Arnold cat map
E1046813
Anosov diffeomorphism
area-preserving map
chaotic map
discrete-time dynamical system
dynamical system
ergodic transformation
hyperbolic dynamical system
measure-preserving transformation
mixing transformation
toral automorphism
The Arnold cat map is a famous example of a chaotic, area-preserving transformation on the torus that illustrates how simple deterministic rules can produce complex, seemingly random behavior.
Statements (56)
| Predicate | Object |
|---|---|
| instanceOf |
Anosov diffeomorphism
ⓘ
area-preserving map ⓘ chaotic map ⓘ discrete-time dynamical system ⓘ dynamical system ⓘ ergodic transformation ⓘ hyperbolic dynamical system ⓘ measure-preserving transformation ⓘ mixing transformation ⓘ toral automorphism ⓘ |
| actsOn | unit square with opposite sides identified ⓘ |
| definedOn |
2-dimensional torus
ⓘ
T^2 ⓘ |
| hasCharacteristicPolynomial | λ^2 - 3λ + 1 ⓘ |
| hasDeterminant | 1 ⓘ |
| hasDomain | [0,1) × [0,1) with opposite sides identified ⓘ |
| hasEigenvalues |
(3+√5)/2
ⓘ
(3-√5)/2 ⓘ |
| hasMatrixRepresentation | [[2,1],[1,1]] modulo 1 GENERATED ⓘ |
| hasProperty |
area-preserving
ⓘ
chaotic ⓘ deterministic ⓘ ergodic with respect to Lebesgue measure ⓘ exhibits exponential divergence of nearby trajectories ⓘ has dense orbits ⓘ has periodic orbits of arbitrarily large period ⓘ has positive Lyapunov exponents ⓘ has sensitive dependence on initial conditions ⓘ invertible ⓘ is a Bernoulli system (measure-theoretically conjugate to a Bernoulli shift) ⓘ is a prototype of deterministic chaos ⓘ is structurally stable ⓘ linear on the torus ⓘ preserves Lebesgue measure ⓘ topologically mixing ⓘ uniformly hyperbolic ⓘ volume-preserving on the torus ⓘ |
| hasRange | [0,1) × [0,1) with opposite sides identified ⓘ |
| hasTrace | 3 ⓘ |
| hasTypicalVisualization | iterative scrambling and reappearance of a cat image ⓘ |
| illustrates |
how simple deterministic rules can produce complex behavior
ⓘ
mixing of phase space ⓘ |
| isAlsoKnownAs |
Arnold’s cat map
NERFINISHED
ⓘ
cat map NERFINISHED ⓘ |
| isAreaPreservingBecause | its defining matrix has determinant 1 ⓘ |
| isDefinedByFormula |
(x',y') = (2x + y, x + y) mod 1
ⓘ
(x',y') = A(x,y) mod 1 with A = [[2,1],[1,1]] ⓘ |
| named mathematical object in dynamical systems theory},{ | Vladimir Arnold NERFINISHED ⓘ |
| relatedTo |
Anosov systems
ⓘ
Lyapunov exponents NERFINISHED ⓘ ergodic theory ⓘ mixing in Hamiltonian systems ⓘ symbolic dynamics ⓘ |
| usedFor |
pedagogical examples in dynamical systems courses
ⓘ
studying properties of hyperbolic toral automorphisms ⓘ visual demonstrations of chaos ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.