dynachaos — reproduction gallery

Numerical reproductions of dynamical systems and chaos phenomena. back to the paper · View on GitHub

Circle Map: Devil's Staircase and Arnold Tongues

Devil's staircase of the circle map: rotation number rises in flat mode-locking steps.
devils_staircase.png

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.
arnold_tongues.png

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.
staircase_zoom.png

Zoomed view of the devil's staircase, showing self-similar mode-locking steps at finer scale.

Transition to Chaos in Coupled Maps

Parameter-plane survey of the coupled logistic map showing symmetry breaking and regions of chaos.
phase_diagram.png

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.
attractors.png

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.
basins.png

Basin of attraction of the coupled logistic map, showing a striped boundary between two mirror-image cycles.

Torus Doubling

Torus doubling in Map (I): a simple torus, a doubled torus, and the collapse to chaos.
map_I_attractors.png

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.
map_IV_attractors.png

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.
map_IV_lyapunov.png

Lyapunov spectrum of Map (IV), showing near-zero exponents through the doubling cascade until chaos sets in.

Quasiperiodic Oscillation and Phase Locking

Attractor portraits of the delayed logistic map, tracing a torus growing, locking, and breaking down into chaos.
attractors.png

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.
lyapunov_vs_D.png

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.
locking_sequence.png

Close-up sequence of attractors resolving the delayed logistic map's transition from locking to chaos.

Three-Torus and the Double Staircase

Lyapunov spectrum of the coupled delayed logistic map, distinguishing quasiperiodic motion, locking, and chaos.
lyapunov_vs_DB.png

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.
xz_projections.png

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.
double_staircase.png

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.
double_staircase_zoom.png

Zoomed view of the double devil's staircase, showing locking plateaus in finer detail.

Torus Fractalization

A smooth torus in the delayed logistic map develops wrinkles at finer and finer scales as it fractalizes.
fractal_attractors.png

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.
correlation_dimension.png

Correlation dimension of the attractor rising from a smooth torus toward a fractalized one.

Spatiotemporal Intermittency

Spacetime diagrams of spatiotemporal intermittency in three coupled map lattice models.
spacetime_diagrams.png

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.
comoving_lyapunov.png

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.
correlation_decay.png

Spatial correlation decay and finite-size convergence in the logistic coupled map lattice.

Pattern Dynamics in Coupled Map Lattices

Activity map of the logistic coupled map lattice across its nonlinearity and coupling parameters.
phase_diagram.png

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.
space_amplitude.png

Space-amplitude snapshots of the logistic coupled map lattice in five representative pattern regimes.

Globally Coupled Maps

Mean-square deviation of the mean field in a globally coupled map, failing to shrink with system size.
gcm_msd.png

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.
gcm_distribution.png

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.
gcm_clusters.png

Cluster states in a globally coupled map, showing a partially ordered regime.

Largest Lyapunov exponent of the site dynamics of a globally coupled map against the nonlinearity, for three system sizes.
gcm_lyapunov_max.png

Largest Lyapunov exponent of the site dynamics of a globally coupled map against the nonlinearity, for three system sizes.

Finite-size Lyapunov exponent of the mean field of a heterogeneous globally coupled map: the microscopic exponent below a scale that shrinks with system size, a slower collective exponent above it.
gcm_collective_fsle.png

Finite-size Lyapunov exponent of the mean field of a heterogeneous globally coupled map: the microscopic exponent below a scale that shrinks with system size, a slower collective exponent above it.

Chaos Diagnostics

The 0-1 test statistic for the logistic map, near 0 in periodic windows and near 1 in chaotic bands.
test01_sweep.png

The 0-1 test statistic for the logistic map, near 0 in periodic windows and near 1 in chaotic bands.

Permutation entropy for the logistic and delayed logistic maps, low in regular windows and high in irregular ones.
permutation_entropy.png

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.
complexity_entropy_plane.png

Complexity-entropy plane locations for the logistic and delayed logistic maps.

Recurrence quantification measures tracking the delayed logistic map's torus-to-chaos transition.
rqa_measures.png

Recurrence quantification measures tracking the delayed logistic map's torus-to-chaos transition.

SALI and GALI_2 to GALI_4 over 10^6 iterates for a torus, a partially locked orbit and a chaotic orbit of the coupled delayed logistic map, on a logarithmic iteration axis.
alignment_three_torus.png

SALI and GALI_2 to GALI_4 over 10^6 iterates for a torus, a partially locked orbit and a chaotic orbit of the coupled delayed logistic map, on a logarithmic iteration axis.

ROC curves of the 0-1 test, permutation entropy and the RQA measures DET and LAM against the sign of the largest Lyapunov exponent, for the logistic, delayed logistic and coupled delayed logistic maps.
agreement_roc.png

ROC curves of the 0-1 test, permutation entropy and the RQA measures DET and LAM against the sign of the largest Lyapunov exponent, for the logistic, delayed logistic and coupled delayed logistic maps.

Routes to Intermittency

Type-I intermittency: a tangent-bifurcation channel, laminar-length statistics, and a Lorenz reinjection channel.
type_i_intermittency.png

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.
on_off_intermittency.png

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_ii_intermittency.png

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.
type_iii_intermittency.png

Type-III intermittency: a period-doubling return map, escape episodes, and reinjection statistics.

Spatiotemporal intermittency in a coupled map lattice: turbulent-fraction onset and power-law, exponential and truncated power-law fits to the laminar-length tail at three couplings.
sti_spine.png

Spatiotemporal intermittency in a coupled map lattice: turbulent-fraction onset and power-law, exponential and truncated power-law fits to the laminar-length tail at three couplings.