Anti-integrability

Example: Standard map

Written as a single difference equation, the standard map is given as

${\displaystyle x_{t+1}-2x_{t}+x_{t-1}=\lambda \sin {x_{t}}}$,

for ${\displaystyle x_{t}\in \mathbb {R} }$ modulo ${\displaystyle 2\pi }$ and ${\displaystyle \lambda >0}$. The Lagrangian of this difference equation takes the form

${\displaystyle L(x_{t},x_{t+1})=K(x_{t}-x_{t+1})-V(x_{t})=(x_{t}-x_{t+1})^{2}-2\lambda \cos {x_{t}}.}$

In accordance with the principle of least action, trajectories of the standard map lie at the critical points of the sum of Lagrangians, i.e., where

${\displaystyle {\frac {\partial }{\partial x_{n}}}\sum _{t}L(x_{t},x_{t+1})=0}$

for any ${\displaystyle n\in \mathbb {Z} }$. This condition on the Lagrangian above reproduces the standard map.

The anti-integrable limit of the standard map is the limit ${\displaystyle \lambda \to \infty }$. To find this limit, we rescale the parameter as ${\displaystyle \lambda ={\frac {1}{\epsilon }}}$ so that it becomes

${\displaystyle \epsilon (x_{t+1}-2x_{t}+x_{t-1})=\sin {x_{t}}.}$

The corresponding Lagrangian is now

${\displaystyle L(x_{t},x_{t+1})=\epsilon (x_{t}-x_{t+1})^{2}-2\cos {x_{t}}}$.

With this rescaling, the anti-integrable limit is now the limit ${\displaystyle \epsilon \to 0}$. Applying this limit and the principle of least action leads to the non-deterministic, implicit relation

${\displaystyle \sin {x_{t}}=0,}$

with solutions ${\displaystyle m\pi }$ for integer ${\displaystyle m}$. Valid anti-integrable states then come in the form

${\displaystyle \{\ldots ,m_{-1}\pi ,m_{0}\pi ,m_{1}\pi ,\ldots \}\quad {\text{for}}\quad m_{n}\in \mathbb {Z} ,}$

and the dynamics reduce to the shift operator on these states.

Arguments using the implicit function theorem and contraction mapping theorem can then be made to show that anti-integrable states can persist for small ${\displaystyle \epsilon >0}$, as discussed in Aubry and Abramovici's formative work.

The definition above was generalized by Aubry in 1995 to include all discrete maps. This generalized definition provides an alternative, and arguably easier, approach to implement anti-integrability, as can be seen in the example of the logistic map below.

Example: Logistic map

Following the work of Chen, the logistic map,

${\displaystyle x_{t+1}=\mu x_{t}(1-x_{t}),\quad x\in \mathbb {R} ,\quad \mu >0,}$

has an anti-integrable limit ${\displaystyle \mu \to \infty }$. To see this, rewrite the map using the rescaling ${\displaystyle \mu ={\frac {1}{\epsilon }}}$ to obtain

${\displaystyle x_{t}(1-x_{t})=\epsilon x_{t+1}.}$

The singular anti-integrable limit ${\displaystyle \mu \to \infty }$ now corresponds with ${\displaystyle \epsilon \to 0}$.

When ${\displaystyle \epsilon =0}$, the map becomes the non-deterministic, implicit relation ${\displaystyle x_{t}(1-x_{t})=0}$ and has the solutions ${\displaystyle x_{t}=0}$ or ${\displaystyle x_{t}=1}$ for every ${\displaystyle t\in \mathbb {Z} }$. Each valid anti-integrable state is associated with a symbolic sequence, ${\displaystyle s=\{\ldots ,s_{-1},s_{0},s_{1},\ldots \}}$, where ${\displaystyle s_{t}\in \{0,1\},}$ for integer ${\displaystyle t}$. Thus, at the anti-integrable limit, the 'dynamics' become a shift on these two symbols.

The anti-integrable states at ${\displaystyle \epsilon =0}$ persist for small ${\displaystyle \epsilon >0}$, which Chen proves analytically with an implicit function theorem argument.