Hopf bifurcation for a 3D filippov system

Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis

Volumn 16, Issue 6, 2009, Pages 759-775

  Akhmet, M U ^a   Arugaslan D ^a   Turan, M ^b  

^a MIDDLE EAST TECHNICAL UNIVERSITY   (Turkey)

^b ATILIM UNIVERSITY   (Turkey)

Author keywords

Center manifold; Filippov systems; Hopf bifurcation; Impulsive differential equations

Indexed keywords

EID: 73249146222     PISSN: 12013390     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (19)

1. E. Akahn and M. U. Akhmet, The principles of B-smooth discontinuous flows, Math. Comput. Simulat., (49) 981-995 (2005).
2. M. U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis, (60) 163-178 (2005).
3. M. Akhmetov, Asymptotic representation of solutions of regularly perturbed systems of differential equations with a nonclassical right-hand side, Ukrainian Mathematical Journal, (43) 1209-1214 (1991).
4. M. Akhmetov, On the expansion of solutions to differential equations with discontinuous right-hand side in a series in initial data and parameters, Ukrainian Mathematical Journal, (45) 786-789 (1993).
5. M. Akhmetov and N.A. Perestyuk, Differential properties of solutions and integral surfaces of nonlinear impulse systems, Differential Equations, (28) 445-453 (1992).
6. M. U. Akhmet, On the reduction principle for differential equations with piecewise constant argument of generalized type, Journal of Mathematical Analysis and Applications, 336 (2007) 646-663.
7. N. N. Bautin and E. A. Leontovich, Methods and rules for the qualitative study of dynamical systems on the plane, (Russian) Nauka, Moscow, (1990).
8. B. Coll, A. Gasull and R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appi, (253) 671-690 (2001).
9. A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, Dordrecht, (1988).
10. C. Henry, Differential equations with discontinuous righthand side for planning procedure, J. Econom. Theory, (4) 541-551 (1972).
11. A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations (3) 546-570 (1967).
12. M. Kunze and T. Küpper, Qualitative bifurcation analysis of a non-smooth friction- oscillator model, Z. Angew. Math. Phys., (48) 87-101 (1997).
13. M.Kunze, Non-Smooth Dynamical Systems, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, New York, (2000).
14. T. Küpper and S. Moritz, Generalized Hopf bifurcation for non-smooth planar systems, Phil. Trans. R. Soc. bond., (359) 2483-2496 (2001).
15. Y. Zou, T. Küpper and W.-J. Beyn, Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlinear Sciences, (16) 159-177 (2006).
16. N. Minorsky, Nonlinear Oscillations, Van Nostrand, New York, (1962).
17. H. E. Nusse and J. A. Yorke, Border-collision bifurcations including "period two to period three" for piecewise smooth systems. Phys. D, (57) 39-57 (1992).
18. H. E. Nusse and J. A. Yorke, Border-collision bifurcations for piecewise smooth one- dimensional maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg., (5) 189-207 (1995).
19. V. Pliss, A reduction principle in the theory of stability of motion, Izv. Akad. Nauk SSSR Ser. Mat, (28) 1297-1324 (1964).