Resonance and nonlinearity: A survey

Abstract

RESONANCE AND NONLINEARITY: A SURVEY REZONANS I NELINIJNIST\: OHLQDThis paper surveys recent results about nonresonant and resonant periodically forced nonlinear oscillators. This includes the existence of periodic, unbounded or bounded solutions for bounded nonlinear perturbations of linear and of piecewise linear oscillators, as well as of some classes of planar Hamiltonian systems.Navedeno ohlqd ostannix rezul\tativ wodo nerezonansnyx i rezonansnyx periodyçno zbudΩuvanyx nelinijnyx oscylqtoriv: isnuvannq periodyçnyx neobmeΩenyx abo obmeΩenyx rozv'qzkivdlq obmeΩenyx nelinijnyx zburen\ linijnyx ta kuskovo-linijnyx oscylqtoriv, a takoΩ deqkyx klasiv ploskyx hamil\-tonovyx system.

Example of a suspension bridge ODE model exhibiting chaotic dynamics: A topological approach

“…Now, we fix θ as above and discuss (H * r ). As previously observed, condition (15) in (H * r ) is always satisfied by taking sufficiently large. This ends our verification.…”

“…We observe that for any choice of 0 < η 1 < η 2 and m 2, it is always possible to find (sufficiently large) in order to have (15) satisfied. This is obvious from the fact that T is strictly increasing on [0, +∞) and therefore (H * r ) will be satisfied for…”

“…Our theorems may have some possible interest also from the point of view of the study of periodically forced second order asymmetric oscillators (see [15] for an interesting survey about the state of the art about this topic). In fact, our function g(s) := ks + is in critical position with respect to the periodic Dancer-Fučik spectrum.…”

Chaotic dynamics in periodically forced asymmetric ordinary differential equations

“…Our results are strongly related also to the study of periodically forced asymmetric oscillators (see [24] for a recent survey on this subject).…”

Chaos in periodically perturbed planar Hamiltonian systems using linked twist maps

“…which have been widely studied in the past thirty years (see [14] for a recent survey on this topic).…”