RK4 integration of ẋ=−y−z, ẏ=x+ay, ż=b+z(x−c); period-doubling cascade as c grows.
About this model
The Rössler system is a three-dimensional continuous dynamical system ẋ = −y − z, ẏ = x + a y, ż = b + z(x − c) that produces a famous single-scroll chaotic attractor for classic parameters (often a = b = 0.2, c ≈ 5.7). This simulator integrates the ODEs with RK4 and lets you raise c through a period-doubling cascade from periodic orbits into chaos. Trajectories are plotted in phase space so folding and stretching of the attractor become visible. The model is deterministic ODE chaos — no noise, no delay equations, and no Poincaré-section analysis beyond what the time series implies. Sweep c (and related parameters as exposed) to watch period doubling and the onset of the strange attractor.
Who it's for: Nonlinear dynamics, chaos theory, and applied math / physics courses on continuous-time attractors.
Key terms
Rössler attractor
Chaos
Period doubling
RK4 integration
Strange attractor
Phase space
How it works
The Rössler system (1976) is a minimal 3-D ODE with a single quadratic nonlinearity *xz* — far simpler than Lorenz, yet it produces a smooth ribbon-like strange attractor at the canonical (a, b, c) = (0.2, 0.2, 5.7). As c grows, you can scroll through a textbook period-doubling cascade (period-1 → 2 → 4 → chaos): pick the presets to feel it. Integration is RK4 with adjustable step.
Key equations
ẋ = −y − z
ẏ = x + a y
ż = b + z (x − c)
Frequently asked questions
What is a period-doubling cascade?
As a parameter such as c increases, a stable periodic orbit can lose stability and be replaced by an orbit of twice the period, then fourfold, and so on. The cascade accumulates at a finite parameter value beyond which chaos appears.
Why use RK4 instead of a simple Euler step?
Chaotic trajectories amplify local errors. A fixed-step RK4 method keeps short-term integration accurate enough for a clear attractor plot; crude Euler steps smear or destabilize the geometry for the same step size.
Is sensitivity to initial conditions a bug?
No — it is the definition of deterministic chaos. Two nearby seeds separate exponentially on the attractor while remaining confined to a bounded fractal-looking set in phase space.