Jellyfish exist

Abstract: We show the existence of infinitely many geometrically distinct homothetic expanders (jellyfish) for the elastic flow, epicyclic shrinkers for the curve diffusion flow, and epicyclic expanders for the ideal flow.

• The paper establishes infinite families of homothetic expanders for elastic flow and epicyclic shrinkers/expanders for curve diffusion and ideal flows.
• It employs an ODE perturbation strategy combined with dihedral gluing to meet seam and curvature jet conditions for smooth, self-similar solutions.
• The results expand the known dynamics of curvature flows and suggest practical implications for physical simulations and geometric analysis.

Existence and Construction of Homothetic Expanders and Shrinkers in Curvature Flows

This paper establishes the existence of infinitely many geometrically distinct homothetic expanders (termed "jellyfish") for elastic flow, along with families of epicyclic shrinkers for curve diffusion flow and epicyclic expanders for ideal flow. The context is the gradient descent flows of planar curves with respect to natural geometric energies: elastic energy, curve diffusion energy, and ideal energy. The work employs an ODE perturbation strategy combined with careful symmetry constraints to yield a rich taxonomy of new self-similar solutions not previously documented.

Figure 1: Free elastic flow jellyfish expander ($p/q=1/7$, $\omega=1$, $\varepsilon=0.13290$).

Curvature Flows and Self-Similar Solutions

The study analyzes three fundamental curvature-driven gradient flows for planar curves:

• Elastic flow: Gradient flow of elastic energy $E[\gamma]=\frac12\int k^2\,ds$, evolving by $\partial_t \gamma = -(k_{ss}+\frac12 k^3)\nu$.
• Curve diffusion flow: Gradient flow of length in $H^{-1}$, evolving by $\partial_t \gamma = -k_{ss}\nu$.
• Ideal flow: Gradient flow of ideal energy $I[\gamma]=\frac12\int k_s^2\,ds$, evolving by $$\partial_t \gamma = (k_{ssss} + k^2 k_{ss} - \frac12 k k_s^2)\nu$$.

These flows are known for basic self-similar solutions (circles, lemniscates), but the existence of higher-order homothetic solutions (expanders/shrinkers with nontrivial topology/symmetry) was unresolved.

Construction Strategy: Fundamental Arcs and Symmetry Closure

The primary innovation is a two-stage construction:

1. ODE Boundary Value Problem: Begin with a stationary base arc (half-period of Euler’s elastica for elastic flow, semicircle for epicyclic families). Introduce perturbations parameterized by $\varepsilon$ (distance from trivial solution) and solve for arcs satisfying both a radial seam (orthogonality) condition and a curvature jet (for higher-order flows) condition.
2. Dihedral Gluing: Employ discrete group symmetries via reflection and rotation to glue multiple copies of the fundamental arc into a closed curve. For each symmetry order $m$, the seam angle $\overline\theta$ is tuned via $\varepsilon$ so that $m$ glued copies close smoothly.

This method is proven to yield, for all sufficiently large $m$, an infinite family of closed homothetic solutions with prescribed symmetry.

Elastic Flow Jellyfish Expanders

For the elastic flow, the construction yields jellyfish expanders: closed curves with dihedral symmetry, assembled from arcs perturbed off the elastica. These are immortal solutions, expanding self-similarly as predicted by scaling properties of the flow.

Key result (Theorem): For $m$ greater than a threshold $m_0$, there exists a jellyfish expander with dihedral symmetry of order $m$.

Figure 2: Closed jellyfish-like expanders for the free elastic flow that do not possess dihedral symmetry. The indicated point is the centre of expansion.

Epicyclic Shrinkers/Expanders: Curve Diffusion and Ideal Flow

In the curve diffusion flow case, the construction yields epicyclic shrinkers: closed, self-similarly shrinking curves parametrized by symmetry order $q$ and rotation index $p$. For ideal flow, a similar method produces epicyclic expanders, despite the higher-order jet matching needed at arc seams. In each, the seam and jet conditions are reduced via explicit calculation and use of the Implicit Function Theorem.

Theorems: For all $q > q_0$ (curve diffusion) and all $q$ large (ideal flow), there exist infinitely many closed self-similar shrinkers/expanders with corresponding symmetry.

Geometric Distinctness and Symmetry Classification

The paper rigorously demonstrates that no additional hidden symmetry axes exist in these constructions, so each element in the infinite family is geometrically distinct (non-equivalent under similarity/group action). Symmetry orders, rotation indices, and total tangent turning are used as invariants.

Numerical Observations and Open Problems

The analytical families constructed do not exhaust possible homothetic solutions. Numerical simulations indicate the existence of exotic non-symmetric solutions (e.g., one-sided jellyfish). Complete classification, even for fixed rotation number, remains formidable. The conjecture is put forward that only multiply-covered circles and lemniscates are stable attractors in these flows, and that generic solutions converge exponentially fast to them.

Implications and Future Directions

The discovery of infinite families of new self-similar solutions—beyond the classical circles and lemniscates—expands the dynamical repertoire and complexity of higher-order geometric flows. Practically, these could inform design and stabilization strategies for physical systems (e.g., biological membranes, filament evolution) governed by analogous flows. Theoretically, they provoke new questions in dynamical systems, stability theory, and geometric analysis, especially concerning basin structure and attractor classification. Future work may involve the rigorous construction of "exotic" non-symmetric solutions, full classification in terms of invariants, and exploration of possible applications in image processing, computer graphics and physical simulation.

Conclusion

This paper proves the existence of infinitely many geometrically distinct homothetic expanders and shrinkers in elastic, curve diffusion, and ideal flows, using a perturbative ODE method with symmetry closure. The results reshape the landscape of known solutions for curvature flows, revealing vast new families of self-similar, closed planar curves whose symmetry, expansion/shrinking modes, and geometric properties are determined by analytic and group-theoretic data. The implications for theory and applications are significant, stimulating further study into the structure, stability, and classification of solutions to geometric evolution equations.

(2601.21227)