The Rossler system provides one of the simplest examples of chaos in three dimensions:

$\dot{x} = -y - z, \quad \dot{y} = x + ay, \quad \dot{z} = b + z(x - c)$

For parameters $a = 0.2$, $b = 0.2$, $c = 5.7$, the system exhibits a strange attractor with simpler topology than Lorenz—trajectories spiral outward in a plane, then return via a twist. Despite its simplicity, the system displays:

• Positive Lyapunov exponent
• Sensitive dependence on initial conditions
• Period-doubling route to chaos as $c$ increases
• Fractal attractor dimension

The Rossler attractor serves as a pedagogical example, more tractable than Lorenz while retaining essential chaotic features.

Chua's circuit, an electronic oscillator with a nonlinear resistor, generates chaos through simple electronic components:

$\dot{x} = \alpha(y - x - f(x)), \quad \dot{y} = x - y + z, \quad \dot{z} = -\beta y$

where $f(x)$ is a piecewise-linear function modeling the nonlinear resistor. The double scroll attractor has two lobes, with trajectories spiraling around unstable fixed points in each lobe and irregularly switching between them.

Chua's circuit was historically significant as an easily constructed physical system exhibiting chaos, allowing experimental verification of theoretical predictions about strange attractors and bifurcations.

From the Lorenz attractor, one can construct a one-dimensional return map by recording successive maxima of the $z$ coordinate: $z_{n+1} = f(z_n)$. Near the critical parameter values, this Lorenz map has the form:

$f(z) = \begin{cases} m_1 z & z < z_c \\ m_2(z - z_c) & z \geq z_c \end{cases}$

with slopes $m_1, m_2 > 1$ and a discontinuity at $z_c$. The map exhibits:

• Sensitive dependence (both slopes exceed 1)
• Dense orbits and topological transitivity
• Positive topological entropy

This reduction reveals that the three-dimensional flow's complexity can be captured by a one-dimensional map, facilitating rigorous analysis.

The baker's map is a canonical model for chaotic mixing, mimicking kneading dough:

$B(x, y) = \begin{cases} (2x, y/2) & 0 \leq x < 1/2 \\ (2x-1, (y+1)/2) & 1/2 \leq x < 1 \end{cases}$

This map:

• Stretches horizontally by factor 2 (expanding direction)
• Compresses vertically by factor 2 (contracting direction)
• Cuts and stacks like a baker folding dough

The baker's map is exactly solvable, with Lyapunov exponents $\lambda_1 = \ln 2$ (positive, stretching) and $\lambda_2 = -\ln 2$ (negative, contraction). It preserves area overall but creates fractal structure through repeated stretching and folding—the paradigm for hyperbolic chaos.