This work develops a perturbation approach to the estimation and propagation of states in nonlinear dynamical systems with uncertainties in both the process model and output measurements. The state estimation error and statistical moments are propagated in time through a regular perturbation method in an artificial ordering parameter λ, while the state updating occurs through a linear update eq...

A numerical order verification technique is applied to demonstrate that the asymptotic expansions of solutions of the Duffing equation obtained respectively by the Lindstedt-Poincaré(LP) method and the modified Lindstedt-Poincaré(MLP) method are uniformly valid for small parameter values. A numerical comparison of error shows that the MLP method is valid whereas the LP method is invalid for lar...

In this paper, we study dynamics in delayed van der Pol–Duffing equation, with particular attention focused on nonresonant double Hopf bifurcation. Both multiple time scales and center manifold reduction methods are applied to obtain the normal forms near a double Hopf critical point. A comparison between these two methods is given to show their equivalence. Bifurcations are classified in a two...

We investigate the impact of nonlinear damping on the dynamics of a nanomechanical doubly clamped beam. The beam is driven into nonlinear regime and the response is measured by a displacement detector. For data analysis we introduce a nonlinear damping term to Duffing equation. The experiment shows conclusively that accounting for nonlinear damping effects is needed for correct modeling of the ...

We derive the exact bifurcation diagram of the Duffing oscillator with parametric noise thanks to the analytical study of the associated Lyapunov exponent. When the fixed point is unstable for the underlying deterministic dynamics, we show that the system undergoes a noise-induced reentrant transition in a given range of parameters. The fixed point is stabilised when the amplitude of the noise ...

In the last decade we have witnessed exciting technological advances in the fabrication and control of microelectromechanical and nanoelectromechanical systems (MEMS&NEMS) [16, 19, 26, 54, 55]. Such systems are being developed for a host of nanotechnological applications, such as highly sensitive mass [25, 34, 67], spin [56], and charge detectors [17, 18], as well as for basic research in the m...

Chaos theory has a manifold variety of applications in science and engineering. There are many systems in nature with several stable states, which are separated by energy barriers. When the system can move along the stable states, its dynamics can become quite complex. A simple mechanical model that depicts some of these complex dynamical features is the famous Duffing double-well oscillator (1...

We study dynamics of the Duffing–Van der Pol driven oscillator. Periodic steady-state solutions of the corresponding equation are determined within the Krylov-Bogoliubov-Mitropolsky approach to yield dependence of amplitude on forcing frequency as an implicit function, referred to as resonance curve or amplitude profile. Equations for singular points of resonance curves are solved exactly. We i...