systems.builtin.deterministic.discrete.HenonMap

systems.builtin.deterministic.discrete.HenonMap(*args, **kwargs)

The Hénon Map: Paradigm of 2D dissipative chaos and strange attractors.

Physical System:

The Hénon map was introduced by Michel Hénon (1976) as a simplified model for the Poincaré section of the Lorenz equations. It represents a generic quadratic area-contracting map that captures essential features of dissipative chaos.

Mathematical Form: The defining equations are remarkably simple:

x[n+1] = 1 - a·x[n]² + y[n]
y[n+1] = b·x[n]

Despite this simplicity, the map exhibits: - Chaotic attractors (strange attractors) - Period-doubling cascades - Fractal basin boundaries - Sensitive dependence on initial conditions - Homoclinic bifurcations

Physical Interpretation: While originally a mathematical abstraction, the Hénon map can model: 1. Poincaré sections of 3D continuous flows 2. Population dynamics with two age classes 3. Chemical reactions with two coupled species 4. Economic models with production and capital 5. Fluid mixing and turbulence onset

The key feature is DISSIPATION: phase space volume contracts, leading to attractors (unlike conservative Hamiltonian systems).

Dissipation and Attractors: The Jacobian determinant is: det(J) = -b

For |b| < 1, the map is area-contracting: - Phase space volume shrinks by factor |b| each iteration - Asymptotically, dynamics confined to attractor - Attractor has dimension < 2 (fractal)

This contrasts with: - Logistic map (1D): Dimension 0 or 1 attractors - Standard map (conservative): No attractors, volume-preserving - Hénon map (2D dissipative): Dimension ~1.26 attractor (fractal!)

State Space:

State: x[n] = [x[n], y[n]] First coordinate: - x: “Position” or primary variable * Unbounded in principle: -∞ < x < ∞ * Typically confined to bounded region by attractor * For standard parameters: -1.5 < x < 1.5

Second coordinate:
- y: "Velocity" or secondary variable
  * Coupled to previous x value: y[n] = b·x[n-1]
  * Acts as delayed feedback
  * For standard parameters: -0.4 < y < 0.4

Phase Space Structure: The (x, y) plane contains: - Attractors (where orbits converge) - Basins of attraction (initial conditions leading to each attractor) - Basin boundaries (separating different basins) - Saddle points and unstable manifolds

Control: u[n] (optional, for controlled Hénon map) - Can represent: * Perturbation to x dynamics: x[n+1] = 1 - a·x[n]² + y[n] + u[n] * External forcing or feedback control * Stabilization of unstable periodic orbits - Standard map: u = 0 (autonomous)

Output: y[n] = [x[n], y[n]] - Full state measurement (both coordinates) - In practice, may observe only x (partial observation) - Time-delay embedding can reconstruct attractor from x alone

Dynamics:

The Hénon map is defined by:

x[n+1] = 1 - a·x[n]² + y[n]
y[n+1] = b·x[n]

Key Mathematical Properties:

Quadratic nonlinearity:
- The x² term creates folding in phase space
- Enables period-doubling and chaos
- Simplest polynomial map with strange attractor
Area contraction (dissipation): Jacobian: J = | -2ax 1 | | b 0 |

det(J) = -b (constant, independent of state!)

For |b| < 1:
- Area contracts by factor |b| per iteration
- Volume of phase space → 0 as n → ∞
- Dynamics attracted to lower-dimensional set
Invertibility: The map has a unique inverse (for b ≠ 0):

x[n] = y[n+1]/b y[n] = x[n+1] - 1 + a·(y[n+1]/b)²

This allows backward iteration (unlike logistic map).
Time-reversal symmetry (special case): For b = -1, map is time-reversible (area-preserving) This is the limit connecting to Hamiltonian systems.

Parameters:

a : float, default=1.4 Nonlinearity parameter (controls folding strength)

**Physical Meaning:**
- Controls strength of quadratic nonlinearity
- Larger |a| → stronger folding
- Determines attractor complexity

**Parameter Regimes:**

**a = 0:**
- Linear map: x[n+1] = 1 + y[n], y[n+1] = b·x[n]
- Simple dynamics, no chaos
- All orbits converge or diverge simply

**0 < a < 0.368:**
- Stable fixed point
- All trajectories converge to single point
- No oscillations or complexity

**a ≈ 0.368:**
- Flip bifurcation (period-doubling)
- Fixed point becomes unstable
- Period-2 orbit appears

**0.368 < a < 1.06:**
- Periodic attractors (period-2, 4, 8, ...)
- Period-doubling cascade
- Feigenbaum scaling (universal constants)
- Resembles logistic map behavior

**a ≈ 1.06:**
- Onset of chaos
- Accumulation point of period-doubling cascade
- Transition from periodic to chaotic

**1.06 < a < 1.4:**
- Chaotic attractor (strange attractor)
- Positive Lyapunov exponent
- Fractal structure
- Sensitive dependence on initial conditions

**a = 1.4 (canonical value):**
- Classic Hénon attractor
- Well-studied strange attractor
- Fractal dimension ≈ 1.26
- Hausdorff dimension ≈ 1.261
- Correlation dimension ≈ 1.25

**a > 1.4:**
- Attractor may fragment or disappear
- Escape to infinity possible
- Multiple attractors may coexist
- Complex basin boundaries

**a >> 1:**
- Most orbits escape to infinity
- Bounded attractors rare
- Chaotic scattering regime

b : float, default=0.3 Dissipation parameter (controls area contraction)

**Physical Meaning:**
- Controls phase space volume contraction rate
- |det(J)| = |b| = volume contraction factor
- Larger |b| → slower contraction, less dissipation
- Smaller |b| → faster contraction, more dissipation

**Special Values:**

**b = 0:**
- Extreme dissipation (collapse to 1D)
- y coordinate decouples: y[n+1] = 0
- Effectively becomes 1D logistic-like map
- Loses 2D structure

**0 < b < 1:**
- Standard dissipative regime
- Area-contracting (det(J) < 1)
- Attractors exist (typical case)
- b = 0.3 is canonical value

**b = 1:**
- Borderline case (area-preserving)
- Conservative limit
- No attractors (like Standard Map)
- Unlikely for physical dissipative systems

**b = -1:**
- Area-preserving AND time-reversible
- Special mathematical interest
- Connects to Hamiltonian dynamics
- Not typical for dissipative systems

**-1 < b < 0:**
- Orientation-reversing
- Still dissipative (|b| < 1)
- Can have strange attractors
- Less commonly studied

**|b| > 1:**
- Area-expanding
- No bounded attractors
- Orbits typically escape
- Not physical for dissipative systems

dt : float, default=1.0 Time step between iterations - Usually normalized to 1 - Represents sampling period - Only affects interpretation, not dynamics

Equilibria and Fixed Points:

Fixed points satisfy: x* = 1 - a·x² + y y* = b·x*

Substituting second into first: x* = 1 - a·x² + b·x a·x² - (b-1)·x + 1 = 0

Solutions (for standard b = 0.3, a varies): x* = [(b-1) ± √((b-1)² - 4a)] / (2a)

Fixed points exist if: (b-1)² ≥ 4a

For b = 0.3: 0.49 ≥ 4a a ≤ 0.1225

For a > 0.1225, fixed points are complex (no real equilibria).

Typical scenario (a = 1.4, b = 0.3): - No real fixed points - System has no equilibria - All dynamics are transient or on attractor - Attractor is only long-term behavior

Stability of fixed points: When fixed points exist, linearization gives: λ₁, λ₂ = eigenvalues of Jacobian

Fixed point is stable if |λ₁|, |λ₂| < 1.

Strange Attractor:

The Hénon attractor (a = 1.4, b = 0.3) is a STRANGE ATTRACTOR:

Definition: A strange attractor is an attracting set that: 1. Is an attractor (nearby trajectories approach it) 2. Is strange (exhibits sensitive dependence on IC) 3. Has fractal structure (non-integer dimension)

Properties of Hénon attractor: - Fractal dimension: D ≈ 1.26 (between 1D and 2D) - Correlation dimension: d_c ≈ 1.25 - Lyapunov exponents: λ₁ ≈ 0.42, λ₂ ≈ -1.62 - Lyapunov dimension: D_L = 1 + λ₁/|λ₂| ≈ 1.26 - Information dimension: Similar to fractal dimension

Visual structure: - Appears as collection of parallel curves (bands) - Actually infinitely many curves (fractal layering) - Self-similar structure at all scales - Gaps between bands contain saddle points

Unstable manifolds: - Curves connecting unstable fixed points/periodic orbits - Create complicated tangles (homoclinic tangles) - Fold back on themselves infinitely - Define skeleton of attractor

Chaos Characterization:

Lyapunov Exponents: For canonical Hénon map (a = 1.4, b = 0.3): λ₁ ≈ 0.42 (positive → chaos) λ₂ ≈ -1.62 (negative → dissipation) λ₁ + λ₂ = ln|b| ≈ -1.20 (contraction rate)

The positive Lyapunov exponent confirms: - Exponential divergence of nearby trajectories - Sensitive dependence on initial conditions - Chaotic dynamics (deterministic but unpredictable)

Basin of Attraction: The set of initial conditions leading to the attractor. For Hénon map: - Basin is typically simply connected (single piece) - But can have fractal boundary - Some parameters → multiple attractors with intertwined basins

Periodic Orbits: Embedded in attractor are infinitely many unstable periodic orbits: - Period-1, 2, 3, 4, … orbits - All are unstable (saddles) - Dense in the attractor - Can be used for control (OGY method)

Bifurcations and Route to Chaos:

As ‘a’ increases (with b = 0.3 fixed), the Hénon map undergoes:

1. Fixed point stability loss (a ≈ 0.37): - Flip bifurcation (period-doubling) - Stable fixed point → period-2 orbit

2. Period-doubling cascade (0.37 < a < 1.06): - Period-2 → period-4 → period-8 → … - Geometric convergence (Feigenbaum constants) - δ ≈ 4.669… (universal) - α ≈ 2.502… (universal)

3. Onset of chaos (a ≈ 1.06): - Accumulation point of cascade - Lyapunov exponent becomes positive - Attractor dimension becomes fractal

4. Fully developed chaos (1.06 < a < 1.4): - Strange attractor forms - Sensitive dependence maximizes - Periodic windows may appear

5. Attractor crisis (a > 1.4): - Attractor may collide with unstable manifold - Sudden destruction or enlargement - Possible escape to infinity

This sequence is the period-doubling route to chaos, one of three universal routes (along with intermittency and quasi-periodicity).

Control Objectives:

1. Chaos Control (OGY Method): Goal: Stabilize unstable periodic orbits embedded in attractor Method: Apply small parameter perturbations - Locate periodic orbit (e.g., period-1) - Compute local stable/unstable directions - Apply control when trajectory near orbit - Perturbation: u[n] = -K·(x[n] - x_orbit)

2. Chaos Synchronization: Goal: Make two Hénon maps follow same trajectory Applications: Secure communication, pattern recognition Method: Couple systems or drive-response configuration

3. Targeting: Goal: Steer trajectory to specific location on attractor Useful for optimization and search

4. Anti-control (Chaotification): Goal: Make periodic system chaotic Applications: Mixing, encryption, avoiding resonance

5. Basin of attraction enlargement: Goal: Expand region attracting to desired attractor Important when multiple attractors coexist

Numerical Considerations:

Attractor Visualization: Standard method: 1. Choose initial condition (often random) 2. Discard first ~1000 iterations (transient) 3. Plot next 10,000+ points 4. Result shows attractor structure

Transient Behavior: - First iterations may not reflect attractor - Discard typically 100-1000 initial points - Longer transients near bifurcations

Numerical Precision: - Double precision (64-bit) sufficient - Chaos amplifies roundoff errors exponentially - Long-term trajectories unreliable (>1000 iterations) - Statistical properties converge faster than trajectories

Basin of Attraction Computation: - Grid initial conditions in (x, y) plane - Iterate each until convergence or escape - Color-code by attractor reached - Reveals basin structure and boundaries

Dimension Estimation: Several algorithms available: - Box-counting dimension (covers attractor with boxes) - Correlation dimension (uses point correlations) - Lyapunov dimension (from Lyapunov exponents) - Information dimension (uses entropies)

All give similar values (~1.26) for canonical Hénon attractor.

Example Usage:

Create canonical Hénon map

system = HenonMap(a=1.4, b=0.3)

Visualize strange attractor

x0 = np.array([0.1, 0.1]) result = system.simulate( … x0=x0, … u_sequence=None, … n_steps=10000 … )

Plot attractor (discard transient)

import plotly.graph_objects as go states_attractor = result[‘states’][1000:, :] # Skip first 1000

fig = go.Figure() fig.add_trace(go.Scatter( … x=states_attractor[:, 0], … y=states_attractor[:, 1], … mode=‘markers’, … marker=dict(size=1, color=‘blue’, opacity=0.5), … name=‘Hénon Attractor’ … ))

fig.update_layout( … title=f’Hénon Attractor (a={system.a}, b={system.b})‘, … xaxis_title=’x’, … yaxis_title=‘y’, … width=800, … height=600, … plot_bgcolor=‘white’ … ) fig.show()

Compute Lyapunov exponents

lyap_exp = system.compute_lyapunov_exponents( … x0=np.array([0.1, 0.1]), … n_iterations=50000 … ) print(f”Lyapunov exponents: λ₁ = {lyap_exp[0]:.4f}, λ₂ = {lyap_exp[1]:.4f}“) print(f”Sum: λ₁ + λ₂ = {sum(lyap_exp):.4f} (should ≈ ln|b| = {np.log(abs(system.b)):.4f})“)

if lyap_exp[0] > 0: … print(“System is CHAOTIC”) … # Estimate attractor dimension … D_L = 1 + lyap_exp[0] / abs(lyap_exp[1]) … print(f”Lyapunov dimension: D_L ≈ {D_L:.3f}“)

Sensitive dependence on initial conditions

x0_a = np.array([0.1, 0.1]) x0_b = np.array([0.1 + 1e-10, 0.1]) # Tiny difference

result_a = system.simulate(x0_a, None, n_steps=50) result_b = system.simulate(x0_b, None, n_steps=50)

divergence = np.linalg.norm( … result_a[‘states’] - result_b[‘states’], … axis=1 … )

fig = go.Figure() fig.add_trace(go.Scatter( … x=np.arange(51), … y=divergence, … mode=‘lines+markers’, … name=‘Distance’ … )) fig.update_yaxes(type=‘log’) fig.update_layout( … title=‘Exponential Divergence (Sensitive Dependence)’, … xaxis_title=‘Iteration n’, … yaxis_title=‘||x_a - x_b||’, … ) fig.show()

Generate bifurcation diagram (varying ‘a’)

bifurcation_data = system.generate_bifurcation_diagram( … parameter=‘a’, … param_range=(0.8, 1.5), … n_points=500, … n_transient=500, … n_samples=200 … )

fig = go.Figure() fig.add_trace(go.Scatter( … x=bifurcation_data[‘param’], … y=bifurcation_data[‘x’], … mode=‘markers’, … marker=dict(size=0.5, color=‘black’), … name=‘Bifurcation’ … )) fig.update_layout( … title=‘Hénon Map Bifurcation Diagram’, … xaxis_title=‘a (nonlinearity parameter)’, … yaxis_title=’x*’, … width=1000, … height=600 … ) fig.show()

Explore basin of attraction

basin_data = system.compute_basin_of_attraction( … x_range=(-2, 2), … y_range=(-2, 2), … resolution=400, … max_iterations=100 … )

fig = go.Figure(data=go.Heatmap( … z=basin_data[‘converged’], … x=basin_data[‘x_grid’][0, :], … y=basin_data[‘y_grid’][:, 0], … colorscale=‘RdBu’, … showscale=True … )) fig.update_layout( … title=‘Basin of Attraction (blue=converges, red=escapes)’, … xaxis_title=‘x₀’, … yaxis_title=‘y₀’, … width=800, … height=800 … ) fig.show()

Compare multiple parameter values

from plotly.subplots import make_subplots

a_values = [0.9, 1.1, 1.3, 1.4] fig = make_subplots( … rows=2, cols=2, … subplot_titles=[f’a = {a}’ for a in a_values] … )

for idx, a_val in enumerate(a_values): … row = idx // 2 + 1 … col = idx % 2 + 1 … … sys_temp = HenonMap(a=a_val, b=0.3) … res_temp = sys_temp.simulate( … x0=np.array([0.1, 0.1]), … u_sequence=None, … n_steps=5000 … ) … … states_plot = res_temp[‘states’][1000:, :] … … fig.add_trace( … go.Scatter( … x=states_plot[:, 0], … y=states_plot[:, 1], … mode=‘markers’, … marker=dict(size=0.5), … showlegend=False … ), … row=row, col=col … )

fig.update_xaxes(title_text=‘x’) fig.update_yaxes(title_text=‘y’) fig.update_layout( … title_text=‘Hénon Map: Transition to Chaos’, … height=800, … width=1000 … ) fig.show()

Estimate correlation dimension

corr_dim = system.estimate_correlation_dimension( … x0=np.array([0.1, 0.1]), … n_points=5000, … n_transient=1000 … ) print(f”Correlation dimension: d_c ≈ {corr_dim:.3f}“) print(f”(Theoretical for Hénon attractor: ≈ 1.25)“)

Physical Insights:

Why 2D is Different from 1D: - 1D maps (logistic): Attractors are points or periodic cycles - 2D maps (Hénon): Attractors can be STRANGE (fractal curves) - Dimension 1 < D < 2 possible (impossible in 1D!) - Richer dynamics: homoclinic tangles, cantori, etc.

Dissipation Creates Attractors: - Area contraction (det(J) = -b with |b| < 1) crucial - Without dissipation: no attractors (like Standard Map) - Dissipation “squeezes” phase space onto lower-dimensional set - Competition: folding (nonlinearity) vs contraction (dissipation)

Folding Mechanism: The x² term creates quadratic folding: - Phase space is stretched along one direction - Then folded back on itself - Repeated stretching + folding → fractal structure - This is the “baker’s transformation” in disguise

Connection to Continuous Systems: Hénon map approximates Poincaré section of: - Lorenz system (chaotic convection) - Duffing oscillator (forced nonlinear pendulum) - Other 3D continuous flows with strange attractors

Discrete map captures essential features while being: - Easier to compute (no integration) - Easier to analyze (algebraic not differential) - Still exhibiting all chaos phenomena

Universality: The period-doubling cascade in Hénon map obeys same scaling laws (Feigenbaum constants) as logistic map, forced pendulum, and countless other systems. This is UNIVERSAL - a profound discovery showing chaos theory has quantitative predictions!

Fractal Basin Boundaries: When multiple attractors coexist, basin boundaries can be fractal: - Impossibly complicated boundary structure - Arbitrarily close ICs can lead to different attractors - “Riddled basins” - attractor’s basin has holes everywhere - Final-state uncertainty for practical systems

Common Pitfalls:

Not discarding transient: First hundreds of iterations may not show attractor Always skip initial points before plotting
Wrong parameter values: Using a > 1.5 or b > 1 can cause escape to infinity Canonical values (a=1.4, b=0.3) are well-behaved
Insufficient points: Need thousands of points to see attractor structure Fractal detail requires high point density
Confusing with Hamiltonian systems: Hénon map is DISSIPATIVE (area-contracting) Has attractors (unlike Standard Map) Lyapunov exponents sum to negative value
Expecting simple fixed points: For typical parameters, no real fixed points exist All dynamics are transient or on attractor Cannot linearize around equilibrium (none exists!)
Ignoring basin of attraction: Not all ICs converge to same attractor Some ICs escape to infinity Basin can have complex structure

Extensions and Variations:

Generalized Hénon map: x[n+1] = c - a·x[n]^p + y[n] y[n+1] = b·x[n] Different exponents p change dynamics
Lozi map: x[n+1] = 1 - a·|x[n]| + y[n] y[n+1] = b·x[n] Piecewise-linear version (easier analysis)
Coupled Hénon maps: Network of interacting Hénon maps Models spatiotemporal chaos
3D Hénon map: Extension to three dimensions Hyperchaos (multiple positive Lyapunov exponents)
Delayed Hénon map: Include time delays in coupling Richer bifurcation structure
Noisy Hénon map: Add stochastic perturbations Studies chaos-noise interaction

Attributes

Name	Description
a	Nonlinearity parameter.
area_contraction_rate	Phase space area contraction rate (\|det(J)\| = \|b\|).
b	Dissipation parameter.

Methods

Name	Description
classify_regime	Classify dynamical regime based on parameters.
compute_basin_of_attraction	Compute basin of attraction.
compute_jacobian	Compute Jacobian matrix at state x.
compute_lyapunov_exponents	Compute both Lyapunov exponents using QR decomposition method.
define_system	Define Hénon map dynamics.
estimate_correlation_dimension	Estimate correlation dimension using Grassberger-Procaccia algorithm.
generate_bifurcation_diagram	Generate bifurcation diagram by varying a parameter.
setup_equilibria	Set up fixed points if they exist.

classify_regime

systems.builtin.deterministic.discrete.HenonMap.classify_regime()

Classify dynamical regime based on parameters.

Returns

Name	Type	Description
	str	Regime description

Examples

>>> system = HenonMap(a=1.4, b=0.3)
>>> print(system.classify_regime())
'strange_attractor'

compute_basin_of_attraction

systems.builtin.deterministic.discrete.HenonMap.compute_basin_of_attraction(
    x_range=(-2, 2),
    y_range=(-2, 2),
    resolution=400,
    max_iterations=100,
    escape_radius=1000.0,
)

Compute basin of attraction.

Grids initial conditions and determines which converge to attractor vs escape to infinity.

Parameters

Name	Type	Description	Default
x_range	tuple	Range of x initial conditions	`(-2, 2)`
y_range	tuple	Range of y initial conditions	`(-2, 2)`
resolution	int	Grid resolution (points per dimension)	`400`
max_iterations	int	Iterations before declaring convergence/escape	`100`
escape_radius	float	Radius beyond which orbit considered escaped	`1000.0`

Returns

Name	Type	Description
	dict	Dictionary with: - ‘x_grid’: Meshgrid of x values - ‘y_grid’: Meshgrid of y values - ‘converged’: Boolean array (True=converged, False=escaped)

Examples

>>> system = HenonMap(a=1.4, b=0.3)
>>> basin = system.compute_basin_of_attraction(resolution=300)
>>>
>>> import plotly.graph_objects as go
>>> fig = go.Figure(data=go.Heatmap(
...     z=basin['converged'].astype(int),
...     x=basin['x_grid'][0, :],
...     y=basin['y_grid'][:, 0],
...     colorscale='RdBu'
... ))
>>> fig.show()

compute_jacobian

systems.builtin.deterministic.discrete.HenonMap.compute_jacobian(x)

Compute Jacobian matrix at state x.

For Hénon map: J = | -2ax 1 | | b 0 |

Parameters

Name	Type	Description	Default
x	np.ndarray	State [x, y]	required

Returns

Name	Type	Description
	np.ndarray	Jacobian matrix (2×2)

Notes

Determinant is constant: det(J) = -b (independent of state!) This gives constant area contraction rate.

compute_lyapunov_exponents

systems.builtin.deterministic.discrete.HenonMap.compute_lyapunov_exponents(
    x0,
    n_iterations=50000,
    n_transient=1000,
)

Compute both Lyapunov exponents using QR decomposition method.

Parameters

Name	Type	Description	Default
x0	np.ndarray	Initial state [x₀, y₀]	required
n_iterations	int	Number of iterations for averaging	`50000`
n_transient	int	Number of initial iterations to discard	`1000`

Returns

Name	Type	Description
	np.ndarray	Array of two Lyapunov exponents [λ₁, λ₂] λ₁: Largest (expansion rate) λ₂: Smallest (contraction rate)

Notes

For dissipative systems: λ₁ + λ₂ = ln|det(J)| = ln|b|

For canonical Hénon (a=1.4, b=0.3): λ₁ ≈ 0.42 (positive → chaos) λ₂ ≈ -1.62 (negative → dissipation) λ₁ + λ₂ ≈ -1.20 ≈ ln(0.3)

Examples

>>> system = HenonMap(a=1.4, b=0.3)
>>> lyap = system.compute_lyapunov_exponents(
...     x0=np.array([0.1, 0.1]),
...     n_iterations=100000
... )
>>> print(f"λ₁ = {lyap[0]:.4f}, λ₂ = {lyap[1]:.4f}")
>>> print(f"Sum = {sum(lyap):.4f} (should be ln|b| = {np.log(abs(system.b)):.4f})")

define_system

systems.builtin.deterministic.discrete.HenonMap.define_system(
    a=1.4,
    b=0.3,
    dt=1.0,
    use_controlled_version=False,
)

Define Hénon map dynamics.

Parameters

Name	Type	Description	Default
a	float	Nonlinearity parameter (controls folding)	`1.4`
b	float	Dissipation parameter (controls area contraction)	`0.3`
dt	float	Time step between iterations (usually 1.0)	`1.0`
use_controlled_version	bool	If True, adds control input u[n] to x dynamics	`False`

estimate_correlation_dimension

systems.builtin.deterministic.discrete.HenonMap.estimate_correlation_dimension(
    x0,
    n_points=5000,
    n_transient=1000,
    r_min=0.001,
    r_max=1.0,
    n_radii=50,
)

Estimate correlation dimension using Grassberger-Procaccia algorithm.

The correlation dimension d_c characterizes the fractal structure of the attractor.

Parameters

Name	Type	Description	Default
x0	np.ndarray	Initial state	required
n_points	int	Number of points on attractor to use	`5000`
n_transient	int	Points to discard as transient	`1000`
r_min	float	Minimum radius for correlation sum	`0.001`
r_max	float	Maximum radius	`1.0`
n_radii	int	Number of radii to test	`50`

Returns

Name	Type	Description
	float	Estimated correlation dimension

Notes

For canonical Hénon attractor: d_c ≈ 1.25

Examples

>>> system = HenonMap(a=1.4, b=0.3)
>>> d_c = system.estimate_correlation_dimension(
...     x0=np.array([0.1, 0.1]),
...     n_points=10000
... )
>>> print(f"Correlation dimension: {d_c:.3f}")

generate_bifurcation_diagram

systems.builtin.deterministic.discrete.HenonMap.generate_bifurcation_diagram(
    parameter='a',
    param_range=(0.8, 1.5),
    n_points=500,
    n_transient=500,
    n_samples=200,
    x0=None,
)

Generate bifurcation diagram by varying a parameter.

Parameters

Name	Type	Description	Default
parameter	str	Parameter to vary (‘a’ or ‘b’)	`'a'`
param_range	tuple	Range of parameter values (min, max)	`(0.8, 1.5)`
n_points	int	Number of parameter values to test	`500`
n_transient	int	Iterations to discard (transient)	`500`
n_samples	int	Points to sample after transient	`200`
x0	Optional[np.ndarray]	Initial condition (None = use [0.1, 0.1])	`None`

Returns

Name	Type	Description
	dict	Dictionary with: - ‘param’: Parameter values - ‘x’: x-coordinate values - ‘y’: y-coordinate values (optional)

Examples

>>> system = HenonMap()
>>> bifurc_data = system.generate_bifurcation_diagram(
...     parameter='a',
...     param_range=(0.8, 1.5),
...     n_points=1000
... )
>>>
>>> import plotly.graph_objects as go
>>> fig = go.Figure()
>>> fig.add_trace(go.Scatter(
...     x=bifurc_data['param'],
...     y=bifurc_data['x'],
...     mode='markers',
...     marker=dict(size=0.3, color='black')
... ))
>>> fig.update_layout(title='Hénon Bifurcation Diagram')
>>> fig.show()

setup_equilibria

systems.builtin.deterministic.discrete.HenonMap.setup_equilibria()

Set up fixed points if they exist.

Fixed points satisfy: x* = 1 - a·x² + y y* = b·x*

Substituting: a·x² - (b-1)·x + 1 = 0

Real solutions exist only if (b-1)² ≥ 4a.

# systems.builtin.deterministic.discrete.HenonMap { #cdesym.systems.builtin.deterministic.discrete.HenonMap } ```python systems.builtin.deterministic.discrete.HenonMap(*args, **kwargs) ``` The Hénon Map: Paradigm of 2D dissipative chaos and strange attractors. ## Physical System: {.doc-section .doc-section-physical-system} The Hénon map was introduced by Michel Hénon (1976) as a simplified model for the Poincaré section of the Lorenz equations. It represents a generic quadratic area-contracting map that captures essential features of dissipative chaos. **Mathematical Form:** The defining equations are remarkably simple: x[n+1] = 1 - a·x[n]² + y[n] y[n+1] = b·x[n] Despite this simplicity, the map exhibits: - Chaotic attractors (strange attractors) - Period-doubling cascades - Fractal basin boundaries - Sensitive dependence on initial conditions - Homoclinic bifurcations **Physical Interpretation:** While originally a mathematical abstraction, the Hénon map can model: 1. **Poincaré sections** of 3D continuous flows 2. **Population dynamics** with two age classes 3. **Chemical reactions** with two coupled species 4. **Economic models** with production and capital 5. **Fluid mixing** and turbulence onset The key feature is DISSIPATION: phase space volume contracts, leading to attractors (unlike conservative Hamiltonian systems). **Dissipation and Attractors:** The Jacobian determinant is: det(J) = -b For |b| < 1, the map is area-contracting: - Phase space volume shrinks by factor |b| each iteration - Asymptotically, dynamics confined to attractor - Attractor has dimension < 2 (fractal) This contrasts with: - Logistic map (1D): Dimension 0 or 1 attractors - Standard map (conservative): No attractors, volume-preserving - Hénon map (2D dissipative): Dimension ~1.26 attractor (fractal!) ## State Space: {.doc-section .doc-section-state-space} State: x[n] = [x[n], y[n]] First coordinate: - x: "Position" or primary variable * Unbounded in principle: -∞ < x < ∞ * Typically confined to bounded region by attractor * For standard parameters: -1.5 < x < 1.5 Second coordinate: - y: "Velocity" or secondary variable * Coupled to previous x value: y[n] = b·x[n-1] * Acts as delayed feedback * For standard parameters: -0.4 < y < 0.4 **Phase Space Structure:** The (x, y) plane contains: - Attractors (where orbits converge) - Basins of attraction (initial conditions leading to each attractor) - Basin boundaries (separating different basins) - Saddle points and unstable manifolds Control: u[n] (optional, for controlled Hénon map) - Can represent: * Perturbation to x dynamics: x[n+1] = 1 - a·x[n]² + y[n] + u[n] * External forcing or feedback control * Stabilization of unstable periodic orbits - Standard map: u = 0 (autonomous) Output: y[n] = [x[n], y[n]] - Full state measurement (both coordinates) - In practice, may observe only x (partial observation) - Time-delay embedding can reconstruct attractor from x alone ## Dynamics: {.doc-section .doc-section-dynamics} The Hénon map is defined by: x[n+1] = 1 - a·x[n]² + y[n] y[n+1] = b·x[n] **Key Mathematical Properties:** 1. **Quadratic nonlinearity:** - The x² term creates folding in phase space - Enables period-doubling and chaos - Simplest polynomial map with strange attractor 2. **Area contraction (dissipation):** Jacobian: J = | -2ax 1 | | b 0 | det(J) = -b (constant, independent of state!) For |b| < 1: - Area contracts by factor |b| per iteration - Volume of phase space → 0 as n → ∞ - Dynamics attracted to lower-dimensional set 3. **Invertibility:** The map has a unique inverse (for b ≠ 0): x[n] = y[n+1]/b y[n] = x[n+1] - 1 + a·(y[n+1]/b)² This allows backward iteration (unlike logistic map). 4. **Time-reversal symmetry (special case):** For b = -1, map is time-reversible (area-preserving) This is the limit connecting to Hamiltonian systems. ## Parameters: {.doc-section .doc-section-parameters} a : float, default=1.4 Nonlinearity parameter (controls folding strength) **Physical Meaning:** - Controls strength of quadratic nonlinearity - Larger |a| → stronger folding - Determines attractor complexity **Parameter Regimes:** **a = 0:** - Linear map: x[n+1] = 1 + y[n], y[n+1] = b·x[n] - Simple dynamics, no chaos - All orbits converge or diverge simply **0 < a < 0.368:** - Stable fixed point - All trajectories converge to single point - No oscillations or complexity **a ≈ 0.368:** - Flip bifurcation (period-doubling) - Fixed point becomes unstable - Period-2 orbit appears **0.368 < a < 1.06:** - Periodic attractors (period-2, 4, 8, ...) - Period-doubling cascade - Feigenbaum scaling (universal constants) - Resembles logistic map behavior **a ≈ 1.06:** - Onset of chaos - Accumulation point of period-doubling cascade - Transition from periodic to chaotic **1.06 < a < 1.4:** - Chaotic attractor (strange attractor) - Positive Lyapunov exponent - Fractal structure - Sensitive dependence on initial conditions **a = 1.4 (canonical value):** - Classic Hénon attractor - Well-studied strange attractor - Fractal dimension ≈ 1.26 - Hausdorff dimension ≈ 1.261 - Correlation dimension ≈ 1.25 **a > 1.4:** - Attractor may fragment or disappear - Escape to infinity possible - Multiple attractors may coexist - Complex basin boundaries **a >> 1:** - Most orbits escape to infinity - Bounded attractors rare - Chaotic scattering regime b : float, default=0.3 Dissipation parameter (controls area contraction) **Physical Meaning:** - Controls phase space volume contraction rate - |det(J)| = |b| = volume contraction factor - Larger |b| → slower contraction, less dissipation - Smaller |b| → faster contraction, more dissipation **Special Values:** **b = 0:** - Extreme dissipation (collapse to 1D) - y coordinate decouples: y[n+1] = 0 - Effectively becomes 1D logistic-like map - Loses 2D structure **0 < b < 1:** - Standard dissipative regime - Area-contracting (det(J) < 1) - Attractors exist (typical case) - b = 0.3 is canonical value **b = 1:** - Borderline case (area-preserving) - Conservative limit - No attractors (like Standard Map) - Unlikely for physical dissipative systems **b = -1:** - Area-preserving AND time-reversible - Special mathematical interest - Connects to Hamiltonian dynamics - Not typical for dissipative systems **-1 < b < 0:** - Orientation-reversing - Still dissipative (|b| < 1) - Can have strange attractors - Less commonly studied **|b| > 1:** - Area-expanding - No bounded attractors - Orbits typically escape - Not physical for dissipative systems dt : float, default=1.0 Time step between iterations - Usually normalized to 1 - Represents sampling period - Only affects interpretation, not dynamics ## Equilibria and Fixed Points: {.doc-section .doc-section-equilibria-and-fixed-points} Fixed points satisfy: x* = 1 - a·x*² + y* y* = b·x* Substituting second into first: x* = 1 - a·x*² + b·x* a·x*² - (b-1)·x* + 1 = 0 Solutions (for standard b = 0.3, a varies): x* = [(b-1) ± √((b-1)² - 4a)] / (2a) **Fixed points exist if:** (b-1)² ≥ 4a For b = 0.3: 0.49 ≥ 4a a ≤ 0.1225 For a > 0.1225, fixed points are complex (no real equilibria). **Typical scenario (a = 1.4, b = 0.3):** - No real fixed points - System has no equilibria - All dynamics are transient or on attractor - Attractor is only long-term behavior **Stability of fixed points:** When fixed points exist, linearization gives: λ₁, λ₂ = eigenvalues of Jacobian Fixed point is stable if |λ₁|, |λ₂| < 1. ## Strange Attractor: {.doc-section .doc-section-strange-attractor} The Hénon attractor (a = 1.4, b = 0.3) is a STRANGE ATTRACTOR: **Definition:** A strange attractor is an attracting set that: 1. Is an attractor (nearby trajectories approach it) 2. Is strange (exhibits sensitive dependence on IC) 3. Has fractal structure (non-integer dimension) **Properties of Hénon attractor:** - **Fractal dimension:** D ≈ 1.26 (between 1D and 2D) - **Correlation dimension:** d_c ≈ 1.25 - **Lyapunov exponents:** λ₁ ≈ 0.42, λ₂ ≈ -1.62 - **Lyapunov dimension:** D_L = 1 + λ₁/|λ₂| ≈ 1.26 - **Information dimension:** Similar to fractal dimension **Visual structure:** - Appears as collection of parallel curves (bands) - Actually infinitely many curves (fractal layering) - Self-similar structure at all scales - Gaps between bands contain saddle points **Unstable manifolds:** - Curves connecting unstable fixed points/periodic orbits - Create complicated tangles (homoclinic tangles) - Fold back on themselves infinitely - Define skeleton of attractor ## Chaos Characterization: {.doc-section .doc-section-chaos-characterization} **Lyapunov Exponents:** For canonical Hénon map (a = 1.4, b = 0.3): λ₁ ≈ 0.42 (positive → chaos) λ₂ ≈ -1.62 (negative → dissipation) λ₁ + λ₂ = ln|b| ≈ -1.20 (contraction rate) The positive Lyapunov exponent confirms: - Exponential divergence of nearby trajectories - Sensitive dependence on initial conditions - Chaotic dynamics (deterministic but unpredictable) **Basin of Attraction:** The set of initial conditions leading to the attractor. For Hénon map: - Basin is typically simply connected (single piece) - But can have fractal boundary - Some parameters → multiple attractors with intertwined basins **Periodic Orbits:** Embedded in attractor are infinitely many unstable periodic orbits: - Period-1, 2, 3, 4, ... orbits - All are unstable (saddles) - Dense in the attractor - Can be used for control (OGY method) ## Bifurcations and Route to Chaos: {.doc-section .doc-section-bifurcations-and-route-to-chaos} As 'a' increases (with b = 0.3 fixed), the Hénon map undergoes: **1. Fixed point stability loss (a ≈ 0.37):** - Flip bifurcation (period-doubling) - Stable fixed point → period-2 orbit **2. Period-doubling cascade (0.37 < a < 1.06):** - Period-2 → period-4 → period-8 → ... - Geometric convergence (Feigenbaum constants) - δ ≈ 4.669... (universal) - α ≈ 2.502... (universal) **3. Onset of chaos (a ≈ 1.06):** - Accumulation point of cascade - Lyapunov exponent becomes positive - Attractor dimension becomes fractal **4. Fully developed chaos (1.06 < a < 1.4):** - Strange attractor forms - Sensitive dependence maximizes - Periodic windows may appear **5. Attractor crisis (a > 1.4):** - Attractor may collide with unstable manifold - Sudden destruction or enlargement - Possible escape to infinity This sequence is the **period-doubling route to chaos**, one of three universal routes (along with intermittency and quasi-periodicity). ## Control Objectives: {.doc-section .doc-section-control-objectives} **1. Chaos Control (OGY Method):** Goal: Stabilize unstable periodic orbits embedded in attractor Method: Apply small parameter perturbations - Locate periodic orbit (e.g., period-1) - Compute local stable/unstable directions - Apply control when trajectory near orbit - Perturbation: u[n] = -K·(x[n] - x_orbit) **2. Chaos Synchronization:** Goal: Make two Hénon maps follow same trajectory Applications: Secure communication, pattern recognition Method: Couple systems or drive-response configuration **3. Targeting:** Goal: Steer trajectory to specific location on attractor Useful for optimization and search **4. Anti-control (Chaotification):** Goal: Make periodic system chaotic Applications: Mixing, encryption, avoiding resonance **5. Basin of attraction enlargement:** Goal: Expand region attracting to desired attractor Important when multiple attractors coexist ## Numerical Considerations: {.doc-section .doc-section-numerical-considerations} **Attractor Visualization:** Standard method: 1. Choose initial condition (often random) 2. Discard first ~1000 iterations (transient) 3. Plot next 10,000+ points 4. Result shows attractor structure **Transient Behavior:** - First iterations may not reflect attractor - Discard typically 100-1000 initial points - Longer transients near bifurcations **Numerical Precision:** - Double precision (64-bit) sufficient - Chaos amplifies roundoff errors exponentially - Long-term trajectories unreliable (>1000 iterations) - Statistical properties converge faster than trajectories **Basin of Attraction Computation:** - Grid initial conditions in (x, y) plane - Iterate each until convergence or escape - Color-code by attractor reached - Reveals basin structure and boundaries **Dimension Estimation:** Several algorithms available: - Box-counting dimension (covers attractor with boxes) - Correlation dimension (uses point correlations) - Lyapunov dimension (from Lyapunov exponents) - Information dimension (uses entropies) All give similar values (~1.26) for canonical Hénon attractor. ## Example Usage: {.doc-section .doc-section-example-usage} >>> # Create canonical Hénon map >>> system = HenonMap(a=1.4, b=0.3) >>> >>> # Visualize strange attractor >>> x0 = np.array([0.1, 0.1]) >>> result = system.simulate( ... x0=x0, ... u_sequence=None, ... n_steps=10000 ... ) >>> >>> # Plot attractor (discard transient) >>> import plotly.graph_objects as go >>> states_attractor = result['states'][1000:, :] # Skip first 1000 >>> >>> fig = go.Figure() >>> fig.add_trace(go.Scatter( ... x=states_attractor[:, 0], ... y=states_attractor[:, 1], ... mode='markers', ... marker=dict(size=1, color='blue', opacity=0.5), ... name='Hénon Attractor' ... )) >>> >>> fig.update_layout( ... title=f'Hénon Attractor (a={system.a}, b={system.b})', ... xaxis_title='x', ... yaxis_title='y', ... width=800, ... height=600, ... plot_bgcolor='white' ... ) >>> fig.show() >>> >>> # Compute Lyapunov exponents >>> lyap_exp = system.compute_lyapunov_exponents( ... x0=np.array([0.1, 0.1]), ... n_iterations=50000 ... ) >>> print(f"Lyapunov exponents: λ₁ = {lyap_exp[0]:.4f}, λ₂ = {lyap_exp[1]:.4f}") >>> print(f"Sum: λ₁ + λ₂ = {sum(lyap_exp):.4f} (should ≈ ln|b| = {np.log(abs(system.b)):.4f})") >>> >>> if lyap_exp[0] > 0: ... print("System is CHAOTIC") ... # Estimate attractor dimension ... D_L = 1 + lyap_exp[0] / abs(lyap_exp[1]) ... print(f"Lyapunov dimension: D_L ≈ {D_L:.3f}") >>> >>> # Sensitive dependence on initial conditions >>> x0_a = np.array([0.1, 0.1]) >>> x0_b = np.array([0.1 + 1e-10, 0.1]) # Tiny difference >>> >>> result_a = system.simulate(x0_a, None, n_steps=50) >>> result_b = system.simulate(x0_b, None, n_steps=50) >>> >>> divergence = np.linalg.norm( ... result_a['states'] - result_b['states'], ... axis=1 ... ) >>> >>> fig = go.Figure() >>> fig.add_trace(go.Scatter( ... x=np.arange(51), ... y=divergence, ... mode='lines+markers', ... name='Distance' ... )) >>> fig.update_yaxes(type='log') >>> fig.update_layout( ... title='Exponential Divergence (Sensitive Dependence)', ... xaxis_title='Iteration n', ... yaxis_title='||x_a - x_b||', ... ) >>> fig.show() >>> >>> # Generate bifurcation diagram (varying 'a') >>> bifurcation_data = system.generate_bifurcation_diagram( ... parameter='a', ... param_range=(0.8, 1.5), ... n_points=500, ... n_transient=500, ... n_samples=200 ... ) >>> >>> fig = go.Figure() >>> fig.add_trace(go.Scatter( ... x=bifurcation_data['param'], ... y=bifurcation_data['x'], ... mode='markers', ... marker=dict(size=0.5, color='black'), ... name='Bifurcation' ... )) >>> fig.update_layout( ... title='Hénon Map Bifurcation Diagram', ... xaxis_title='a (nonlinearity parameter)', ... yaxis_title='x*', ... width=1000, ... height=600 ... ) >>> fig.show() >>> >>> # Explore basin of attraction >>> basin_data = system.compute_basin_of_attraction( ... x_range=(-2, 2), ... y_range=(-2, 2), ... resolution=400, ... max_iterations=100 ... ) >>> >>> fig = go.Figure(data=go.Heatmap( ... z=basin_data['converged'], ... x=basin_data['x_grid'][0, :], ... y=basin_data['y_grid'][:, 0], ... colorscale='RdBu', ... showscale=True ... )) >>> fig.update_layout( ... title='Basin of Attraction (blue=converges, red=escapes)', ... xaxis_title='x₀', ... yaxis_title='y₀', ... width=800, ... height=800 ... ) >>> fig.show() >>> >>> # Compare multiple parameter values >>> from plotly.subplots import make_subplots >>> >>> a_values = [0.9, 1.1, 1.3, 1.4] >>> fig = make_subplots( ... rows=2, cols=2, ... subplot_titles=[f'a = {a}' for a in a_values] ... ) >>> >>> for idx, a_val in enumerate(a_values): ... row = idx // 2 + 1 ... col = idx % 2 + 1 ... ... sys_temp = HenonMap(a=a_val, b=0.3) ... res_temp = sys_temp.simulate( ... x0=np.array([0.1, 0.1]), ... u_sequence=None, ... n_steps=5000 ... ) ... ... states_plot = res_temp['states'][1000:, :] ... ... fig.add_trace( ... go.Scatter( ... x=states_plot[:, 0], ... y=states_plot[:, 1], ... mode='markers', ... marker=dict(size=0.5), ... showlegend=False ... ), ... row=row, col=col ... ) >>> >>> fig.update_xaxes(title_text='x') >>> fig.update_yaxes(title_text='y') >>> fig.update_layout( ... title_text='Hénon Map: Transition to Chaos', ... height=800, ... width=1000 ... ) >>> fig.show() >>> >>> # Estimate correlation dimension >>> corr_dim = system.estimate_correlation_dimension( ... x0=np.array([0.1, 0.1]), ... n_points=5000, ... n_transient=1000 ... ) >>> print(f"Correlation dimension: d_c ≈ {corr_dim:.3f}") >>> print(f"(Theoretical for Hénon attractor: ≈ 1.25)") ## Physical Insights: {.doc-section .doc-section-physical-insights} **Why 2D is Different from 1D:** - 1D maps (logistic): Attractors are points or periodic cycles - 2D maps (Hénon): Attractors can be STRANGE (fractal curves) - Dimension 1 < D < 2 possible (impossible in 1D!) - Richer dynamics: homoclinic tangles, cantori, etc. **Dissipation Creates Attractors:** - Area contraction (det(J) = -b with |b| < 1) crucial - Without dissipation: no attractors (like Standard Map) - Dissipation "squeezes" phase space onto lower-dimensional set - Competition: folding (nonlinearity) vs contraction (dissipation) **Folding Mechanism:** The x² term creates quadratic folding: - Phase space is stretched along one direction - Then folded back on itself - Repeated stretching + folding → fractal structure - This is the "baker's transformation" in disguise **Connection to Continuous Systems:** Hénon map approximates Poincaré section of: - Lorenz system (chaotic convection) - Duffing oscillator (forced nonlinear pendulum) - Other 3D continuous flows with strange attractors Discrete map captures essential features while being: - Easier to compute (no integration) - Easier to analyze (algebraic not differential) - Still exhibiting all chaos phenomena **Universality:** The period-doubling cascade in Hénon map obeys same scaling laws (Feigenbaum constants) as logistic map, forced pendulum, and countless other systems. This is UNIVERSAL - a profound discovery showing chaos theory has quantitative predictions! **Fractal Basin Boundaries:** When multiple attractors coexist, basin boundaries can be fractal: - Impossibly complicated boundary structure - Arbitrarily close ICs can lead to different attractors - "Riddled basins" - attractor's basin has holes everywhere - Final-state uncertainty for practical systems ## Common Pitfalls: {.doc-section .doc-section-common-pitfalls} 1. **Not discarding transient:** First hundreds of iterations may not show attractor Always skip initial points before plotting 2. **Wrong parameter values:** Using a > 1.5 or b > 1 can cause escape to infinity Canonical values (a=1.4, b=0.3) are well-behaved 3. **Insufficient points:** Need thousands of points to see attractor structure Fractal detail requires high point density 4. **Confusing with Hamiltonian systems:** Hénon map is DISSIPATIVE (area-contracting) Has attractors (unlike Standard Map) Lyapunov exponents sum to negative value 5. **Expecting simple fixed points:** For typical parameters, no real fixed points exist All dynamics are transient or on attractor Cannot linearize around equilibrium (none exists!) 6. **Ignoring basin of attraction:** Not all ICs converge to same attractor Some ICs escape to infinity Basin can have complex structure ## Extensions and Variations: {.doc-section .doc-section-extensions-and-variations} 1. **Generalized Hénon map:** x[n+1] = c - a·x[n]^p + y[n] y[n+1] = b·x[n] Different exponents p change dynamics 2. **Lozi map:** x[n+1] = 1 - a·|x[n]| + y[n] y[n+1] = b·x[n] Piecewise-linear version (easier analysis) 3. **Coupled Hénon maps:** Network of interacting Hénon maps Models spatiotemporal chaos 4. **3D Hénon map:** Extension to three dimensions Hyperchaos (multiple positive Lyapunov exponents) 5. **Delayed Hénon map:** Include time delays in coupling Richer bifurcation structure 6. **Noisy Hénon map:** Add stochastic perturbations Studies chaos-noise interaction ## See Also: {.doc-section .doc-section-see-also} LogisticMap : 1D chaotic map StandardMap : 2D conservative (Hamiltonian) chaos Lorenz : Continuous 3D chaos (Hénon as Poincaré section) ## Attributes | Name | Description | | --- | --- | | [a](#cdesym.systems.builtin.deterministic.discrete.HenonMap.a) | Nonlinearity parameter. | | [area_contraction_rate](#cdesym.systems.builtin.deterministic.discrete.HenonMap.area_contraction_rate) | Phase space area contraction rate (\|det(J)\| = \|b\|). | | [b](#cdesym.systems.builtin.deterministic.discrete.HenonMap.b) | Dissipation parameter. | ## Methods | Name | Description | | --- | --- | | [classify_regime](#cdesym.systems.builtin.deterministic.discrete.HenonMap.classify_regime) | Classify dynamical regime based on parameters. | | [compute_basin_of_attraction](#cdesym.systems.builtin.deterministic.discrete.HenonMap.compute_basin_of_attraction) | Compute basin of attraction. | | [compute_jacobian](#cdesym.systems.builtin.deterministic.discrete.HenonMap.compute_jacobian) | Compute Jacobian matrix at state x. | | [compute_lyapunov_exponents](#cdesym.systems.builtin.deterministic.discrete.HenonMap.compute_lyapunov_exponents) | Compute both Lyapunov exponents using QR decomposition method. | | [define_system](#cdesym.systems.builtin.deterministic.discrete.HenonMap.define_system) | Define Hénon map dynamics. | | [estimate_correlation_dimension](#cdesym.systems.builtin.deterministic.discrete.HenonMap.estimate_correlation_dimension) | Estimate correlation dimension using Grassberger-Procaccia algorithm. | | [generate_bifurcation_diagram](#cdesym.systems.builtin.deterministic.discrete.HenonMap.generate_bifurcation_diagram) | Generate bifurcation diagram by varying a parameter. | | [setup_equilibria](#cdesym.systems.builtin.deterministic.discrete.HenonMap.setup_equilibria) | Set up fixed points if they exist. | ### classify_regime { #cdesym.systems.builtin.deterministic.discrete.HenonMap.classify_regime } ```python systems.builtin.deterministic.discrete.HenonMap.classify_regime() ``` Classify dynamical regime based on parameters. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|--------------------| | | str | Regime description | #### Examples {.doc-section .doc-section-examples} ```python >>> system = HenonMap(a=1.4, b=0.3) >>> print(system.classify_regime()) 'strange_attractor' ``` ### compute_basin_of_attraction { #cdesym.systems.builtin.deterministic.discrete.HenonMap.compute_basin_of_attraction } ```python systems.builtin.deterministic.discrete.HenonMap.compute_basin_of_attraction( x_range=(-2, 2), y_range=(-2, 2), resolution=400, max_iterations=100, escape_radius=1000.0, ) ``` Compute basin of attraction. Grids initial conditions and determines which converge to attractor vs escape to infinity. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |----------------|--------|------------------------------------------------|-----------| | x_range | tuple | Range of x initial conditions | `(-2, 2)` | | y_range | tuple | Range of y initial conditions | `(-2, 2)` | | resolution | int | Grid resolution (points per dimension) | `400` | | max_iterations | int | Iterations before declaring convergence/escape | `100` | | escape_radius | float | Radius beyond which orbit considered escaped | `1000.0` | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|-------------------------------------------------------------------------------------------------------------------------------------------------| | | dict | Dictionary with: - 'x_grid': Meshgrid of x values - 'y_grid': Meshgrid of y values - 'converged': Boolean array (True=converged, False=escaped) | #### Examples {.doc-section .doc-section-examples} ```python >>> system = HenonMap(a=1.4, b=0.3) >>> basin = system.compute_basin_of_attraction(resolution=300) >>> >>> import plotly.graph_objects as go >>> fig = go.Figure(data=go.Heatmap( ... z=basin['converged'].astype(int), ... x=basin['x_grid'][0, :], ... y=basin['y_grid'][:, 0], ... colorscale='RdBu' ... )) >>> fig.show() ``` ### compute_jacobian { #cdesym.systems.builtin.deterministic.discrete.HenonMap.compute_jacobian } ```python systems.builtin.deterministic.discrete.HenonMap.compute_jacobian(x) ``` Compute Jacobian matrix at state x. For Hénon map: J = | -2ax 1 | | b 0 | #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|------------|---------------|------------| | x | np.ndarray | State [x, y] | _required_ | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|------------|-----------------------| | | np.ndarray | Jacobian matrix (2×2) | #### Notes {.doc-section .doc-section-notes} Determinant is constant: det(J) = -b (independent of state!) This gives constant area contraction rate. ### compute_lyapunov_exponents { #cdesym.systems.builtin.deterministic.discrete.HenonMap.compute_lyapunov_exponents } ```python systems.builtin.deterministic.discrete.HenonMap.compute_lyapunov_exponents( x0, n_iterations=50000, n_transient=1000, ) ``` Compute both Lyapunov exponents using QR decomposition method. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------------|------------|-----------------------------------------|------------| | x0 | np.ndarray | Initial state [x₀, y₀] | _required_ | | n_iterations | int | Number of iterations for averaging | `50000` | | n_transient | int | Number of initial iterations to discard | `1000` | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|------------|-------------------------------------------------------------------------------------------------------| | | np.ndarray | Array of two Lyapunov exponents [λ₁, λ₂] λ₁: Largest (expansion rate) λ₂: Smallest (contraction rate) | #### Notes {.doc-section .doc-section-notes} For dissipative systems: λ₁ + λ₂ = ln|det(J)| = ln|b| For canonical Hénon (a=1.4, b=0.3): λ₁ ≈ 0.42 (positive → chaos) λ₂ ≈ -1.62 (negative → dissipation) λ₁ + λ₂ ≈ -1.20 ≈ ln(0.3) #### Examples {.doc-section .doc-section-examples} ```python >>> system = HenonMap(a=1.4, b=0.3) >>> lyap = system.compute_lyapunov_exponents( ... x0=np.array([0.1, 0.1]), ... n_iterations=100000 ... ) >>> print(f"λ₁ = {lyap[0]:.4f}, λ₂ = {lyap[1]:.4f}") >>> print(f"Sum = {sum(lyap):.4f} (should be ln|b| = {np.log(abs(system.b)):.4f})") ``` ### define_system { #cdesym.systems.builtin.deterministic.discrete.HenonMap.define_system } ```python systems.builtin.deterministic.discrete.HenonMap.define_system( a=1.4, b=0.3, dt=1.0, use_controlled_version=False, ) ``` Define Hénon map dynamics. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |------------------------|--------|---------------------------------------------------|-----------| | a | float | Nonlinearity parameter (controls folding) | `1.4` | | b | float | Dissipation parameter (controls area contraction) | `0.3` | | dt | float | Time step between iterations (usually 1.0) | `1.0` | | use_controlled_version | bool | If True, adds control input u[n] to x dynamics | `False` | ### estimate_correlation_dimension { #cdesym.systems.builtin.deterministic.discrete.HenonMap.estimate_correlation_dimension } ```python systems.builtin.deterministic.discrete.HenonMap.estimate_correlation_dimension( x0, n_points=5000, n_transient=1000, r_min=0.001, r_max=1.0, n_radii=50, ) ``` Estimate correlation dimension using Grassberger-Procaccia algorithm. The correlation dimension d_c characterizes the fractal structure of the attractor. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |-------------|------------|--------------------------------------|------------| | x0 | np.ndarray | Initial state | _required_ | | n_points | int | Number of points on attractor to use | `5000` | | n_transient | int | Points to discard as transient | `1000` | | r_min | float | Minimum radius for correlation sum | `0.001` | | r_max | float | Maximum radius | `1.0` | | n_radii | int | Number of radii to test | `50` | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|---------------------------------| | | float | Estimated correlation dimension | #### Notes {.doc-section .doc-section-notes} For canonical Hénon attractor: d_c ≈ 1.25 #### Examples {.doc-section .doc-section-examples} ```python >>> system = HenonMap(a=1.4, b=0.3) >>> d_c = system.estimate_correlation_dimension( ... x0=np.array([0.1, 0.1]), ... n_points=10000 ... ) >>> print(f"Correlation dimension: {d_c:.3f}") ``` ### generate_bifurcation_diagram { #cdesym.systems.builtin.deterministic.discrete.HenonMap.generate_bifurcation_diagram } ```python systems.builtin.deterministic.discrete.HenonMap.generate_bifurcation_diagram( parameter='a', param_range=(0.8, 1.5), n_points=500, n_transient=500, n_samples=200, x0=None, ) ``` Generate bifurcation diagram by varying a parameter. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |-------------|------------------------|-------------------------------------------|--------------| | parameter | str | Parameter to vary ('a' or 'b') | `'a'` | | param_range | tuple | Range of parameter values (min, max) | `(0.8, 1.5)` | | n_points | int | Number of parameter values to test | `500` | | n_transient | int | Iterations to discard (transient) | `500` | | n_samples | int | Points to sample after transient | `200` | | x0 | Optional\[np.ndarray\] | Initial condition (None = use [0.1, 0.1]) | `None` | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|---------------------------------------------------------------------------------------------------------------| | | dict | Dictionary with: - 'param': Parameter values - 'x': x-coordinate values - 'y': y-coordinate values (optional) | #### Examples {.doc-section .doc-section-examples} ```python >>> system = HenonMap() >>> bifurc_data = system.generate_bifurcation_diagram( ... parameter='a', ... param_range=(0.8, 1.5), ... n_points=1000 ... ) >>> >>> import plotly.graph_objects as go >>> fig = go.Figure() >>> fig.add_trace(go.Scatter( ... x=bifurc_data['param'], ... y=bifurc_data['x'], ... mode='markers', ... marker=dict(size=0.3, color='black') ... )) >>> fig.update_layout(title='Hénon Bifurcation Diagram') >>> fig.show() ``` ### setup_equilibria { #cdesym.systems.builtin.deterministic.discrete.HenonMap.setup_equilibria } ```python systems.builtin.deterministic.discrete.HenonMap.setup_equilibria() ``` Set up fixed points if they exist. Fixed points satisfy: x* = 1 - a·x*² + y* y* = b·x* Substituting: a·x*² - (b-1)·x* + 1 = 0 Real solutions exist only if (b-1)² ≥ 4a.