On Hopf bifurcations of piecewise planar Hamiltonian systems. (English) Zbl 1222.34042
This paper studies generalized Hopf bifurcations for piecewise planar Hamiltonian systems having a discontinuity curve given by a straight line. More concretely, the authors consider autonomous systems of the form
\[ \dot x=H_y(x,y),\quad \dot y=-H_x(x,y), \]
where \(H(x,y)=H^+(x,y)\) for \(x>0\) and \(H(x,y)=H^-(x,y)\) for \(x<0\) with \(H^\pm\) analytic functions on the whole plane. Moreover, they assume that the system has a monodromic generalized singular point on \(x=0\). Recall that these points can be of three types: focus-focus, focus-parabolic or parabolic-parabolic according to the behaviors of both Hamiltonian systems near the singular point. The main results of the paper study lower and upper cyclicities of the generalized singular point under some assumptions on \(H^\pm\). One of these hypotheses is that both \(H^\pm(0,y)\) are polynomials.
\[ \dot x=H_y(x,y),\quad \dot y=-H_x(x,y), \]
where \(H(x,y)=H^+(x,y)\) for \(x>0\) and \(H(x,y)=H^-(x,y)\) for \(x<0\) with \(H^\pm\) analytic functions on the whole plane. Moreover, they assume that the system has a monodromic generalized singular point on \(x=0\). Recall that these points can be of three types: focus-focus, focus-parabolic or parabolic-parabolic according to the behaviors of both Hamiltonian systems near the singular point. The main results of the paper study lower and upper cyclicities of the generalized singular point under some assumptions on \(H^\pm\). One of these hypotheses is that both \(H^\pm(0,y)\) are polynomials.
Reviewer: Armengol Gasull (Barcelona)
MSC:
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34A36 | Discontinuous ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 37J20 | Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems |
References:
| [1] | di Bernardo, M.; Budd, C. J.; Champneys, A. R.; Kowalczyk, P., Piecewise-smooth Dynamical Systems, Theory and Applications (2008), Springer-Verlag: Springer-Verlag London · Zbl 1146.37003 |
| [2] | Coll, B.; Gasull, A.; Prohens, R., Degenerate Hopf bifurcation in discontinuous planar systems, J. Math. Anal. Appl., 253, 671-690 (2001) · Zbl 0973.34033 |
| [3] | Filippov, A. F., Differential Equation with Discontinuous Righthand Sides (1988), Kluwer Academic Pub.: Kluwer Academic Pub. Dordrecht, The Netherlands · Zbl 0664.34001 |
| [4] | Han, M., On Hopf cyclicity of planar systems, J. Math. Anal. Appl., 245, 404-422 (2000) · Zbl 1054.34052 |
| [5] | Han, M.; Zhang, W., On Hopf bifurcation in non-smooth Planar systems, J. Differential Equations, 248, 2399-2416 (2010) · Zbl 1198.34059 |
| [6] | Kunze, M., Non-Smooth Dynamical Systems (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0965.34026 |
| [7] | Liu, X.; Han, M., Hopf bifurcation for nonsmooth liénard systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19, 7, 2401-2405 (2009) · Zbl 1176.34047 |
| [8] | Shafaravich, I. R., Basic Algebraic Geometry (1974), Springer-Verlag: Springer-Verlag New York · Zbl 0284.14001 |
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