# Construction of Mappings for Hamiltonian Systems and Their Applns

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Construction of mappings for Hamiltonian systems and their applications. Approximation Algorithms for Complex Systems. Inherently parallel algorithms in feasibility and optimization and their applications. Circuits and Systems for Wireless Communications. Approximation algorithms for complex systems. This approach is most ideal to study the long-term evolution of a system, especially in cases where the system exhibits chaotic behavior caused by exponential divergency of orbits with close initial coor- dinates in phase space.

In spite of the extensive use of symplectic maps for many Hamiltonian problems during the last four decades, the derivation of generic symplectic maps from given Hamiltonian equations still remains somehow elusive. There are several approaches to construct symplectic maps from the continuous for- mulation of systems.

## Symplectic integrator

One approach is based on the a priori assumption that the map has a symplectic form and the generating functions associated with the map are found from the equations of motion Lichtenberg and Lieber- man, Another method to construct symplectic maps is based on the assumption that a time-periodic perturbation acting on the integrable system may be replaced by periodic delta functions, which is equivalent to adding fast oscillating terms to the Hamiltonian Wisdom, ; Zaslavsky, ; Sagdeev et al.

Integration of the equations of motion along delta functions gives symplectic maps with the time-step equal to the period of perturbation.

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In particular, this method was used by Chirikov to derive the celebrated standard map Chirikov, ; Lichtenberg and Lieberman, Par- ticularly, they do not establish more general forms of the maps, estimate their accuracy, and establish relations between variables of the original system and of the mapping. Recently in Abdullaev , a mathematically rigorous method to derive symplectic maps has been developed.

Based on the Hamilton—Jacobi theory and the classical perturbation theory, it allows one to construct sym- plectic mappings for generic Hamiltonian systems in a rigorous and consistent way. The method is based on the Hamilton—Jacobi method and pertur- bation theory of classical mechanics. This book compresses 13 chapters.

Application of mapping methods to study physical problems described by Hamiltonian systems are given in Chaps.

In the second chapter we have presented the methods of classical perturbation theory. Time-dependent perturbation theory that constitutes the basis for the construction of symplectic mappings has been also reiter- ated in this chapter. The current methods to construct the symplectic maps for generic Hamiltonian problems are discussed in the third chapter.

## Papers by Yuri B. Suris

The Hamilton—Jacobi method or the method of canonical transformation to con- struct Hamiltonian mappings is presented in the fourth chapter. Map- pings near separatrix of Hamiltonian systems are constructed in Chap. The construction of mappings near separatrix is illustrated in Chap. In Chap. The rescaling invariance proper- ties of Hamiltonian systems near the hyperbolic saddle points are discussed in Chap.

Particularly, in Chap. The book is intended for postgraduate students and researchers, physi- cists, and astronomers working in the areas of Hamiltonian dynamics and chaos, and its applications to plasma physics, hydrodynamics, celestial me- chanics, dynamical astronomy, and accelerator physics. Readers are supposed to be familiar with the meth- ods of classical mechanics on the level of Chaps.

Wolf, Professor Ulrich Samm and Professor Gert Eilenberger who invited me to this project and supported my activities in these areas. Pro- fessor G.

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Fruitful cooperations with Dr. Marcin Jakubowski, Mr. Armin Kaleck, Dr.