# David M. Grobman

**David Grobman** (Russian: Давид Матвеевич Гробман; November 16, 1922, Moscow, USSR - March 31, 2007, Palo Alto, USA) - Soviet mathematician, Doctor of Physics and Mathematics (1966), professor (1982). He made a significant contribution to the development of the theory of differential equations and its applications to the analysis of dynamical systems (the Hartman-Grobman theorem). D. Grobman belongs to the first generation of Soviet programmers. He was a lead software developer for computers M-2 and M-5. He is one of the founders of the national school algorithmic modeling and diagnostics of computer digital units. He has developed a number of original methods in this area. He was the head and lead developer of automated software system for modeling, synthesis and analysis of tests used in INEUM (Russian: Институт Электронных Управляющих Машин, the Institute of Electronic Computing Machines) for diagnosing errors in digital units of M-5 computer, ASVT-M computer series, various models of SM computer series. Published 80 scientific papers, including 10 in Proceedings of the USSR Academy of Sciences DAN USSR...^{[1]}

## Scientific activity

### Differential Equations

Mathematical interests of D. Grobman lay in the theory of nonlinear dynamic systems. This is how a well-known professor of mathematics V.V. Nemytsky in the article “Mathematics in the USSR for forty years (1917-1957)” ^{[2]} characterized D. Grobman: “We turn to the comparison method for nonlinear systems. This topic, starting from the works of A.M. Lyapunova, I.G. Petrovsky, has always been successfully developed by Soviet mathematicians. The last decade is no exception. A.A.Shestakov and A.U.Paivin, V.A.Yakubovich and D.M.Grobman obtained outstanding results. It should be noted that the theorem that was formulated and proved by D.M. Grobman, hereinafter referred to as the Grobman-Hartman theorem, is classical and represents a significant contribution to the development of the qualitative theory of differential equations and the theory of dynamic systems” ^{[3]}

The Grobman-Hartman Theorem specifies conditions for nonlinear dynamic system to behave as linear in the neighborhood of equilibrium point: “only when the equilibrium is not hyperbolic do the nonlinear term have to be accounted for in analyzing the stability properties of the equilibrium point.”^{[4]}

“The situation surrounding the Hartman-Grobman Theorem suggests that there is something special about systems having nonhyperbolic equilibrium points, and that for such cases there may well be a third ^{[5]} manifold of initial points for which the nonlinear terms cannot be neglected in analyzing the behavior of nonlinear system – even locally.”

D. Grobman is a coauthor of the «Theory of Lyapunov characteristic exponents» book^{[6]}

### Simulation, control and diagnostics of digital devices

Before solving the problems of technical diagnostics of Computer Digital Units, D. Grobman took an active part in the design of the M-5 computer, in particular, in specifying the system of commands. He developed programming techniques for this machine and led the work on programming automation. In addition, his works in computational mathematics found applications in the field of econometrics and power industry.

D. Grobman together with I.S. Brook defined the framework for the department of technical diagnostics of Computer Digital Units. It included the creation of algorithmic methods for synthesis and analysis of tests for digital units, as well as the design and development of hardware for testing and diagnostics ^{[7]}

D. Grobman authored numerous seminal articles. He was the scientific editor of the collective volumes of Scientific Papers on testing and diagnostics published by INEUM, in which he and his colleagues formulated the principles for creating automated software systems for modeling, synthesizing and analyzing the diagnostic tests for digital devices. He was a scientific adviser for multiple doctoral students, 12 of which awarded PhD degree. ^{[8]}

## References

- ↑ Encyclopedia of Electronic Computing Machines (Russian: Энциклопедия вычислительной техники)
- ↑ https://vdoc.pub/documents/1917-1957-6br22bhb1350
- ↑ B.M. Basok. D. Grobman’s Department of Technical Diagnostics and his school. Journal of the History of science and technology, #5, 2008 (Б.М. Басок. Отдел диагностического контроля Д.М. Гробмана и его школа. "История науки и техники", #5, 2008) - https://www.computer-museum.ru/articles/predpriyatiya-i-nii/2140/?sphrase_id=686693
- ↑ J.L. Casti. Five More Golden Rules, p. 54-60. John Willey & Sons, NY, 2000.
- ↑ Center Manifold. The two others are the stable and unstable manifolds. The Center Manifold Theorem, which is a step towards proving Hopf Bifurcation Theorem shows “that the question of whether initial states lying on the center manifold are attracted back to the equilibrium cannot be answered by appealing solely to information contained in the linear approximation to the system – even for initial states arbitrary close to the equilibrium.”[5]
- ↑ B.Bylov, R.Vinograd, D.Grobman, V.Nemytsky. Theory of Lyapunov characteristic exponents, Nauka, M., 1966 (Б.Ф.Былов, Р.Э.Виноград, Д.М.Гробман, В.В.Немыцкий. Теория показателей Ляпунова, Наука, М., 1966)
- ↑ B. M. Basok. INEUM: development of software and hardware for diagnostic control and verification of computer systems. Virtual computer museum. (In Russinr: Б. М. Басок. ИНЭУМ: разработка программно-аппаратных средств диагностического контроля и верификации вычислительных систем. Виртуальный компьютерный музей.)
- ↑ B. M. Basok. David Grobman's 100th anniversary. Virtual computer museum. (In Russinr: Б. М. Басок. 100 лет Давиду Матвеевичу ГробмануВиртуальный компьютерный музей.)

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