Circle Map: Devil's Staircase and Arnold Tongues
Numerical reproductions of dynamical systems and chaos phenomena. View on GitHub
Devil's staircase of the circle map: rotation number rises in flat mode-locking steps.
Rotation number over the circle-map parameter plane; the uniform wedges are frequency-locked Arnold tongues.
Zoomed view of the devil's staircase, showing self-similar mode-locking steps at finer scale.
Parameter-plane survey of the coupled logistic map showing symmetry breaking and regions of chaos.
Attractor portraits of the coupled logistic map along its symmetry-breaking route.
Basin of attraction of the coupled logistic map, showing a striped boundary between two mirror-image cycles.
Torus doubling in Map (I): a simple torus, a doubled torus, and the collapse to chaos.
Torus-doubling cascade of Map (IV): a fourfold torus, an eightfold torus, and chaos.
Lyapunov spectrum of Map (IV), showing near-zero exponents through the doubling cascade until chaos sets in.
Attractor portraits of the delayed logistic map, tracing a torus growing, locking, and breaking down into chaos.
Lyapunov exponents of the delayed logistic map, marking the shift from quasiperiodic motion to chaos.
Close-up sequence of attractors resolving the delayed logistic map's transition from locking to chaos.
Lyapunov spectrum of the coupled delayed logistic map, distinguishing quasiperiodic motion, locking, and chaos.
Attractor projections of the coupled delayed logistic map moving through a resonance web, locking, and chaos.
Double devil's staircase of the modulated circle map, showing locking plateaus in both rotation numbers.
Zoomed view of the double devil's staircase, showing locking plateaus in finer detail.
A smooth torus in the delayed logistic map develops wrinkles at finer and finer scales as it fractalizes.
Correlation dimension of the attractor rising from a smooth torus toward a fractalized one.
Spacetime diagrams of spatiotemporal intermittency in three coupled map lattice models.
Co-moving Lyapunov exponent of the logistic coupled map lattice, whose zero crossings mark propagation speeds.
Spatial correlation decay and finite-size convergence in the logistic coupled map lattice.
Activity map of the logistic coupled map lattice across its nonlinearity and coupling parameters.
Space-amplitude snapshots of the logistic coupled map lattice in five representative pattern regimes.
Mean-square deviation of the mean field in a globally coupled map, failing to shrink with system size.
Distribution of the mean field in a globally coupled map, with variance that does not narrow as system size grows.
Cluster states in a globally coupled map, showing a partially ordered regime.
Collective Lyapunov exponent of the mean field in a globally coupled map, marking collective chaos.
The 0-1 test statistic for the logistic map, near 0 in periodic windows and near 1 in chaotic bands.
SALI time series for the coupled delayed logistic map across regimes from quasiperiodic motion to chaos.
Permutation entropy for the logistic and delayed logistic maps, low in regular windows and high in irregular ones.
Complexity-entropy plane locations for the logistic and delayed logistic maps.
Recurrence quantification measures tracking the delayed logistic map's torus-to-chaos transition.
Type-I intermittency: a tangent-bifurcation channel, laminar-length statistics, and a Lorenz reinjection channel.
On-off intermittency near a blowout onset, showing laminar epochs, bursts, and their statistics.
Type-II intermittency, illustrated with a normal-form spiral orbit and its laminar-length statistics.
Type-III intermittency: a period-doubling return map, escape episodes, and reinjection statistics.
Spatiotemporal intermittency in a coupled map lattice: turbulent-fraction onset and an exponential laminar cluster-size tail.