Algebraic Cycles and Limit-Cycle Challenge
S13
smale derived
smale derived
External: externally open
τ response: further investigation
For polynomial vector fields in the plane, can one bound or classify the number and arrangement of limit cycles as a function of polynomial degree?
Current τ response
See the paired Algebraic Cycles and Limit-Cycle Challenge — Challenge Response on the Results lane for the program's current response status, registry evidence, verification route, and external-review boundary.
Current status: further investigation.
Challenge statement
For polynomial vector fields in the plane, can one bound or classify the number and arrangement of limit cycles as a function of polynomial degree?
Why this challenge is in the ledger
Links algebraic geometry, dynamical systems, cycles, phase-space topology, and symbolic-to-dynamic transition.
τ-facing burden
Show whether τ-cycle structure, address resolution, or basin decomposition gives a new route to bounding or classifying limit cycles.
First reviewer questions
- Does τ provide an upper bound, a classification scheme, or only a phase-space vocabulary?
- Can τ handle known hard examples and bifurcations?
- Is the τ account compatible with standard polynomial vector-field theory?
Source anchors
Source anchors are background references, not endorsements of Panta Rhei claims.