Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth

<p style='text-indent:20px;'>We prove the existence and multiplicity of subharmonic solutions for bounded coupled Hamiltonian systems. The nonlinearities are assumed to satisfy Landesman-Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is based on phase plane analysis and a higher dimensional version of the Poincaré-Birkhoff twist theorem by Fonda and Ureña. The results obtained generalize the previous works for scalar second-order differential equations or relativistic equations to higher dimensional systems.</p>

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Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

Advances in Nonlinear Analysis ◽

10.1515/anona-2017-0040 ◽

2017 ◽

Vol 8 (1) ◽

pp. 583-602 ◽

Cited By ~ 3

Author(s):

Alessandro Fonda ◽

Rodica Toader

Keyword(s):

Hamiltonian Systems ◽

Zero Eigenvalue ◽

Birkhoff Theorem ◽

Subharmonic Solutions ◽

Existence And Multiplicity ◽

Volterra Systems

Abstract We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman–Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincaré–Birkhoff Theorem. Different situations, including Lotka–Volterra systems, or systems with singularities, are also illustrated.

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Periodic perturbations of Hamiltonian systems

Advances in Nonlinear Analysis ◽

10.1515/anona-2015-0122 ◽

2016 ◽

Vol 5 (4) ◽

Cited By ~ 4

Author(s):

Alessandro Fonda ◽

Maurizio Garrione ◽

Paolo Gidoni

Keyword(s):

Fixed Point ◽

Hamiltonian Systems ◽

Coupled Systems ◽

Multiplicity Results ◽

Periodic Perturbations ◽

Existence And Multiplicity ◽

Weakly Coupled ◽

Higher Dimensional ◽

Dimensional Version ◽

Weakly Coupled Systems

AbstractWe prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

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Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers

Advanced Nonlinear Studies ◽

10.1515/ans-2021-2134 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Shuang Wang ◽

Dingbian Qian

Keyword(s):

Hamiltonian Systems ◽

Rotation Number ◽

Phase Plane Analysis ◽

Cauchy Problems ◽

Multiplicity Results ◽

Birkhoff Theorem ◽

Subharmonic Solutions ◽

Large Solutions ◽

Plane Analysis ◽

Rotation Numbers

Abstract We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems J ⁢ z ′ = ∇ ⁡ H ⁢ ( t , z ) {Jz^{\prime}=\nabla H(t,z)} from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on ∂ ⁡ H ∂ ⁡ x ⁢ ( t , x , y ) {\frac{\partial H}{\partial x}(t,x,y)} , uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.

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