We examine the Lyness mapping (an integrable Nth-order discrete system which can be generated from a one-dimensional reduction of the Hirota-Miwa equation) from the point of view of deautonomisation. We show that only the N=2 case can be deautonomised when one works with the standard form of the mapping. However it turns out that deautonomisation is possible for arbitrary N when one considers the derivative form of the Lyness mapping. The deautonomisation of the derivative of the N=2 case leads to a result we have never met before: the secular dependence in the coefficients of the mapping enters through two different exponential terms instead of just a single one. As a consequence, it turns out that a limit of this multiplicative dependence towards an additive one is possible without modifying the dependent variable. Finally, the analysis of the `late' singularity confinement of the N=2 case leads to a novel realisation of the full-deautonomisation principle: the dynamical degree is not given (as is customary) simply by the solution of some linear or multiplicative equation, but is present in the growth of the non-linear (and non-integrable) late-confinement conditions.

We study the singularities of some selected integrable lattice equations and show they all admit three types of singularities: one of finite and two of infinite extent. In particular, we show that all the equations we study possess a recently discovered, strip-like, singularity which is known under the moniker of ”taishi”. We study in detail the interaction of these taishi with the remaining two types of singularities and show that the rich behaviour first obtained in the case of the Korteweg-deVries equation is also present for other equations. Moreover, we find that in all cases this behaviour can be encoded in terms of simple symbolic dynamics which are just the dynamics governing a Box Ball cellular automaton.

The deautonomisation of birational maps that have the singularity confinement property, i.e. the construction of nonautonomous versions of such maps that preserve the singularity properties of the original, has proven crucial in our understanding of the mathematical properties behind the integrability of second order maps. For example, the deautonomisation procedure led directly to the development of a general theory of discrete Painlevé equations, and it seems highly likely it will play a crucial role in any future theory of higher dimensional Painlevé equations as well. Generally speaking however, higher order integrable mappings may have non-confined singularities and it is important to understand if, and how, deautonomisation should work for such mappings. In this paper we explore different deautonomisation scenarios on a series of carefully constructed higher order mappings, integrable as well as non-integrable, that possess non-confined singularities and we challenge some common assumptions regarding the co-dimensionality of the singular loci that might play a role in the deautonomisation process. Along the way we also propose a novel procedure to calculate the growth of the multiplicities of singularities that appear in so-called anticonfined singularity patterns, based on an ultradiscrete version of the mapping.

We present various discretisation techniques that mark a departure from the classical algorithms of numerical analysis, in the aim of preserving the physical behaviour of the solutions to the differential systems we are discretising, even for discretisation steps that are not very small. These techniques are inspired by the principles laid down by Mickens and are close to those proposed by Hirota and Kahan. Detailed applications of these methods are presented for the logistic equation and for the Lotka-Volterra system. We discuss the importance of preserving positivity whenever the latter is an expected property of the solution and give a practical rule for achieving this. Possible problems arising from the discretisations we propose are also discussed. A section is devoted to discretisations of integrable systems aimed at preserving integrability. Finally we conclude this review with a discussion of the possible relevance of a discrete space-time.

Over the past 30 years, the special limiting procedure known as ultradiscretisation has become the tool of choice in the field of infinite dimensional integrable systems for constructing cellular automata that exhibit solitonic behaviour, as e.g. in the Korteweg-de Vries equation. A lesser known fact is that, in many cases, ultradiscretisation can also be used to construct cellular automata that retain the essential dynamical features (such as the existence of limit cycles etc.) of a dynamical system expressed in terms of ordinary differential equations (ODEs). In its standard application, the ultradiscretisation procedure relies on the prior construction of a `good' discretisation of the dynamical system at hand, that shares the essential dynamical features of the ODE and which is sign-free, making it amenable to the ultradiscrete limit. In this paper we show, on a simple but generic model, that even if one starts from a discrete system with different dynamical features than the ODE one obtains as its continuum limit, the ultradiscrete limit can be tweaked such that the dynamics of the resulting cellular automaton is closer to that of the continuum limit than to that of the discrete model itself.

The symmetry group of a (discrete) Painlevé equation provides crucial information on the properties of the equation. In this paper, we argue against the commonly held belief that the symmetry group of a given equation is solely determined by its surface type as given in the famous Sakai classification. We will dispel this misconception by using a specific example of a d-PII equation, which corresponds to a half-translation on the root lattice dual to its surface-type root lattice but becomes a genuine translation on a sub-lattice thereof that corresponds to its real symmetry group. The latter fact is shown in two different ways, first by a brute force calculation, and then through the use of normalizer theory, which we believe to be an extremely useful tool for this purpose. We finish the paper with the analysis of a sub-case of our main example, which arises in the study of gap probabilities for Freud unitary ensembles, and the symmetry group of which is even further restricted due to the appearance of a nodal curve on the surface on which the equation is regularized.

We derive a class of discrete Painlev\'e equations associated with the affine Weyl group E$_7^{(1)}$. The method used is the deautonomisation of a QRT mapping belonging to the canonical form VI (according to the classification of said mappings). An equation of such a form was the first instance of a symmetric -- in QRT parlance -- discrete analogue of the Painlev\'e VI equation. In this paper we present an exhaustive derivation of all the discrete Painlev\'e equations of this class. This is made possible thanks to previous studies that established the proper lengths of singularity patterns that are compatible with integrablity, and which were already successfully applied to the study of discrete Painlev\'e equations associated to the affine Weyl group E$_8^{(1)}$. Given that, from the latter, one can obtain by degeneration the equations related to E$_7^{(1)}$, we decided to link the results of the present study to those of the aforementioned ones. It turns out that a bridge from E$_8^{(1)}$ to E$_7^{(1)}$ exists in almost all cases, with one exception where, while in the former case a discrete Painlev\'e equation does exist, in the latter we find a mapping with only periodic coefficients, devoid of secular dependence.

In this paper we give an explanation of a number of observations relating to degree growth of birational mappings of the plane and their deautonomisation by singularity confinement. These observations are of a link between two a priori unrelated notions: firstly the dynamical degree of the mapping and secondly the evolution of parameters required for its singularity structure to remain unchanged under a sufficiently general deautonomisation. We explain this correspondence for a large class of birational mappings of the plane via the spaces of initial conditions for their deautonomised versions. We show that even for non-integrable mappings in this class, the surfaces forming these spaces have effective anticanonical divisors and one can define a period map parametrising them, similar to that in the theory of rational surfaces associated with discrete Painlevé equations. This provides a bridge between the evolution of coefficients in the deautonomised mapping and the induced dynamics on the Picard lattice which encode the dynamical degree.

We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings based on Halburd’s method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer arithmetic with additions and, in some cases, multiplications by small integers. It is therefore extremely fast. Several examples of integrable and non-integrable mappings are presented. In the latter case, the dynamical degree we obtain from our method is always in agreement with that calculated by previously known methods.

We examine a collection of integrable and non-integrable birational mappings of order higher than two and show that the methods developed for calculating the degree growth of second order mappings can be transposed to higher-order systems. In particular, Halburd's method which allows one to obtain explicitly the degree growth of a second order mapping, based on the patterns of its singularities, will be shown to work perfectly well in the higher-order case. The only precaution one must take is to choose initial conditions that are specifically adapted to the method. In the case of non-integrable mappings, one can apply either the express variant of Halburd's method or the full-deautonomisation approach and we show that both lead to the (same) exact expression for the dynamical degree of the mapping.

We study the link between the degree growth of integrable birational mappings of order higher than two and their singularity structures. The higher order mappings we use in this study are all obtained by coupling mappings that are integrable through spectral methods, typically belonging to the QRT family, to a variety of linearisable ones. We show that by judiciously choosing these linearisable mappings, it is possible to obtain higher order mappings that exhibit the maximal degree growth compatible with integrability, i.e. for which the degree grows as a polynomial of order equal to the order of the mapping. In all the cases we analysed, we found that maximal degree growth was associated with the existence of an unconfining singularity pattern. Several cases with submaximal growth but which still possess unconfining singularity patterns are also presented. In many cases the exact degrees of the iterates of the mappings were obtained by applying a method due to Halburd, based on the preimages of specific values that appear in the singularity patterns of the mapping, but we also present some examples where such a calculation appears to be impossible.

We present an application of the full-deautonomisation method to a class of secondorder mappings which, using an ancillary variable, can be cast into a form that greatly facilitates the study of their singularities. The ancillary approach was originally introduced to make it possible to construct discrete Painlevé equations associated with the affine Weyl group E (1) 8 by deautonomising a QRT mapping. The full-deautonomisation method has been shown to offer a practical technique for calculating the exact dynamical degree of a mapping, whereby allowing the detection of discrete integrability using only singularity analysis. We study the confinement property for a given singularity, for a wide class of mappings that includes the autonomous limit of the standard additive Painlevé equation with E (1) 8 symmetry. This leads to a class of non-autonomous mappings, which can be integrable or not, for which we obtain their exact dynamical degrees. The case of a non-confining singularity is also analysed and again we obtain the corresponding dynamical degrees.

We study the singularities of a modified lattice Korteweg-deVries (KdV) equation and show that it admits three families of singularities, with analogous properties to those found in the lattice KdV equation. The first family consists of localised singularities which can occupy an arbitrarily large domain but which are, nevertheless, always confined. The second family consists of one or more lines extending all the way from the south-west to the north-east on the plane, involving a single finite value that depends on the parameter that appears in the equation. We argue that the infinite extent of this singularity is not incompatible with the confinement property or with the integrability of the equation. The third family consists of horizontal strips in which the product of values on vertically adjacent lattice sites is equal to 1. In the case of the lattice KdV equation this type of singularity was dubbed `taishi'. The taishi for the modified lattice KdV equation can interact with singularities of the other two families, giving rise to very rich {and quite intricate} singularity structures. Nonetheless, these interactions can be described in a compact way through the formulation of a symbolic representation of the dynamics that is similar to, but in a sense, simpler than that for the KdV case. We give an interpretation of this symbolic representation in terms of a Box&Ball system related to the ultradiscrete mKdV equation. These results show that taishi-type singularities are not limited to the lattice KdV equation, but might very well be a general feature of integrable lattice equations with deep connections to other facets of their integrability.

We study the structure of singularities in the discrete Korteweg-deVries (d-KdV) equation. Four different types of singularities are identified. The first type corresponds to localised, `confined', singularities, the confinement constraints for which provide the integrability conditions for generalisations of d-KdV. Two other types of singularities are of infinite extent and consist of oblique lines of infinities, possibly alternating with lines of zeros. The fourth type of singularity corresponds to horizontal strips where the product of the values on vertically adjacent points is equal to 1. (A vertical version of this singularity with product equal to $-1$ on horizontally adjacent sites also exists). Due to its orientation this singularity can, in fact, interact with the other types. This leads to an extremely rich structure for the singularities of d-KdV, which is studied in detail in this paper. Given the important role played by the fourth type of singularity we decided to give it a special name: taishi (the origin of which is explained in the text). The taishi do not exist for nonintegrable extensions of d-KdV, which explains the relative paucity of singularity structures in the nonintegrable case: the second and third type of singularities that correspond to oblique lines still exist and the localised singularities of the integrable case now become unconfined, leading to semi-infinite lines of infinities alternating with zeros.

We derive the discrete Painlevé equations associated to the affine Weyl group E_8^(1) that can be represented by an (in the QRT sense) "asymmetric" trihomographic system. The method used in this paper is based on singularity confinement. We start by obtaining all possible singularity patterns for a general asymmetric trihomographic system and discard those patterns which cannot lead to confined singularities. Working with the remaining ones we implement the confinement conditions and derive the corresponding discrete Painlevé equations, which involve two variables. By eliminating either of these variables we obtain a "symmetric" equation. Examining all these equations of a single variable, we find that they coincide exactly with those derived in previous works of ours, thereby establishing the completeness of our results.

We present a simple model for describing the dynamics of the interaction between a homogeneous population or society, and the natural resources and reserves that the society needs for its survival. The model is formulated in terms of ordinary differential equations, which are subsequently discretised, the discrete system providing a natural integrator for the continuous one. An ultradiscrete, generalised cellular automaton-like, model is also derived. The dynamics of our simple, three-component, model are particularly rich exhibiting either a route to a steady state or an oscillating, limit cycle-type regime or to a collapse. While these dynamical behaviours depend strongly on the choice of the details of the model, the important conclusion is that a collapse or near collapse, leading to the disappearance of the population or to a complete transfiguration of its societal model, is indeed possible.

The discrete KdV (dKdV) equation, the pinnacle of discrete integrability, is often thought to possess the singularity confinement property because it confines on an elementary quadrilateral. Here we investigate the singularity structure of the dKdV equation through reductions of the equation, obtained for initial conditions on a staircase with height 1, and show that it is much more subtle than one might assume. We first study the singularities for the mappings obtained after reduction and contrast these with the singularities that arise in non-integrable generalizations of those mappings. We then show that the so-called 'express method' for obtaining dynamical degrees for second order mappings can be succesfully applied to all the higher order mappings we derived. Finally, we use the information obtained on the singularity structure of the reductions to describe an important subset of singularity patterns for the dKdV equation and we present an example of a non-confining pattern and explain why its existence does not contradict the integrability of the dKdV equation.

In this paper we present a rigorous method for deciding whether a birational three point mapping that has the singularity confinement property is integrable or not, based only on the structure of its (confined) singularity patterns. We also explain how the exact value of the dynamical degree for such a mapping may be deduced from the singularity patterns.