Draft:Anti-integrability

Definition 1

A symplectic dynamical system with trajectories ${\displaystyle \{x_{t}\}}$ and discrete time ${\displaystyle t\in \mathbb {Z} }$ is anti-integrable when its action (or generating function) ${\displaystyle \sum _{t}L(x_{t},x_{t+1})}$ can be written as ${\displaystyle \sum _{t}V(x_{t})}$.

The dynamics of the map governing ${\displaystyle x_{t}\mapsto x_{t+1}}$ are considered integrable when the potential energy vanishes, where solutions can be found via the law of conservation of momentum. In contrast, the anti-integrable limit can be interpreted as the kinetic energy vanishing. Following the principle of least action, trajectories of the map at the anti-integrable limit must lie at critical points of the potential and the 'dynamics' become non-deterministic, reducing to the shift operator acting on the set of these critical points. Trajectories at this limit, referred to as anti-integrable states, can then be analytically or numerically continued toward integrability. As Aubry pointed out, this process is similar to perturbing an integrable system via KAM theory. KAM theory is used to find regular solutions that persist away from integrability, when more and more solutions become irregular. In contrast, the theory of anti-integrability is used to find irregular solutions that persist away from the anti-integrable limit, when more and more solutions become regular.

Example: Standard Map

The standard map can be written as a two-dimensional map,

${\displaystyle {\begin{pmatrix}p_{t+1}\\x_{t+1}\end{pmatrix}}={\begin{pmatrix}p_{t}+\lambda \sin {x_{t}}\\x_{t}+p_{t+1}\end{pmatrix}},}$

for ${\displaystyle p_{t},x_{t}\in \mathbb {R} }$ modulo ${\displaystyle 2\pi }$ and ${\displaystyle \lambda >0}$. The map can also be written as a single difference equation,

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

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 ${\displaystyle n\in \mathbb {Z} }$. Note that this condition reproduces the difference equation above.

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

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

The corresponding Lagrangian is now

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

The anti-integrable limit now corresponds with 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 ${\displaystyle \epsilon >0}$.^[5]

Definition 2

Consider a one parameter ${\displaystyle \epsilon }$-continuous family of deterministic dynamical systems ${\displaystyle X_{t+1}=F_{\epsilon }(X_{t})}$. The limit ${\displaystyle \epsilon \to 0}$ is called the anti-integrable limit when

• The system can be defined as an implicit dynamical system, i.e., there exists a function ${\displaystyle G_{\epsilon }(X,Y)}$ which depends continuously on ${\displaystyle \epsilon }$ such that the implicit equation ${\displaystyle G_{\epsilon }(X,Y)=0}$ is equivalent to ${\displaystyle Y=F_{\epsilon }(X)}$ for ${\displaystyle \epsilon \neq 0}$ and such that the limit ${\displaystyle G_{0}(X,Y)}$ is defined.
• The solutions ${\displaystyle \{X_{t}\}}$ of the implicit equations ${\displaystyle G_{0}(X_{t+1},X_{t})=0}$ for all ${\displaystyle t}$ form a discrete set which can be characterized by an infinite sequence ${\displaystyle \{s_{t}\}\in \Sigma ^{\mathbb {Z} }}$ called a coding (or symbolic) sequence ${\displaystyle \{s_{t}\}\to \{X_{t}\}}$ where ${\displaystyle s_{t}}$ belongs to a discrete set ${\displaystyle \Sigma }$ of codes (or symbols).

In this definition, the need for an implicit relation at the anti-integrable limit and the utilization of symbolic dynamics is more explicit.

Example: Logistic Map

The picture of the anti-integrable limit and the concept of anti-integrability are particularly clear with the logistic map,

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

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

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

The singular limit ${\displaystyle \mu \to \infty }$ for the logistic map 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, ${\displaystyle s=\{\ldots ,s_{-1},s_{0},s_{1},\ldots \}}$, in the space

${\displaystyle \Sigma =\{s:s_{t}\in \{0,1\},t\in \mathbb {Z} \}.}$

Let ${\displaystyle \sigma$ :\Sigma \to \Sigma } be the shift map where ${\displaystyle \sigma (s)_{t}=s_{t+1}}$. 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 can be proved analytically with an implicit function theorem argument^[7].