Russian Journal of Mathematical Physics
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
Main page
About this project
Software
Classifications
Links
Terms of Use

Search papers
Search references

RSS
Current issues
Archive issues
What is RSS






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Russ. J. Math. Phys., 2013, Volume 20, Issue 1, Pages 25–32 (Mi rjmph8)  

Generic fractal structure of finite parts of trajectories of piecewise smooth hamiltonian systems

R. Hildebranda, L. V. Lokoutsievskiyb, M. I. Zelikinb

a Laboratory Jean Kuntzmann, University Grenoble 1 / CNRS, 51 rue des Mathématiques, BP 53, 38041 Grenoble cedex 09, France
b Dept. of Mechanics and Mathematics, Moscow State University, 119991 Moscow, Russia

Abstract: Piecewise smooth Hamiltonian systems with tangent discontinuity are studied. A new phenomenon is discovered, namely, the generic chaotic behavior of finite parts of trajectories. The approach is to consider the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It turns out that, near second-order singular points lying on a discontinuity stratum of codimension two, the system of Poisson brackets is reduced to the Hamiltonian system of the Pontryagin Maximum Principle. The corresponding optimization problem is studied and the topological structure of its optimal trajectories is constructed (optimal synthesis). The synthesis contains countably many periodic solutions on the quotient space by the scale group and a Cantor-like set of nonwandering points (NW) having fractal Hausdorff dimension. The dynamics of the system is described by a topological Markov chain. The entropy is evaluated, together with bounds for the Hausdorff and box dimension of (NW).

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00986-a
Russian Academy of Sciences - Federal Agency for Scientific Organizations
This work was supported by the Russian Foundation for Basic Research (grant no. 11-01-00986-a) and by the program “Mathematical Theory of Control” of the Presidium of the Russian Academy of Sciences.


DOI: https://doi.org/10.1134/S1061920813010032


Bibliographic databases:

Received: 03.12.2012
Language:

Linking options:
  • http://mi.mathnet.ru/eng/rjmph8

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Number of views:
    This page:51

     
    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2022