When we attempt to control a linear system in which some noise has been added, typically we need a control higher or equal to the amount of noise added. When we have a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient with the exception of a set of points of zero Lebesgue measure. The action of an external noise makes all trajectories escape even faster. Attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task. Here we show, however, that this goal is indeed possible, based on a geometrical property found typically in this situation: the existence of a horseshoe. The horseshoe implies that there exists what we call safe sets, which assures that there is a general strategy that allows one to keep trajectories inside that region with a control smaller than noise. We call this type of control partial control of chaos [1,2] that allows one to keep the trajectories of a dynamical system close to the saddle even in presence of a noise stronger than the applied control. In this talk recent progress and new results on this control strategy are presented. ??This is joint work with James A Yorke (USA), Samuel Zambrano and Juan Sabuco (Spain).
|
Miguel A. F. Sanjuan
Nonlinear Dynamics, Chaos and Complex Systems Group
Departamento de Fisica, Universidad Rey Juan Carlos
Tulip¨˘n s/n,28933 M¨Žstoles, Madrid, Spain
miguel.sanjuan@urjc.es
|
Prof. Miguel A. F. Sanjuan received a Bachellor Degree in Physics by University of Valladolid, Spain, in 1981, where he was granted the Outstanding Graduation Honor for Undergraduate Studies and a PhD Degree by National University at a Distance (UNED), Madrid, Spain in 1990, on Nonlinear Dynamics and Chaos. He is Full Professor of Physics at the Universidad Rey Juan Carlos, Madrid, Spain, where he has been the Founder of the Department of Physics. He is currently Head of the Department of Physics and Head of the Nonlinear Dynamics, Chaos and Complex Systems Research Group. He was elected Foreign Member of the Lithuanian Academy of Sciences in the areas of Physics and Mechanical Engineering within the Technical Sciences Division on March 18 2008. Member of Editorial Boards of six international journals (Communications in Nonlinear Science and Numerical Simulation, Discrete and Continuous Dynamical Systems B, Journal of Vibroengineering, Journal of Nonlinear Systems and Applications, International Journal of Bifurcation and Chaos and Mathematical Problems in Engineering). Supervisor of 12 PhD Theses in Nonlinear Dynamics, Chaos and Complex Systems. Participation in Thesis committees in Spain, France and Cameroon. Referee on Nonlinear Dynamics, Chaos and Complex Systems on more than 40 scientific journals. Principal Investigator and Researcher in more than 20 competitive funded projects. Author and Co-author of more than 100 scientific articles, edited 4 books and more than 100 communications to conferences. Around 50 invited lectures in seminars and conferences in Spain, Europe, USA, China, Japan, India, Cameroon, Mexico, Chile, Brazil. Fellowship of the Japan Society for the Promotion of Science. University of Tokyo. Coauthor of one review for Reviews of Modern Physics. Its impact factor is higher than Nature and Science. Scientific Evaluator of Research Agencies in Spain, Chile, Argentine, France, Georgia and INTAS-EU. Organizer of different Meetings, Seminars and Conferences in Nonlinear Dynamics. Secretary of the Specialized Group on Physics of Life Sciences. Spanish Physics Royal Society, since April 2004. Member of many Scientific Committees for evaluation and research. He has participated in many popularizing research activities and author of a blog on Complexity. Member of Organizing and Scientific Committees of many conferences. Visiting Research Associate of the Institute for Physical Sciences and Technology of the University of Maryland. Visiting Research Associate of the University of Tokyo. Visiting Researcher in Germany, Lithuania, Italy, USA, Japan, Portugal. Currently Visiting Research Professor at Beijing Jiaotong University from the "Key Invitation Program for Top-Level Experts" from the "State Administration of Foreign Experts Affairs" of the Chinese Government. August-January 2011.
|