Coupling linearity and twist: an extension of the Poincaré–Birkhoff theorem for Hamiltonian systems

Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model

“…Indeed, their flexibility is one of the advantages of the application of topological methods. The employment of topological tools to generalize existence results for ODEs to systems of differential equations obtained by suitable coupling of the original equation has been recently applied in various frameworks, for instance in [6,19,20,21]. We remark that such results are not necessarily restricted to small perturbations: indeed, as in our case, they apply also to larger suitable couplings.…”

Periodic solutions of nearly integrable Hamiltonian systems bifurcating from infinite-dimensional tori

“…From now on, we will take ⃗ ξ i (t), ⃗ I i (t), ⃗ z i (t) identically equal to zero when i > J . Concerning the "approximating system", we will need the following slight modification of [7,Corollary 2.3]. Let us consider the finite-dimensional Hamiltonian system…”

“…The above result was recently extended in [7] for systems of the type ⎧ ⎪ ⎨ ⎪ ⎩ φ = ∇K(I) + ε∇ I P (t, φ, I, z) − İ = ε∇ φ P (t, φ, I, z) J ż = Az + ε∇ z P (t, φ, I, z) , (1.4) where J = (…”

“…The spaces X and Z may be infinitedimensional, finite-dimensional, or even reduced to {0}. If X is finite-dimensional, the cases Z = {0} and Z finite-dimensional correspond to the settings in [6] and [7], respectively. However, if X or Z are infinitedimensional, we will be able to prove the bifurcation of at least one periodic orbit from an invariant torus, which can also be infinite-dimensional.…”

Existence of a periodic solution for superlinear second order ODEs

“…They considered the case f (t, 0) = 0, hence assuming x ≡ 0 as a first periodic solution, and obtained infinitely many others, rotating around the first one, by iterate applications of the Poincaré-Birkhoff Theorem. We refer to [7,2] for generalizations on the plane and to [3,13,12] for the extension to systems of differential equations. Outside the Hamiltonian case, such results are in general no longer true, since the constant null solution may be the unique T -periodic solution, cf.…”