Dynamical systems and ergodic theory Hamiltonian dynamics Example

Standard map

Builds on Circle Kolmogorov–Arnold–Moser theorem Chaos theory

Example

Let $K \in \R$ be a parameter. The standard map, or Chirikov–Taylor map, acts on points $(\theta, p)$ of the cylinder, where $\theta$ is taken modulo $2\pi$ . On the torus, $p$ is also taken modulo $2\pi$ . The map is $p_{n+1} = p_n + K \sin \theta_n, \qquad \theta_{n+1} = \theta_n + p_{n+1}.$ Its Jacobian matrix with respect to $(p_n, \theta_n)$ is $\begin{pmatrix} 1 & K \cos \theta_n \\ 1 & 1 + K \cos \theta_n \end{pmatrix},$ whose determinant is $1 + K \cos \theta_n - K \cos \theta_n = 1$ . So the map preserves area and orientation, and it is a symplectomorphism of the plane form $dp \wedge d\theta$ .

Remarks

For $K = 0$ the map is integrable: $p$ is constant and each circle $p = c$ is invariant, with rotation by the angle $c$ . For small $K \ne 0$ , KAM theory for twist maps shows that the invariant circles with sufficiently irrational rotation numbers survive, slightly deformed. These circles wrap around the cylinder and prevent $p$ from growing without bound. As $K$ grows the circles break; numerically, the last one disappears near $K \approx 0.9716$ , found by Greene, and beyond that $p$ can drift without bound. For large $K$ most orbits look chaotic, with islands of stable motion around periodic orbits.

The map models a kicked rotor: a rotor that receives a periodic kick whose strength depends on its angle.

Reading path

To understand this page from the ground up, read these 70 pages in order. The foundations come first.

Action-angle coordinates concept
Chaos theory concept
Circle concept
Element definition
Hamiltonian system concept
Integrable system concept
Proposition concept
Real numbers object
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Topological space definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Function definition
Complement definition
Basis of a topology definition
Subspace topology definition
Equivalence relation definition
Preimage of a set definition
Product topology definition
Codomain definition
Function composition definition
Injective function definition
Metric space definition
Equivalence class definition
Surjective function definition
Open ball definition
Quotient set definition
Bijective function definition
Metric topology definition
Quotient topology definition
Inverse function definition
Open set definition
Continuous map of topological spaces definition
Hausdorff space definition
Open cover definition
Closed set definition
Homeomorphism definition
Universal property of the quotient topology statement
Compact space definition
Complement of an open set is closed statement
Continuity via closed sets statement
Compact set definition
Compactness via open covers statement
Continuous image of a compact set is compact statement
Compact subsets of a Hausdorff space are closed statement
Closed subsets of compact sets are compact statement
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Circle as a quotient of the real line example
Torus example
Kolmogorov–Arnold–Moser theorem statement
Standard map example

Tag	`8TEV7CSX2H`
URI	`m2://8TEV7CSX2H`
Kind	example
Writing	W0 stub
MSC	37E40, 37J11, 70K55
Instance of	Symplectomorphism
Uses	Circle, Kolmogorov–Arnold–Moser theorem, Chaos theory
Same as	Wikidata Q2257676 Wikipedia Standard map

Standard map

Example

Remarks

Reading path

Further reading

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