Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

Tobia Dondè, Fabio Zanolin

Abstract


In this paper we prove the existence of multiple periodic (harmonic and subharmonic) solutions for a class of planar Hamiltonian systems which includes the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincaré-Birkhoff fixed point theorem as well as some refinements on the side of the theory of topological horseshoes. A Duffing-type equation and an exponential nonlinearity case are studied as applications.

Keywords


Poincaré-Birkhoff theorem; bend-twist maps; topological horseshoes; periodic solutions; complex oscillations

Full Text:

PREVIEW FULL TEXT

References


A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: a rotation number approach, Adv. Nonlinear Stud. 11 (2011), no. 1, 77–103.

A. Boscaggin and M. Garrione, Sign-changing subharmonic solutions to unforced equations with singular φ-Laplacian, Differential and Difference Equations with Applications, Springer Proc. Math. Stat., vol. 47, Springer, New York, 2013, pp. 321–329.

T. Burton and R. Grimmer, On continuability of solutions of second order differential equations, Proc. Amer. Math. Soc. 29 (1971), 277–283.

C.V. Coffman and D.F. Ullrich, On the continuation of solutions of a certain nonlinear differential equation, Monatsh. Math. 71 (1967), 385–392.

T.R. Ding and F. Zanolin, Periodic solutions of Duffing’s equations with superquadratic potential, J. Differential Equations 97 (1992), no. 2, 328–378.

A. Fonda and A.J. Ureña, A higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 3, 679–698.

V.K. Le and K. Schmitt, Minimization problems for noncoercive functionals subject to constraints, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4485–4513.

A. Margheri, C. Rebelo and F. Zanolin, Maslov index, Poincaré–Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, J. Differential Equations 183 (2002), no. 2, 342–367.

D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill’s equations, Adv. Nonlinear Stud. 4 (2004), no. 1, 71–91.

C. Rebelo, A note on the Poincaré–Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlinear Anal. 29 (1997), no. 3, 291–311.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism