Boundedness in a Piecewise Linear Oscillator and a Variant of the Small Twist Theorem

Boundedness in Nonlinear Oscillations at Resonance

“…(see [2,7,10,13] etc.) In order to prove the existence of such invariant curves for every $< <1, it suffices to verify that for every $< <1, the Poincare map P satisfies all the assumptions of a variant of Moser's small twist theorem which is due to Ortega [16]. In the rest of this part, we will give an expression for (…”

“…As far as we know, this is the first nontrivial result of boundedness in a semilinear case. Recently, further results have appeared in [16] and [10] for semilinear Duffing equations.…”

“…The Poincare map of the transformed system is closed to a so-called twist map in R 2 "D. Then a variant of Moser's twist theorem [16] guarantees the existence of arbitrarily large invariant curves diffeomorphic to circles and surrounding the origin in the (x, x$)-plane. Every such curve is the base of a time-periodic and flow-invariant cylinder in the extended phase space (x, x$, t) # R 2 _R, which confines the solutions in the interior and which leads to a bound of these solutions.…”

Boundedness of solutions for semilinear reversible systems

“…On the other hand, the sublinear and semilinear cases are relatively new fields; see [4], [9], [10], [11], [12], [13], [17], [18], [19] and the references therein for details.…”

Boundedness of Solutions for Duffing's Equations with Semilinear Potentials

“…Later Liu [11] and Ortega [21] improved the result for the cases 1/`a+1/`b ¥ Q and 1/`a+1/`b ¥ R 0 Q, respectively. Further results have appeared in [12,13,22,27] for semilinear Duffing's equation.…”