leading to cross-well chaos
• Hopf bifurcations from periodic to quasi-periodic motion
• Period-doubling cascades to chaos

The Duffing equation is a paradigm for forced nonlinear oscillators, appearing in mechanical systems, electronic circuits, and optical devices.

The Lorenz system $\dot{x} = \sigma(y-x)$, $\dot{y} = x(\rho - z) - y$, $\dot{z} = xy - \beta z$ undergoes several bifurcations as the parameter $\rho$ increases (with $\sigma = 10$, $\beta = 8/3$):

1. $\rho < 1$: Origin is globally stable
2. $\rho = 1$: Pitchfork bifurcation; two new fixed points $C^\pm$ emerge
3. $\rho \approx 1.346$: Subcritical Hopf bifurcation at $C^\pm$
4. $\rho \approx 13.926$: Homoclinic bifurcation; onset of transient chaos
5. $\rho \approx 24.74$: Transition to sustained chaos (Lorenz attractor)
6. $\rho > 24.74$: Chaotic regime with periodic windows

The Lorenz attractor demonstrates how a deterministic system can exhibit seemingly random, unpredictable behavior.

The cusp catastrophe is a codimension-two bifurcation described by the potential:

$V(x; a, b) = \frac{x^4}{4} + \frac{ax^2}{2} + bx$

The equilibria satisfy $\frac{dV}{dx} = x^3 + ax + b = 0$. The bifurcation set in the $(a, b)$ parameter plane consists of:

• Fold curve: $4a^3 + 27b^2 = 0$ (saddle-node bifurcations)

Inside the cusp region, there are three equilibria (two stable, one unstable). Outside, there is one stable equilibrium. Crossing the fold curve causes catastrophic jumps between equilibria. This structure models hysteresis and sudden transitions in systems from structural mechanics to opinion dynamics.