Boundary map explorer

The set-valued map F(M) = B̄_ε(f(M)) with f(x) = (λ₁x₁ + a x₂², λ₂x₂), started from a disc of radius 3. Three views of one system: the sets and noisy trajectories downstairs, the Legendrian Σ_k = β^k(N₁⁺∂M₀) upstairs, and the angle of its tangent inside the contact plane.

a perturbation strength

k iterate shown

λ₁

λ₂

Presets Animate Curve Resample

State space ℝ²

set M_k front on ∂M_k front buried inside cusp (κ = ∞) inflection (κ = 0) noisy trajectories

Legendrian Σ_k ⊂ ℝ²×S¹ drag to rotate

Σ_k, height = θ shadow = front π(Σ_k) vertical tangent

Angle of TΣ_k in the contact plane along Σ_k

π/2: tangent vertical, front cusps 0: tangent horizontal, κ = 0

Lagrangian projection (θ, p) to T*S¹ p = ⟨x, n^⊥⟩

always an immersed curve κ = ∞ κ = 0

Second front (θ, u) in J¹S¹ u = ⟨x, n⟩, the support value

graph of h_M when M is convex cusp here = κ = 0 on the ℝ² front κ = ∞, smooth here

State space ℝ2

Legendrian Σk ⊂ ℝ2×S1 drag to rotate

Angle of TΣk in the contact plane along Σk

Lagrangian projection (θ, p) to T*S1 p = ⟨x, n⊥⟩

Second front (θ, u) in J1S1 u = ⟨x, n⟩, the support value

State space ℝ²

Legendrian Σ_k ⊂ ℝ²×S¹ drag to rotate

Angle of TΣ_k in the contact plane along Σ_k

Lagrangian projection (θ, p) to T*S¹ p = ⟨x, n^⊥⟩

Second front (θ, u) in J¹S¹ u = ⟨x, n⟩, the support value