Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities

We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities. We obtain sharp bounds for h such that the equation has exactly three ordered T-periodic solutions. Moreover, when h is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.Comment: Keywords: Duffing equation; Periodic solution; Stabilit

Infinity of Subharmonics for Asymmetric Duffing Equations with the Lazer–Leach–Dancer Condition

AbstractIn this paper, based on a generalized version of the Poincaré–Birkhoff twist theorem by Franks, we establish the existence of infinitely many subharmonics for the asymmetric Duffing equation with the classical Lazer–Leach–Dancer condition. As a consequence of our result, we obtain a sufficient and necessary condition for existence of arbitrarily large amplitude periodic solutions for a class of asymmetric Duffing equations at resonance

Rate of Decay of Stable Periodic Solutions of Duffing Equations

In this paper, we consider the second-order equations of Duffing type. Bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution. Furthermore, the stability of the unique periodic solution is analyzed; the sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution.Comment: Key words: Periodic solution; Stability; Rate of deca

Existence, uniqueness, and stability of periodic solutions of an equation of Duffing type

We consider a second-order equation of Duffing type. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a periodic solution. Furthermore, the unique periodic solution is asymptotically stable with sharp rate of exponential decay. In particular, for a restoring term independent of the variable $t$, a necessary and sufficient condition is obtained which guarantees the existence and uniqueness of a periodic solution that is stable.Comment: Key words and phrases: Periodic solution, topological degree, stabilit