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.

Version

v2 · 2026-09-19 · History

Domain-specific #

5175

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Dynamical Systems, Chaotic Maps → Mathematics

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.

How would you explain it like I'm…

The Doubling Wrap-Around Game

Imagine a game with two numbers that you play over and over. The first number doubles each turn, and if it gets too big it wraps back around, like a clock hand going past twelve. The second number shrinks a bit each turn and gets a little push that depends on the first number. Because the first number jumps around so wildly, the second one draws a strange, wiggly pattern.

Chaos Driving a Follower

The Kaplan–Yorke map is a rule you repeat over and over on two numbers, x and y. Each step, x is doubled and only the part after the decimal point is kept, so it stays between 0 and 1. This makes x jump around chaotically: two starting values that are nearly the same soon end up far apart. Then y is shrunk or stretched by a fixed amount alpha and given a push that depends on x through a cosine wave. Alpha decides how fast y forgets its past and what shape the final pattern takes.

Doubling-Map Skew Product

The Kaplan–Yorke map sends (x, y) to (2x mod 1, αy + cos(4πx)). The x part is the doubling map: in binary, doubling just shifts the digits left and drops the leading one, which is why tiny differences grow fast (sensitive dependence) and why x can be described by its digit sequence. The y part doesn't affect x at all; it's a linear update driven by x through the cosine term. This one-way setup is called a skew product. The size of α controls how much y contracts, how long it remembers past pushes, and the shape of the attractor. Because doubling eats one binary digit per step, computer simulations with finite precision eventually go wrong in a characteristic way.

 

The Kaplan–Yorke map is the two-dimensional iteration (xₙ₊₁, yₙ₊₁) = (2xₙ mod 1, αyₙ + cos(4πxₙ)). It is a skew product: the base dynamics in x are autonomous and chaotic, while the fiber coordinate y is linearly driven by the base. The doubling map acts as a left shift on binary expansions, yielding Bernoulli-like symbolic dynamics and sensitive dependence on initial conditions. The orbit of x forces y through the cosine term, and |α| sets the contraction rate, the memory of past forcing, and the geometry of the resulting attractor. A characteristic pitfall is finite-precision failure: in floating point, repeated doubling exhausts the mantissa bits, so computed x-orbits collapse in ways the exact map does not. Identifying an instance requires that each iterate apply exactly the modulo-one doubling and the same α-linear cosine-driven y update under a stated numerical representation. Neighboring maps or merely similar notation don't count.

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

1. 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) — outgoing typed relations

• Kaplan–Yorke map is a kind of Iteration Prime

  The dynamical system is defined by repeated application of one nonlinear map.

Hierarchy path (1) — routes to 1 parentless root

• Kaplan–Yorke map → is a kind of Iteration

Neighborhood in Abstraction Space

Kaplan–Yorke map sits in a moderately populated region (51^st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Dynamical Systems & Chaos (17 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 · 2026-10-25