Draft:Anti-integrability

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Submission declined on 6 February 2026 by ChrysGalley (talk). This submission is not adequately supported by reliable sources. Reliable sources are required so that information can be verified. If you need help with referencing, please see Referencing for beginners and Citing sources. This submission does not appear to be written in the formal tone expected of an encyclopedia article. Entries should be written from a neutral point of view, and should refer to a range of independent, reliable, published sources. Please rewrite your submission in a more encyclopedic format. Please make sure to avoid peacock terms that promote the subject. If you would like to continue working on the submission, click on the "Edit" tab at the top of the window. If you have not resolved the issues listed above, your draft will be declined again and potentially deleted. If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors. Please do not remove reviewer comments or this notice until the submission is accepted. Where to get help If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews. If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed. How to improve a draft Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia. Help:Wikitext – how to use the markup Help:Referencing for beginners – how to include references Wikipedia:Article development – how to develop your article Wikipedia:Writing better articles – how to improve your article Wikipedia:Verifiability – make sure your article includes reliable third-party sources You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Add tags to your draft Editor resources Easy tools: Citation bot (help) | Advanced: Fix bare URLs Declined by ChrysGalley 2 days ago. Last edited by Amandahampton64 2 days ago. Reviewer: Inform author. This draft has been resubmitted and is currently awaiting re-review.

• Comment: Several DOI are dead on arrival. Kindly fix. The article would be made more encyclopedic with an introduction which explains the topic in more sensible terms to an intelligent reader.
  Can you also clarify whether you are connected to the subject and its sources? This may need to be declared under WP:COI. ChrysGalley (talk) 16:37, 6 February 2026 (UTC)

The following Wikipedia contributor has declared a personal or professional connection to the subject of this page. Relevant policies and guidelines may include conflict of interest, autobiography, and neutral point of view. Amanda Hampton (talk · contribs) This user has declared a connection. (I am a math professor at Georgia Tech and do research in anti-integrability. I am a co-author on two of the twelve references.)

Anti-integrability is a methodology used to study the chaotic bounds of a dynamical system. A dynamical system is at the anti-integrable limit if it becomes singular and trajectories of the dynamical system no longer have an explicit form. In other words, the system becomes non-deterministic and reduces to an implicit relation with multiple solutions. The significance of the limit is that, under some conditions, trajectories at the anti-integrable limit, referred to as anti-integrable states, can persist onto trajectories of the original dynamical system. Anti-integrability provides a way to rigorously prove the existence of chaotic orbits and can be used to study (often infinite) perturbations of a dynamical system.

Prior to its formalization, the concept of anti-integrability was used to analyze hyperbolic Aubry-Mather sets.^[1], hyperbolic well-ordered Cantor sets^[2], and Cantori of symplectic maps ^[3]. Essentially, it was used as a method to analyze geometric structures within chaotic regimes. It should not be confused with converse- KAM theory.

The term anti-integrability was formalized in the field of solid state physics as a methodology that 'opposed' integrability. It was first defined in the context of discrete Lagrangian systems, or (locally) symplectic maps ^[4]

Trajectories of a discrete Lagrangian/symplectic system, ${\displaystyle \{x_{t}\}}$ for ${\displaystyle t\in \mathbb {Z} }$, can be found via the principle of least action. The action of a trajectory, ${\displaystyle S(\{x_{t}\})}$, can be written as a sum of discrete Lagrangians ${\displaystyle L(x_{t},x_{t+1})}$, which are combinations of kinetic ${\displaystyle K(x_{t}-x_{t+1})}$ and potential ${\displaystyle V(x_{t})}$ energies,

${\displaystyle S(\{x_{t}\})=\sum _{t}L(x_{t},x_{t+1})=\sum _{t}\left(K(x_{t}-x_{t+1})-V(x_{t})\right)}$.

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.

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]

The definition above has been generalized to include all discrete maps^[6]. This generalized definition, arguably, gives an easier approach to implement anti-integrability, as can be seen in the example of the logistic map below.

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.

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].

In the example of the logistic map, the solutions at the anti-integrable limit were also the symbols. This is not always the case. In a study of a three-dimensional map^[8], solutions at an anti-integrable limit are branches of a conic.

Arguments for persistence away from the anti-integrable limit typically utilize the implicit function theorem and/or the contraction mapping theorem. One can also continue anti-integrable states away from the anti-integrable limit using numerical continuation ^[9].

Anti-integrability has also been used to prove the existence of a horseshoe ^{[10] [11]} and to study the development of chaotic attractors over changing parameters ^[12]