Fixed points of the area preserving Poincaré maps on two-manifolds

Wójcik, Klaudiusz

doi:10.1090/proc/13642

Fixed points of the area preserving Poincaré maps on two-manifolds
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Proc. Amer. Math. Soc. 145 (2017), 5223-5233 Request permission

Abstract:

We study the number of fixed points of the area preserving Poincaré map $P$ associated to periodic in time ODE’s on two-manifolds. We prove the fixed point index formula for the Poincaré map based on the method of periodic isolating segments. As the application we show that the $1$-periodic hamiltonian planar system \[ \dot {z}=\overline {z}^n+e^{2\pi i t}\overline {z}^l \] has at least $n+1$ non-zero $1$-periodic solutions provided that $l>n\geq 1$.

References

Richard C. Churchill, Isolated invariant sets in compact metric spaces, J. Differential Equations 12 (1972), 330–352. MR 336763, DOI 10.1016/0022-0396(72)90036-8
Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations 6 (1994), no. 4, 631–637. MR 1303278, DOI 10.1007/BF02218851
Morris W. Hirsch, Topology of fixed point sets of surface homeomorphisms, Houston J. Math. 26 (2000), no. 4, 765–789. MR 1823967
Stephan Pelikan and Edward E. Slaminka, A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds, Ergodic Theory Dynam. Systems 7 (1987), no. 3, 463–479. MR 912377, DOI 10.1017/S0143385700004132
Carl P. Simon, A bound for the fixed-point index of an area-preserving map with applications to mechanics, Invent. Math. 26 (1974), 187–200. MR 353372, DOI 10.1007/BF01418948
Roman Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal. 22 (1994), no. 6, 707–737. MR 1270166, DOI 10.1016/0362-546X(94)90223-2
Roman Srzednicki and Klaudiusz Wójcik, A geometric method for detecting chaotic dynamics, J. Differential Equations 135 (1997), no. 1, 66–82. MR 1434915, DOI 10.1006/jdeq.1996.3222
R. Srzednicki, K. Wójcik, P. Zgliczyński, Fixed point results based on Ważewski method, in R. Brown, M. Furi, L. Górniewicz, B. Jiang (Eds.) Handbook of topological fixed point theory, Kluwer 2004, pp. 903–941.
Francisco R. Ruiz del Portal and José M. Salazar, Fixed point index of iterations of local homeomorphisms of the plane: a Conley index approach, Topology 41 (2002), no. 6, 1199–1212. MR 1923219, DOI 10.1016/S0040-9383(01)00035-0
Klaudiusz Wójcik, On some nonautonomous chaotic system on the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1853–1858. Discrete dynamical systems. MR 1728743, DOI 10.1142/S0218127499001322
Klaudiusz Wójcik and Piotr Zgliczyński, Isolating segments, fixed point index, and symbolic dynamics, J. Differential Equations 161 (2000), no. 2, 245–288. MR 1744149, DOI 10.1006/jdeq.1999.3696
Klaudiusz Wójcik and Piotr Zgliczyński, Isolating segments, fixed point index, and symbolic dynamics. III. Applications, J. Differential Equations 183 (2002), no. 1, 262–278. MR 1917245, DOI 10.1006/jdeq.2001.4099