
Hone, A. (2020). Continued fractions and Hankel determinants from hyperelliptic curves. Communications on Pure and Applied Mathematics [Online]. Available at: https://doi.org/10.1002/cpa.21923.
Following van der Poorten, we consider a family of nonlinear maps which are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g. Using the connection with the classical theory of Jfractions and orthogonal polynomials, we show that in the simplest case g = 1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos4 sequence, which were found in a particular form by Chang, Hu and Xin, We extend these formulae to the higher genus case, and prove that generic Hankel determinants in genus two satisfy a Somos8 relation. Moreover, for all g we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system.

Christodoulidi, H., Hone, A. and Kouloukas, T. (2019). A new class of integrable Lotka–Volterra systems. Journal of Computational Dynamics [Online] 6:223237. Available at: http://dx.doi.org/10.3934/jcd.2019011.
A parameterdependent class of Hamiltonian (generalized) Lotka–Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.

Hay, M., Hone, A., Novikov, V. and Wang, J. (2019). Remarks on certain twocomponent systems with peakon solutions. Journal of Geometric Mechanics [Online] 11:561573. Available at: http://dx.doi.org/10.3934/jgm.2019028.
We consider a Lax pair found by Xia, Qiao and Zhou for a family of twocomponent analogues of the CamassaHolm equation, including an arbitrary function H, and show that this apparent freedom can be removed via a combination of a reciprocal transformation and a gauge transformation, which reduces the system to triangular form. The resulting triangular system may or may not be integrable, depending on the choice of H. In addition, we apply the formal series approach of Dubrovin and Zhang to show that scalar equations of CamassaHolm type with homogeneous nonlinear terms of degree greater than three are not integrable. This article is dedicated to Darryl Holm on his 70th birthday.

Barnes, L. and Hone, A. (2019). Dynamics of conservative peakons in a system of Popowicz. Physics Letters A [Online] 383:406413. Available at: https://doi.org/10.1016/j.physleta.2018.11.015.
We consider a twocomponent Hamiltonian system of partial
differential equations with quadratic nonlinearities introduced by
Popowicz, which has the form of a coupling between the CamassaHolm and DegasperisProcesi equations.
Despite having reductions to these two integrable partial differential equations, the Popowicz
system itself is not integrable. Nevertheless, as one of the authors showed with Irle, it admits
distributional solutions of peaked soliton (peakon) type, with the dynamics of $N$
peakons being determined by a Hamiltonian system on a phase space of dimension $3N$.
As well as the trivial case of a single peakon ($N=1$), the case $N=2$ is Liouville
integrable. We present the explicit solution for the twopeakon dynamics, and describe
some of the novel features of the interaction of peakons in the Popowicz system.

Hone, A. and Varona, J. (2018). Continued fractions and irrationality exponents for modified engel and pierce series. Monatshefte fur Mathematik [Online]:116. Available at: https://doi.org/10.1007/s0060501812441.
An Engel series is a sum of reciprocals of a nondecreasing
sequence (xn) of positive integers, which is such that each term is divisible
by the previous one, and a Pierce series is an alternating sum of the
reciprocals of a sequence with the same property. Given an arbitrary rational
number, we show that there is a family of Engel series which when
added to it produces a transcendental number ? whose continued fraction
expansion is determined explicitly by the corresponding sequence
(xn), where the latter is generated by a certain nonlinear recurrence of
second order. We also present an analogous result for a rational number
with a Pierce series added to or subtracted from it. In both situations (a
rational number combined with either an Engel or a Pierce series), the
irrationality exponent is bounded below by (3 + ?5)/2, and we further
identify infinite families of transcendental numbers ? whose irrationality
exponent can be computed precisely. In addition, we construct the
continued fraction expansion for an arbitrary rational number added to
an Engel series with the stronger property that x2j divides xj+1 for all
j.

Hone, A., Jeffery, L. and Selcoe, R. (2018). On a Family of Sequences Related to Chebyshev Polynomials. Journal of Integer Sequences [Online] 21:18.7.2. Available at: https://cs.uwaterloo.ca/journals/JIS/VOL21/Hone/hone8.pdf.
We consider the appearance of primes in a family of linear recurrence sequences
labelled by a positive integer n. The terms of each sequence correspond to a particular
class of Lehmer numbers, or (viewing them as polynomials in n) dilated versions of the
socalled Chebyshev polynomials of the fourth kind, also known as airfoil polynomials.
We prove that when the value of n is given by a dilated Chebyshev polynomial of the
first kind evaluated at a suitable integer, either the sequence contains a single prime, or no term is prime. For all other values of n, we conjecture that the sequence contains
infinitely many primes, whose distribution has analogous properties to the distribution
of Mersenne primes among the Mersenne numbers. Similar results are obtained for
the sequences associated with negative integers n, which correspond to Chebyshev
polynomials of the third kind, and to another family of Lehmer numbers.

Hone, A. and Ward, C. (2018). On the General Solution of the Heideman–Hogan Family of Recurrences. Proceedings of the Edinburgh Mathematical Society [Online]. Available at: https://doi.org/10.1017/S0013091518000196.
We consider a family of nonlinear rational recurrences of odd order which was introduced
by Heideman and Hogan, and recently rediscovered in the theory of Laurent phenomenon algebras (a
generalization of cluster algebras). All of these recurrences have the Laurent property, implying that for
a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these
particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms
of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an
analogous result for the general solution of each of these recurrences.

Hamad, K., Hone, A., van der Kamp, P. and Quispel, G. (2018). QRT maps and related Laurent systems. Advances in Applied Mathematics [Online] 96. Available at: https://doi.org/10.1016/j.aam.2017.12.006.
In recent work it was shown how recursive factorisation of certain QRT maps leads to Somos4 and Somos5 recurrences with periodic coefficients, and to a fifthorder recurrence with the Laurent property. Here we recursively factorise the 12parameter symmetric QRT map, given by a secondorder recurrence, to obtain a system of three coupled recurrences which possesses the Laurent property. As degenerate special cases, we first derive systems of two coupled recurrences corresponding to the 5parameter multiplicative and additive symmetric QRT maps. In all cases, the Laurent property is established using a generalisation of a result due to Hickerson, and exact formulae for degree growth are found from ultradiscrete (tropical) analogues of the recurrences. For the general 18parameter QRT map it is shown that the components of the iterates can be written as a ratio of quantities that satisfy the same Somos7 recurrence.

Hone, A. and Zullo, F. (2018). A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI. Random Matrices: Theory and Applications [Online] 7:1840001. Available at: https://doi.org/10.1142/S2010326318400014.
We present some observations on the taufunction for the fourth Painlev´e equation.
By considering a Hirota bilinear equation of order four for this taufunction,
we describe the general form of the Taylor expansion around an arbitrary movable
zero. The corresponding Taylor series for the taufunctions of the first and
second Painlev´e equations, as well as that for the Weierstrass sigma function,
arise naturally as special cases, by setting certain parameters to zero.

Hone, A., Kouloukas, T. and Quispel, G. (2017). Some integrable maps and their Hirota bilinear forms. Journal of Physics A: Mathematical and Theoretical [Online] 51. Available at: https://doi.org/10.1088/17518121/aa9b52.
We introduce a twoparameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the HirotaMiwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular twodimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case.

Hone, A., Novikov, V. and Wang, J. (2017). Generalizations of the short pulse equation. Letters in Mathematical Physics [Online] 108:927947. Available at: https://doi.org/10.1007/s1100501710223.
We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation.

Hone, A., Kouloukas, T. and Ward, C. (2017). On reductions of the HirotaMiwa equation. Symmetries, Integrability and Geometry: Methods and Applications [Online] 13:117. Available at: https://dx.doi.org/10.3842/SIGMA.2017.057.
The HirotaMiwa equation (also known as the discrete KP equation, or the octahedron recurrence), is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The HirotaMiwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge transformation), defined by a triple of integers or halfintegers, which produce bilinear ordinary difference equations of Somos/GaleRobinson type. Here it is explained how to obtain Lax pairs and presymplectic structures for these reductions, in order to demonstrate Liouville integrability of some associated maps, certain of which are related to reductions of discrete Toda and discrete KdV equations.

Hone, A., Novikov, V. and Wang, J. (2017). Twocomponent generalizations of the CamassaHolm equation. Nonlinearity [Online] 30:622658. Available at: http://iopscience.iop.org/article/10.1088/13616544/aa5490/meta;jsessionid=0AADAAD96C412EF897587E993641D098.c2.iopscience.cld.iop.org.
A classification of integrable twocomponent systems of nonevolutionary partial differential equations that are analogous to the CamassaHolm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to biHamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered.

Fedorov, Y. and Hone, A. (2016). Sigmafunction solution to the general Somos6 recurrence via hyperelliptic Prym varieties. Journal of Integrable Systems [Online] 1. Available at: https://doi.org/10.1093/integr/xyw012.
We construct the explicit solution of the initial value problem for sequences generated by the general Somos6 recurrence relation, in terms of the Kleinian sigmafunction of genus two. For each sequence there is an associated genus two curve \(X\), such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of \(X\). The construction is based on a Lax pair with a spectral curve \(S\) of genus four admitting an involution \(\sigma\) with two fixed points, and the Jacobian of \(X\) arises as the Prym variety Prym \((S,\sigma)\).

Hone, A. (2016). On the continued fraction expansion of certain Engel series. Journal of Number Theory [Online] 164:269281. Available at: http://dx.doi.org/10.1016/j.jnt.2015.12.024.
An Engel series is a sum of the reciprocals of an increasing sequence of positive integers, which is such that each term is divisible by the previous one. Here we consider a particular class of Engel series, for which each term of the sequence is divisible by the square of the preceding one, and find an explicit expression for the continued fraction expansion of the sum of a generic series of this kind. As a special case, this includes certain series whose continued fraction expansion was found by Shallit. A family of examples generated by nonlinear recurrences with the Laurent property is considered in detail, along with some associated transcendental numbers.

Hone, A. (2015). Curious Continued Fractions, Nonlinear Recurrences and Transcendental Numbers. Journal of Integer Sequences [Online] 18:110. Available at: https://cs.uwaterloo.ca/journals/JIS/VOL18/Hone/hone3.pdf.
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have
the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also integers), appear interlaced in the continued fraction expansion of the sum of the reciprocals of the terms. Using the rapid (double exponential) growth of the terms, for each sequence it is shown that the sum of the reciprocals is a transcendental number.

Hone, A. and Towler, K. (2015). Nonstandard discretization of biological models. Natural Computing [Online] 14:3948. Available at: http://link.springer.com/article/10.1007%2Fs1104701494634#page1.
We consider certain types of discretization schemes for differential equations with quadratic nonlinearities, which were introduced by Kahan, and considered in a broader setting by Mickens. These methods have the property that they preserve important structural features of the original systems, such as the behaviour of solutions near to fixed points, and also, where appropriate (e.g. for certain mechanical systems), the property of being volumepreserving, or preserving a symplectic/Poisson structure. Here we focus on the application of Kahan's method to models of biological systems, in particular to reaction kinetics governed by the Law of Mass Action, and present a general approach to birational discretization, which is applied to population dynamics of LotkaVolterra type.

Hone, A. (2015). Continued fractions for some transcendental numbers. Monatshefte fur Mathematik [Online]. Available at: http://link.springer.com/article/10.1007/s0060501508442?wt_mc=internal.event.1.SEM.ArticleAuthorOnlineFirst.
We consider series of the form $p/q + \sum_{j=2}^\infty 1/x_j$, where $x_1=q$ and the integer sequence $x_n$ satisfies a certain nonautonomous recurrence of second order, which entails that $x_nx_{n+1}$ for n?1. It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.

Hone, A. (2015). Algebraic entropy for algebraic maps. Journal of Physics A: Mathematical and Theoretical [Online] 49. Available at: http://dx.doi.org/10.1088/17518113/49/2/02LT01.
We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Backlund transformations.

Hone, A. and Lafortune, S. (2014). Stability of stationary solutions for nonintegrable peakon equations. Physica D: Nonlinear Phenomena [Online] 269:2836. Available at: http://dx.doi.org/10.1016/j.physd.2013.11.006.
The CamassaHolm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multisoliton solutions  "peakons"  with a peaked shape corresponding to a discontinuous first derivative. There is a oneparameter family of generalized CamassaHolm equations, most of which are not integrable, but which all admit peakon solutions. Numerical studies reported by Holm and Staley indicate changes in the stability of these and other solutions as the parameter varies through the family.
In this article, we describe analytical results on one of these bifurcation phenomena, showing that in a suitable parameter range there are stationary solutions  "leftons"  which are orbitally stable.

Fordy, A. and Hone, A. (2014). Discrete integrable systems and Poisson algebras from cluster maps. Communications in Mathematical Physics [Online] 325:527584. Available at: http://dx.doi.org/10.1007/s002200131867y.
We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutationperiodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skewsymmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure.
Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville–Arnold sense.

Hone, A. and Inoue, R. (2014). Discrete Painlevé equations from Ysystems. Journal of Physics A: Mathematical and Theoretical [Online] 47. Available at: http://iopscience.iop.org/article/10.1088/17518113/47/47/474007/meta;jsessionid=1BAB92959D74F9C50E2861EC9E77E502.c1.
We consider Tsystems and Ysystems arising from cluster mutations applied to quivers that have the property of being periodic under a sequence of mutations. The corresponding nonlinear recurrences for cluster variables (coefficientfree Tsystems) were described in the work of Fordy and Marsh, who completely classified all such quivers in the case of period 1, and characterized them in terms of the skewsymmetric exchange matrix B that defines the quiver. A broader notion of periodicity in general cluster algebras was introduced by Nakanishi, who also described the corresponding Ysystems, and Tsystems with coefficients.
A result of Fomin and Zelevinsky says that the coefficientfree Tsystem provides a solution of the Ysystem. In this paper, we show that in general there is a discrepancy between these two systems, in the sense that the solution of the former does not correspond to the general solution of the latter. This discrepancy is removed by introducing additional nonautonomous coefficients into the Tsystem. In particular, we focus on the period 1 case and show that, when the exchange matrix B is degenerate, discrete Painlev\'e equations can arise from this construction.

Hone, A. and Ward, C. (2014). A family of linearizable recurrences with the Laurent property. Bulletin of the London Mathematical Society [Online] 46:503516. Available at: http://dx.doi.org/10.1112/blms/bdu004.
We consider a family of nonlinear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearizable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrödinger operators is also explained.

Hone, A., van der Kamp, P., Quispel, G. and Tran, D. (2013). Integrability of reductions of the discrete Kortewegde Vries and potential Kortewegde Vries equations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences [Online] 469. Available at: http://dx.doi.org/10.1098/rspa.2012.0747.
We study the integrability of mappings obtained as reductions of the discrete Korteweg–de Vries (KdV) equation and of two copies of the discrete potential KdV (pKdV) equation. We show that the mappings corresponding to the discrete KdV equation, which can be derived from the latter, are completely integrable in the Liouville–Arnold sense. The mappings associated with two copies of the pKdV equation are also shown to be integrable.

Hone, A., Ragnisco, O. and Zullo, F. (2013). Properties of the series solution for Painlevé I. Journal of Nonlinear Mathematical Physics [Online] 20:85100. Available at: http://dx.doi.org/10.1080/14029251.2013.862436.
We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the taufunction around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented.

Chu, D., Zabet, N. and Hone, A. (2011). Optimal Parameter Settings for Information Processing in Gene Regulatory Networks. BioSystems [Online]:182196. Available at: http://www.cs.kent.ac.uk/pubs/2011/3081.
Gene networks can often be interpreted as computational circuits. This article investigates the computational properties of gene regulatory networks defined in terms of the speed and the accuracy of the output of a gene network. It will be shown that there is no single optimal set of parameters, but instead, there is a tradeoff between speed and accuracy. Using the tradeoff it will also be shown how systems with various parameters can be ranked with respect to their computational efficiency. Numerical analysis suggests that the tradeoff can be improved when the output gene is repressing itself, even though the accuracy or the speed of the autoregulated system may be worse than the unregulated system.

Fordy, A. and Hone, A. (2011). Symplectic Maps from Cluster Algebras. Symmetry, Integrability and Geometry: Methods and Applications [Online] 7:112. Available at: http://dx.doi.org/10.3842/SIGMA.2011.091.
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skewsymmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and nonintegrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map.

Hone, A., Lundmark, H. and Szmigielski, J. (2009). Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable CamassaHolm type equation. Dynamics of Partial Differential Equations [Online] 6:253289. Available at: http://www.intlpress.com/DPDE/journal/DPDEv06.php.
Recently Vladimir Novikov found a new integrable analogue of the CamassaHolm equation, admitting peaked soliton (peakon) solutions, which has nonlinear terms that are cubic, rather than quadratic. In this paper, the explicit formulas for multipeakon solutions of Novikov's cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the DegasperisProcesi equation

Hone, A. and Petrera, M. (2009). Threedimensional discrete systems of HirotaKimura type and deformed LiePoisson algebras. Journal of Geometric Mechanics [Online] 1:5585. Available at: http://dx.doi.org/10.3934/jgm.2009.1.55.
Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable biHamiltonian system in three dimensions. The HirotaKimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the twodimensional LotkaVolterra system.
The Euler top is naturally written in terms of the so(3) LiePoisson algebra. Here we consider algebraically integrable systems that are associated with pairs of LiePoisson algebras in three dimensions, as presented by Gümral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and HirotaKimura. We show that the maps thus obtained are also biHamiltonian, with pairs of compatible Poisson brackets that are oneparameter deformations of the original LiePoisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three biHamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd's Diophantine integrability criterion.

Hone, A. and Irle, M. (2009). On the nonintegrability of the Popowicz peakon system. Dynamical Systems and Differential Equations [Online] 2009:359366. Available at: http://dx.doi.org/10.3934/proc.2009.2009.359.
We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a twofield coupling between the CamassaHolm and DegasperisProcesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation with Painlev\'e analysis provides strong evidence that the Popowicz system is nonintegrable. Nevertheless, we are able to construct exact travelling wave solutions in terms of an elliptic integral, together with a degenerate travelling wave corresponding to a single peakon. We also describe the dynamics of Npeakon solutions, which is given in terms of an Hamiltonian system on a phase space of dimension 3N.

Hone, A. and Wang, J. (2008). Integrable peakon equations with cubic nonlinearity. Journal of Physics A: Mathematical and Theoretical [Online] 41. Available at: http://dx.doi.org/10.1088/17518113/41/37/372002.
We present a new integrable partial differential equation found by Vladimir Novikov. Like the CamassaHolm and DegasperisProcesi equations, this new equation admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic. We give a matrix Lax pair for V Novikov's equation, and show how it is related by a reciprocal transformation to a negative flow in the SawadaKotera hierarchy. Infinitely many conserved quantities are found, as well as a biHamiltonian structure. The latter is used to obtain the Hamiltonian form of the finitedimensional system for the interaction of N peakons, and the twobody dynamics (N = 2) is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao.

Timmis, J., Hone, A., Stibor, T. and Clark, E. (2008). Theoretical advances in artificial immune systems. Theoretical Computer Science [Online] 403:1132. Available at: http://dx.doi.org/10.1016/j.tcs.2008.02.011.
Artificial immune systems (AIS) constitute a relatively new area of bioinspired computing. Biological models of the natural immune system, in particular the theories of clonal selection, immune networks and negative selection, have provided the inspiration Cor AIS algorithms. Moreover, such algorithms have been successfully employed in a wide variety of different application areas. However, despite these practical successes, until recently there has been a dearth of theory to justify their Use. In this paper, the existing theoretical work oil AIS is reviewed. After the presentation of a simple example of each of the three main types of AIS algorithm (that is, clonal selection, immune network and negative selection algorithms respectively), details of the theoretical analysis for each of these types are given. Some of the future challenges in this area are also highlighted.

Hone, A. and Swart, C. (2008). Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences. Mathematical Proceedings of the Cambridge Philosophical Society [Online] 145:6585. Available at: http://dx.doi.org/10.1017/s030500410800114x.
Somos 4 sequences are a family of sequences defined by a fourthorder quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a byproduct, nonperiodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.

Hone, A., Novikov, V. and Verhoeven, C. (2008). An extended HenonHeiles system. Physics Letters A [Online] 372:14401444. Available at: http://dx.doi.org/10.1016/j.physleta.2007.09.063.
We consider an integrable system of partial differential equations possessing a Lax pair with an energydependent Lax operator of second order, of the type described by Antonowicz and Fordy. By taking the travelling wave reduction of this system we show that the integrable case (ii) HenonHeiles system can be extended by adding an arbitrary number of nonpolynomial (rational) terms to the potential.

Common, A. and Hone, A. (2008). Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation. Journal of Physics A: Mathematical and Theoretical [Online] 41:485203. Available at: http://dx.doi.org/10.1088/17518113/41/48/485203.
The YablonskiiVorob'ev polynomials yn(t), which are defined by a second order bilinear differentialdifference equation, provide rational solutions of the Toda lattice. They are also polynomial taufunctions for the rational solutions of the second Painlev\'{e} equation (PII). Here we define twovariable polynomials Yn(t,h) on a lattice with spacing h, by considering rational solutions of the discrete time Toda lattice as introduced by Suris. These polynomials are shown to have many properties that are analogous to those of the YablonskiiVorob'ev polynomials, to which they reduce when h=0. They also provide rational solutions for a particular discretisation of PII, namely the so called {\it alternate discrete} PII, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation (PIII). It is shown that B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is a symplectic map, and the shift in time is also symplectic. Finally we present a Lax pair for the alternate discrete PII, which recovers Jimbo and Miwa's Lax pair for PII in the continuum limit h?0.

Hone, A. (2007). Sigma function solution of the initial value problem for Somos 5 sequences. Transactions of the American Mathematical Society [Online] 359:50195034. Available at: http://dx.doi.org/10.1090/S0002994707042158.
The Somos 5 sequences are a family of sequences defined by a fifth order bilinear recurrence relation with constant coefficients. For particular choices of coefficients and initial data, integer sequences arise. By making the connection with a second order nonlinear mapping with a first integral, we prove that the two subsequences of odd/even index terms each satisfy a Somos 4 (fourth order) recurrence. This leads directly to the explicit solution of the initial value problem for the Somos 5 sequences in terms of the Weierstrass sigma function for an associated elliptic curve.

Hone, A. (2007). Singularity confinement for maps with the Laurent property. Physics Letters A [Online] 361:341 345. Available at: http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVM4M27S0J3&_user=125871&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000010239&_version=1&_urlVersion=0&_userid=125871&md5=03ddab063508d4dd3af4ab7ba9c8bdf1.
The singularity confinement test is very useful for isolating integrable cases of discretetime dynamical systems, but it does not provide a sufficient criterion for integrability. Quite recently a new property of the bilinear equations appearing in discrete soliton theory has been noticed: The iterates of such equations are Laurent polynomials in the initial data. A large class of nonintegrable mappings of the plane are presented which both possess this Laurent property and have confined singularities

Hone, A. (2007). Laurent Polynomials and Superintegrable Maps. Symmetry, Integrability and Geometry: Methods and Applications [Online] 3:118. Available at: http://www.emis.de/journals/SIGMA/2007/022/.
This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos4 recurrences, and on the Laurent property. Subsequently a family of fourthorder recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations.

Hone, A. and Kouloukas, T. (2020). Discrete Hirota reductions associated with the lattice KdV equation. Journal of Physics A: Mathematical and Theoretical [Online]. Available at: http://dx.doi.org/10.1088/17518121/aba1b8.
We study the integrability of a family of birational maps obtained as reductions of the discrete Hirota equation, which are related to travelling wave solutions of the lattice KdV equation. In particular, for reductions corresponding to waves moving with rational speed N/M on the lattice, where N,M are coprime integers, we prove the Liouville integrability of the maps when N + M is odd, and prove various properties of the general case. There are two main ingredients to our construction: the cluster algebra associated with each of the Hirota bilinear equations, which provides invariant (pre)symplectic and Poisson structures; and the connection of the monodromy matrices of the dressing chain with those of the KdV travelling wave reductions.

Hone, A. and Pallister, J. (2020). Linear relations for Laurent polynomials and lattice equations. Nonlinearity [Online]. Available at: http://iopscience.iop.org/09517715.
A recurrence relation is said to have the Laurent property if all of its iterates are Laurent polynomials in the initial values with integer coefficients. Recurrences with this property appear in diverse areas of mathematics and physics, ranging from Lie theory and supersymmetric gauge theories to Teichmuller theory and dimer models. In many cases where such recurrences appear, there is a common structural thread running between these different areas, in the form of Fomin and Zelevinsky's theory of cluster algebras. Laurent phenomenon algebras, as defined by Lam and Pylyavskyy, are an extension of cluster algebras, and share with them the feature that all the generators of the algebra are Laurent polynomials in any initial set of generators (seed).
Here we consider a family of nonlinear recurrences with the Laurent property, referred to as "Little Pi", which was derived by Alman et al. via a construction of periodic seeds in Laurent phenomenon algebras, and generalizes the HeidemanHogan family of recurrences. Each member of the family is shown to be linearizable, in the sense that the iterates satisfy linear recurrence relations with constant coefficients. We derive the latter from linear relations with periodic coefficients, which were found recently by Kamiya et al. from travelling wave reductions of a linearizable lattice equation on a 6point stencil. By making use of the periodic coefficients, we further show that the birational maps corresponding to the Little Pi family are maximally superintegrable.
We also introduce another linearizable lattice equation on the same 6point stencil, and present the corresponding linearization for its travelling wave reductions. Finally, for both of the 6point lattice equations considered, we use the formalism of van der Kamp to construct a broad class of initial value problems with the Laurent property.

Hone, A. (2020). ECM Factorization with QRT Maps. Transactions on Computational Science and Computational Intelligence.
QuispelRobertsThompson (QRT) maps are a family of birational maps of the plane which provide the simplest discrete analogue of an integrable Hamiltonian system, and are associated with elliptic fibrations in terms of biquadratic curves. Each generic orbit of a QRT map corresponds to a sequence of points on an elliptic curve. In this preliminary study, we explore versions of the elliptic curve method (ECM) for integer factorization based on performing scalar multiplication of a point on an elliptic curve by iterating three different QRT maps with particular initial data. Pseudorandom number generation and other possible applications are briefly discussed.

Hone, A. (2020). Efficient ECM Factorization in Parallel with the Lyness Map. In: ISSAC ’20: International Symposium on Symbolic and Algebraic Computation.
The Lyness map is a birational map in the plane which provides one of the simplest discrete analogues of a Hamiltonian system with one degree of freedom, having a conserved quantity and an invariant symplectic form. As an example of a symmetric QuispelRobertsThompson (QRT) map, each generic orbit of the Lyness map lies on a curve of genus one, and corresponds to a sequence of points on an elliptic curve which is one of the fibres in a pencil of biquadratic curves in the plane. Here we present a version of the elliptic curve method (ECM) for integer factorization, which is based on iteration of the Lyness map with a particular choice of initial data. More precisely, we give an algorithm for scalar multiplication of a point on an arbitrary elliptic curve over Q, which is represented by one of the curves in the Lyness pencil. In order to avoid field inversion (I), and require only field multiplication (M), squaring (S) and addition, projective coordinates in P1 × P1 are used. Neglecting multiplication by curve constants (assumed small), each addition of the chosen point uses 2M, while each doubling step requires 15M. We further show that the doubling step can be implemented efficiently in parallel with four processors, dropping the effective cost to 4M. In contrast, the fastest algorithms in the literature use twisted Edwards curves (equivalent to Montgomery curves), which correspond to a subset of all elliptic curves. Scalar muliplication on twisted Edwards curves with suitable small curve constants uses 8M for point addition and 4M+4S for point doubling, both of which can be run in parallel with four processors to yield effective costs of 2M and 1M + 1S, respectively. Thus our scalar multiplication algorithm should require, on average, roughly twice as many multiplications per bit as state of the art methods using twisted Edwards curves. In our conclusions, we discuss applications where the use of Lyness curves may provide potential advantages.

Pallister, J. (2020). Linearisability and Integrability of Discrete Dynamical Systems from Cluster and LP Algebras.
From the bipartite belt of a cluster algebra one may obtain generalisations of frieze patterns. It has been proven that linear relations exist within these frieze patterns if the associated quiver is, up to mutation equivalence, Dynkin or affine. The second chapter of this work is devoted to reproving this fact, for affine D and E types, using alternative methods to the known proof, allowing much more detail. We prove the existence of periodic quantities for affine ADE friezes with periods that mirror the widths of the tubes of their AuslanderReiten quivers. Furthermore we interpret these friezes as discrete dynamical systems, given by a generalised cluster map. We prove the integrability of a reduction of this cluster map for each affine E type and for affine D with an even number of vertices. In our third chapter we consider recurrences that lie beyond cluster algebras, in LP algebras, named because mutation in these algebras has the Laurent property, like cluster algebras. We examine two particular examples of these recurrences and show that they can be linearised. We also show that they can be obtained by reductions of lattice equations. Finally we consider a 2dimensional version ofthe Laurent property and give large sets of initial values such that these lattice equations possess this generalised Laurent property.

Hone, A. and Quispel, G. (2019). Analogues of Kahan’s method for higher order equations of higher degree. In: Shi, Y. ed. Springer Proceedings in Mathematics & Statistics. Springer.
Kahan introduced an explicit method of discretization for systems of first order differential equations with nonlinearities of degree at most two (quadratic vector fields). Kahan's method has attracted much interest due to the fact that it preserves many of the geometrical properties of the original continuous system. In particular, a large number of Hamiltonian systems of quadratic vector fields are known for which their Kahan discretization is a discrete integrable system. In this note, we introduce a special class of explicit orderpreserving discretization schemes that are appropriate for certain systems of ordinary differential equations of higher order and higher degree.