PapersNo Access

Directional Control of Bifurcation into Targeted Trajectory

Vladimir Jaćimović

Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put, b.b., 81000 Podgorica, Montenegro

Search for more papers by this author

and

Nikola Konatar

Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put, b.b., 81000 Podgorica, Montenegro

Search for more papers by this author

https://doi.org/10.1142/S0218127415501540Cited by:0 (Source: Crossref)

Abstract

The field of bifurcation control includes several classes of problems that are actively studied with various motivations and from different points of view. The different statements of the problem are articulated by defining various objectives of applying control to the system, depending on the motivation and the specific problem. In this paper, we propose the objective of bifurcation control that is based on one of the main features of bifurcation phenomena: bifurcation generates possibilities for different scenarios to occur. We show how to find the right directions to control the system in order to ensure the realization of a favorable scenario. Our method is based on a bifurcation theorem for an abstract problem with constraints (inequalities). This theorem is further applied to controlled dynamical systems and two reasonably simple bifurcation control problems are analyzed in order to illustrate our objective and method.

Keywords:

References

E. H. Abed and J. H. Fu , Syst. Contr. Lett. 7 , 11 ( 1986 ) . Crossref, Web of Science, Google Scholar
E. H. Abed and J. H. Fu , Syst. Contr. Lett. 8 , 467 ( 1987 ) . Crossref, Web of Science, Google Scholar
A. A. Agrachev and A. V. Sarychev , Ann. Inst. Henri Poincaré 13 , 635 ( 1996 ) . Crossref, Web of Science, Google Scholar
A. Ambrosetti and G. Prodi , A Primer of Nonlinear Analysis ( Cambridge University Press , 1993 ) . Google Scholar
V. S. Anischenko et al. , Int. J. Bifurcation and Chaos 2 , 633 ( 1992 ) . Link, Google Scholar
A. V. Arutyunov , V. Jaćimović and F. L. Pereira , J. Dyn. Contr. Syst. 9 , 131 ( 2003 ) . Crossref, Web of Science, Google Scholar
A. V. Arutyunov and D. Yu. Karamzin , Comput. Math. Math. Phys. 47 , 312 ( 2007 ) . Crossref, Google Scholar
A. V. Arutyunov , A. F. Izmailov and V. Jaćimović , J. Math. Anal. Appl. 359 , 752 ( 2009 ) . Crossref, Web of Science, Google Scholar
A. V. Arutyunov , D. Yu. Karamzin and F. L. Pereira , J. Optim. Th. Appl. 140 , 391 ( 2009 ) . Crossref, Web of Science, Google Scholar
A. Balanov et al. , Synchronization: From Simple to Complex ( Springer , 2009 ) . Google Scholar
I. I. Blekhman , Synchronization in Science and Technology ( ASME Press , NY , 1988 ) . Google Scholar
L. Chen and X. Yu , IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 46 , 767 ( 1999 ) . Crossref, Web of Science, Google Scholar
G. Chen et al. , Acta Phys. Sin. 8 , 416 ( 1999 ) . Web of Science, Google Scholar
G. Chen , J. L. Moiola and H. O. Wang , Int. J. Bifurcation and Chaos 10 , 511 ( 2000 ) . Link, Web of Science, Google Scholar
V. Dykhta , Sibirsk. Mat. Zh. 35 , 70 ( 1994 ) . Google Scholar
P. Gaspard, Encyclopedia of Nonlinear Science, ed. A. Scott (Routledge, NY, 2005) pp. 808–811. Google Scholar
D. Yu. Karamzin , J. Math. Sci. 139 , 7087 ( 2006 ) . Crossref, Google Scholar
A. Pikovsky , M. Rosenblum and J. Kurths , Synchronization: A Universal Concept in Nonlinear Science , Cambridge Nonlinear Sci. Ser 12 ( Cambridge University Press , UK , 2001 ) . Crossref, Google Scholar
O. E. Rössler , Ann. NY Acad. Sci. 316 , 376 ( 1979 ) . Crossref, Google Scholar
J. Sun and Y. Zhang , Phys. Lett. A 306 , 306 ( 2003 ) . Crossref, Web of Science, Google Scholar
R. B. Vinter and F. L. Pereira , SIAM J. Contr. Optim. 26 , 205 ( 1988 ) . Crossref, Web of Science, Google Scholar
X. Wang , W. Xu and T. Yang , Int. J. Nonlin. Sci. 15 , 64 ( 2013 ) . Google Scholar
Z. Wu and P. Yu , IEEE Trans. Autom. Contr. 51 , 1019 ( 2006 ) . Crossref, Web of Science, Google Scholar
Z. Wu and L. M. Sun , Acta Phys. Sin. 60 , 050504 ( 2011 ) . Crossref, Web of Science, Google Scholar
M. Xu , Y. Wei and J. Wei , El. J. Qual. Th. Diff. Eqs. 63 , 1 ( 2010 ) . Google Scholar
T. Yang , C. M. Yang and L. B. Yang , Phys. Lett. A 232 , 356 ( 2010 ) . Crossref, Google Scholar