Kaplan–Yorke map¶
A two-dimensional skew-product chaotic map coupling the doubling map x↦2x mod 1 to a driven contraction or expansion y↦αy+cos(4πx), with dynamics controlled by one parameter α.
Core Idea¶
The Kaplan–Yorke map sends (x_n,y_n) to (2x_n mod 1, αy_n+cos(4πx_n)), forming a skew product with chaotic base dynamics and a linearly forced transverse coordinate. Binary expansion under the doubling map produces sensitive dependence and symbolic dynamics in x. That orbit drives y through cosine forcing, while |α| determines contraction, memory, and attractor geometry. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Kaplan–Yorke map belongs to dynamical systems and is useful where the analyst can specify points (x,y) in a planar or cylindrical phase space, the doubling map modulo one, a cosine forcing term, parameter α, and discrete iteration, then evaluate each iterate applies the exact modulo-one doubling and the same α-linear cosine-driven y update under a declared numerical representation. The scope is broad within that domain but bounded by the need for each iterate applies the exact modulo-one doubling and the same α-linear cosine-driven y update under a declared numerical representation. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making each iterate applies the exact modulo-one doubling and the same α-linear cosine-driven y update under a declared numerical representation the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Kaplan–Yorke map can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Kaplan–Yorke map. Kaplan–Yorke map compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: points (x,y) in a planar or cylindrical phase space, the doubling map modulo one, a cosine forcing term, parameter α, and discrete iteration. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express each iterate applies the exact modulo-one doubling and the same α-linear cosine-driven y update under a declared numerical representation independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of dynamical systems because they reuse points (x,y) in a planar or cylindrical phase space, the doubling map modulo one, a cosine forcing term, parameter α, and discrete iteration, Binary expansion under the doubling map produces sensitive dependence and symbolic dynamics in x. That orbit drives y through cosine forcing, while |α| determines contraction, memory, and attractor geometry., and type the carrier, state every parameter and convention in the definition, test that each iterate applies the exact modulo-one doubling and the same α-linear cosine-driven y update under a declared numerical representation, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Kaplan–Yorke map Domain-specific
Parents (1) — more general patterns this builds on
-
Kaplan–Yorke map is a kind of Iteration Prime
The proposed strict upward parent is
prime:iteration.
Hierarchy path (1) — routes to 1 parentless root
- Kaplan–Yorke map → Iteration
Neighborhood in Abstraction Space¶
Kaplan–Yorke map sits in a sparse region of the domain-specific corpus (63rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Collective Dynamics & Molecular Operators (6 abstractions)
Nearest neighbors
- Recurrent point — 0.88
- Denjoy's theorem on rotation number — 0.86
- Adiabatic invariant — 0.86
- Strange nonchaotic attractor — 0.85
- Reversible reference system propagation algorithm — 0.85
Computed from structural-signature embeddings · 2026-09-08