Nonlinear normal modes and spectral submanifolds: Existence, uniqueness and use in model reduction

We propose a unified approach to nonlinear modal analysis in dissipative oscillatory systems. This approach eliminates conflicting definitions, covers both autonomous and time-dependent systems, and provides exact mathematical existence, uniqueness and robustness results. In this setting, a nonlinear normal mode (NNM) is a set filled with small-amplitude recurrent motions: a fixed point, a periodic orbit or the closure of a quasiperiodic orbit. In contrast, a spectral submanifold (SSM) is an invariant manifold asymptotic to a NNM, serving as the smoothest nonlinear continuation of a spectral subspace of the linearized system along the NNM. The existence and uniqueness of SSMs turns out to depend on a spectral quotient computed from the real part of the spectrum of the linearized system. This quotient may well be large even for small dissipation, thus the inclusion of damping is essential for firm conclusions about NNMs, SSMs and the reduced-order models they yield.Comment: To appear in Nonlinear Dynamic

Periodic impact behavior of a class of Hamiltonian oscillators with obstacles

AbstractIn this paper, we study the existence of harmonic and subharmonic solutions of a class of non-smooth Hamiltonian systems, then apply its results to the vibration problems{−x″=q(x)|x′|2+g(t)x′+f(t),x(t)>0,x′(t0−)=−x′(t0+),ifx(t0)=0. Infinitely many harmonic and subharmonic bouncing solutions are always obtained if q(x) satisfies some coercive conditions

On the variational principle and applications for a class of damped vibration systems with a small forcing term

This paper is dedicated to studying the existence of periodic solutions to a new class of forced damped vibration systems by the variational method. The advantage of this kind of system is that the coefficient of its second order term is a symmetric $N \times N$ matrix valued function rather than the identity matrix previously studied. The variational principle of this problem is obtained by using two methods: the direct method of the calculus of variations and the semi-inverse method. New existence conditions of periodic solutions are created through several auxiliary functions so that two existence theorems of periodic solutions of the forced damped vibration systems are obtained by using the least action principle and the saddle point theorem in the critical point theory. Our results improve and extend many previously known results

Superlinear damped vibration problems on time scales with nonlocal boundary conditions

This paper studies a class of superlinear damped vibration equations with nonlocal boundary conditions on time scales by using the calculus of variations. We consider the Cerami condition, while the nonlinear term does not satisfy Ambrosetti–Rabinowitz condition such that the critical point theory could be applied. Then we establish the variational structure in an appropriate Sobolev’s space, obtain the existence of infinitely many large energy solutions. Finally, two examples are given to prove our results