##
**Discrete Hamiltonian systems. Difference equations, continued fractions, and Riccati equations.**
*(English)*
Zbl 0860.39001

Kluwer Texts in the Mathematical Sciences. 16. Dordrecht: Kluwer Academic Publishers. xiv, 374 p. (1996).

The material is devoted to the four topics of the title, the common link being an elementary theory of difference equations \((\Delta\text{Es})\). The objective is to help higher-semester students identify, and possibly avoid spurious solutions in numerical computations. No familiarity with advanced calculus and the theory of ordinary differential equations (ODEs) is required. The reasoning is linear, and its main tools are matrix operations. Matrix elements are real or complex-valued.

An introductory chapter considers non-singular real-valued recurrences with constant coefficients (point maps). Their general solutions can be expressed as a sum of eigenfunctions, which are generalized exponentials. In an ‘associated’ \(\Delta\text{E}\) the integer \(n\) is replaced by a continuous independent variable \(t\). Initial points become initial functions. It is not explicitly stated that the corresponding solutions become functionals. Since neighbouring points iterated (mapped) by a recurrence are not interrelated (absence of derivatives, integrals, and other smoothness inducers) the initial functions of the associated \(\Delta\text{Es}\) can be sampled, and the sampling leads to Cauchy solutions of a recurrence. Then \(n\) is replaced by \(t\).

Since it is known that a Riccati ODE can be linearized by a transformation involving a logarithmic derivative, a discrete analogue is used to linearize discrete Riccati \(\Delta\text{Es}\). When a recurrence is nonautonomous (coefficients are functions of \(n)\), then in a nonsingular case explicit solutions can be expressed in the form of a finite product of the coefficients, and in certain cases in the form of a continuous fraction. The procedure is similar to that used by Perron to construct a solution of the hypergeometric ODE.

Hamiltonians arise in connection with optimization problems. They are defined abstractly, and not (as one might expect) in terms of the total energy of the control system. The deduced \(\Delta\text{Es}\) contain both retarded and advanced terms (consequence of the absence of damping, which leads to an irreversible energy loss). For numerical computations this intrinsic property provides a ‘natural’ predictor-corrector scheme. When the coefficients are constants, a solution can be constructed in terms of the eigenfunctions of the associated recurrence. When at least one coefficient is periodic, then a Floquet or Lyapunov method becomes necessary (possibility of ‘internal’ resonances). The former is briefly mentioned, but not elaborated.

The rest of the textbook is devoted to a discrete second variation, and then to sundry topics.

The bibliography is of the ‘sectarian’ type: papers of ‘rival’ authors, contemporary or ancient, are not cited.

It is a matter of educational policy whether students of applied disciplines should be screened off from fundamental mathematical problems, or on the contrary, confronted with them. The authors hold the first view. Two examples suffice to illustrate it:

1. It is not explained why a particular solution of \(x(t) = nx (t-1)\) is \(\Gamma (t+1)\), and not a product of terms containing integers and \(t\)-mod 1.

2. Unequal steps in finite differences are related to fractional iterates. The classical definition of the latter (cf. Birkhoff) is not even mentioned.

An introductory chapter considers non-singular real-valued recurrences with constant coefficients (point maps). Their general solutions can be expressed as a sum of eigenfunctions, which are generalized exponentials. In an ‘associated’ \(\Delta\text{E}\) the integer \(n\) is replaced by a continuous independent variable \(t\). Initial points become initial functions. It is not explicitly stated that the corresponding solutions become functionals. Since neighbouring points iterated (mapped) by a recurrence are not interrelated (absence of derivatives, integrals, and other smoothness inducers) the initial functions of the associated \(\Delta\text{Es}\) can be sampled, and the sampling leads to Cauchy solutions of a recurrence. Then \(n\) is replaced by \(t\).

Since it is known that a Riccati ODE can be linearized by a transformation involving a logarithmic derivative, a discrete analogue is used to linearize discrete Riccati \(\Delta\text{Es}\). When a recurrence is nonautonomous (coefficients are functions of \(n)\), then in a nonsingular case explicit solutions can be expressed in the form of a finite product of the coefficients, and in certain cases in the form of a continuous fraction. The procedure is similar to that used by Perron to construct a solution of the hypergeometric ODE.

Hamiltonians arise in connection with optimization problems. They are defined abstractly, and not (as one might expect) in terms of the total energy of the control system. The deduced \(\Delta\text{Es}\) contain both retarded and advanced terms (consequence of the absence of damping, which leads to an irreversible energy loss). For numerical computations this intrinsic property provides a ‘natural’ predictor-corrector scheme. When the coefficients are constants, a solution can be constructed in terms of the eigenfunctions of the associated recurrence. When at least one coefficient is periodic, then a Floquet or Lyapunov method becomes necessary (possibility of ‘internal’ resonances). The former is briefly mentioned, but not elaborated.

The rest of the textbook is devoted to a discrete second variation, and then to sundry topics.

The bibliography is of the ‘sectarian’ type: papers of ‘rival’ authors, contemporary or ancient, are not cited.

It is a matter of educational policy whether students of applied disciplines should be screened off from fundamental mathematical problems, or on the contrary, confronted with them. The authors hold the first view. Two examples suffice to illustrate it:

1. It is not explained why a particular solution of \(x(t) = nx (t-1)\) is \(\Gamma (t+1)\), and not a product of terms containing integers and \(t\)-mod 1.

2. Unequal steps in finite differences are related to fractional iterates. The classical definition of the latter (cf. Birkhoff) is not even mentioned.

Reviewer: I.Gumowski (Thoiry)

### MSC:

39A10 | Additive difference equations |

39-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to difference and functional equations |

65Q05 | Numerical methods for functional equations (MSC2000) |

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

93C55 | Discrete-time control/observation systems |

39A12 | Discrete version of topics in analysis |