Nonstandard discretisation of Painlevé type Hamiltonian systems
SIDE 13 conference, 11 – 17 November 2018, Fukuoka, JapanDr. Thomas Kecker
University of Portsmouth, School of Mathematics and Physics, United Kingdomthomas.kecker@port.ac.uk
Portsmouth harbour, 23 Oct 2018, 6 pm GMTAbstract
We consider the non-standard discretisation method introduced by Kahan & Li [6], for quadratic vector fields ẋ = F (x), givenby the formula−1hx̃ − x= I − DF (x)F (x).
h2
We apply the method to certain cubic Hamiltonian systems of Painlevé type, investigate the possibility of singularity confinement for the resulting systems and their algebraic entropy, or dynamical degree. This is joint work with Dr. habil. Galina
Filipuk (University of Warsaw, Poland) [2].

Introduction
Discretising a differential equation or system of equations is not a unique process. Althoughmany discretisation schemes are available, in particalar for numerical solutions, they often
do not preserve certain nice properties of the equations. For integrable equations, in addition to having the original system as a continuous limit, we seek for the discretised system
to preserve integrability. The Kahan method was applied to the equations of motion for theEuler top [4] and the Lagrange top [5] by Hirota and Kimura, where it does exactly that, andhence is also known as the Hirota-Kimura method. However, for non-autonomous systems
such as, for example the Painlevé equations, the method is in general not successful, apartfrom some special cases [1].

Dynamical degree
Algebraic entropy was introduced by Hietarinta & Viallet [3] as a measure of complexity
for a discrete map, defined bylog dnlim,n→∞nwhere the dn denote the degrees of rational iterates under the discrete map. Zero algebraic
entropy is associated with integrable discrete maps. The dynamical degree λ is the limit
factor of degree growth of the rational iterates under the discrete map so essentially just theexponential of the algebraic entropy.
Using M ATHEMATICA, we obtain the dynamical degree of the discrete map (2):• when fn = cn,

c 6= α − β,c 6= 0, the dynamical degree is 2

• when fn = (α − β)n, α 6= β, the dynamical degree λ is 1 < λ < 2,
where λ ≈ 1.839... is the largest real root of the polynomial λ4 − 2λ3 + 1.
This is found by generating the sequence of degrees dn of the rational iterates:
1, 2, 4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, . . . , obeying the recursive law
dn = 2dn−1 − dn−4,

d0 = 1,dn = 0,

n < 0.

Conclusions
A cubic Hamiltonian systems related to Painlevé IV
We discretise the following differential system given by a cubic Hamiltonian studied in [7],
1 33H(z, p, q) = p + q + f (z)pq + αp + βq, α, β ∈ C
3q 0 = p2 + f (z)q + α, p0 = −q 2 − f (z)p − β,(1)When f (z) = cz + d is linear, c 6= 0, the system is related to the Painlevé IV equation, andin case c = 0, f (z) = const., it has elliptic solutions. We apply the Kahan method to obtainthe following non-autonomous discrete system:
2(fn+1 − 2)yn2 + 4βyn − (fn + 2)(fn+1 + 2)xn − 2αfn − 4αxn+1 =−4 + fn2 − 4xnyn
2(fn+1 + 2)x2n + 4αxn − (fn − 2)(fn+1 − 2)yn − 2βfn + 4β
yn+1 =.
2−4 + fn − 4xnyn(2)

We will perform a type of singularity confinement test, imposing a condition on fn, anddetermine the dynamical degree of the discrete map in this case.

Due to singularity confinement in the case fn = (α−β)n, cancellations occur in the iterationof the discrete map leading to a reduced dynamical degree compared to the generic case, but
not enough for the map to be integrable. This corresponds to the case f (z) = (α − β)z inthe Hamiltonian system (1). When f (z) = cz the system (1) is related to the fourth Painlevé
equation, but only for the special case c = α − β do we get singularity confinement. Thequestion remains how one can further modify the Kahan discretisation method to obtain an
integrable discrete map.

Acknowledgements
TK would like to thank the oragnisers of SIDE 13 for their generous financial support toattend the conference. Furthermore, travel support by the QJMAM fund (Quaterly Journalof Mechanics and Applied Mathematics) provided by the IMA (Institute of Mathematics
and its Applications, U.K.) is greatly acknowledged. The research was conducted in part
supported by an LMS (London Mathematical Society) ’Research in Pairs’ grant for TK tovisit Galina Filipuk in Warsaw, for which we are grateful.

Singularity confinement
Singularities in the discrete system (2) occur when the denominator vanishes, i.e. when
4xnyn = (fn + 2)(fn − 2). We apply a singularity confinement test in the form
fnxn = − σ + ,
2

fnyn = + σ + ,2σ ∈ {1, −1},||  1.

2(2 + α + β) + fn(α − β + fn + 2fn+1)
xn+1 =+ O(1)2fn
2(2 + α + β) + fn(α − β + fn + 2fn+1)
yn+1 = −+ O(1)2fn
The singularity reappears periodically, unless fn satisfies the condition
α − β + fn − fn+1 = 0=⇒

fn = (α − β)n + γ,γ ∈ C,

that is, when f (z) = (α − β)z + d is a linear function in the original system (1), corresponding to the case when it is related to the Painlevé IV equation. But although (1) is integrable
for any f (z) = cz + d, we only find singularity confinement in the discrete system for aspecial value of c.

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to the fourth Painlevé equation, ZAMM (J. for Applied Mathematics and Mechanics),
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