Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers

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.

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>

Harmonic and subharmonic solutions of quadratic Liénard type systems with sublinearity

<abstract><p>In this paper, we prove the existence of harmonic solutions and infinitely many subharmonic solutions of dissipative second order sublinear differential equations named quadratic Liénard type systems. The method of the proof is based on the Poincaré-Birkhoff twist theorem.</p></abstract>

Coexistence of Random Subharmonic Solutions of Random Impulsive Differential Equations and Inclusions on a Circle

The coexistence of random periodic solutions with various periods (i.e. subharmonics) is proved to random differential equations on a circle with random impulses of all integer orders. One of the theorems is also extended to random differential inclusions on a circle with multivalued deterministic impulses. These results can be roughly characterized as a further application of the randomized Sharkovsky type theorems to random impulsive differential equations and inclusions on a circle.

Chaos in a Periodically Perturbed Second-Order Equation with Signum Nonlinearity

We study the periodically perturbed Duffing-type equation [Formula: see text] and its damped counterpart [Formula: see text] The main feature of our model is the presence of a “signum term” in [Formula: see text] We prove the existence of infinitely many subharmonic solutions as well as the presence of chaotic dynamics for some [Formula: see text]-periodic forcing terms.