this paper presents approximate analytical solutions for nonlinear oscillators using the multi-step homotopy analysis method (msham). the proposed scheme is only a simple modification of the homotopy analysis method, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding problems. several illustrat...

AbstractThe paper involves thorough study of non-linear vibratory oscillators and numerical methodology to analyse and resolute the non-linear dynamical world. The study involves the analysis of non-linear oscillators like the Van der Pol Oscillator and Duffing Oscillator. Application of regular perturbation method in the oscillator is also demonstrated. The equilibrium and stability analysis o...

This paper treats the synchronization of a chain of non linear and uncertain model of non identical oscillators. Using the Lyapunov theory of stability, a dynamical controller guarantying the oscillators synchronization is determined. The problem of synchronization is transformed in a problem of asymptotic stabilization for a nonlinear system and then is formulated as a system of linear matrix ...

Relaxation oscillators can usually be represented as a feedback system with hysteresis. The relay relaxation oscillator consists of relay hysteresis and a linear system in feedback. The objective of this work is to study the existence of periodic orbits and the dynamics of coupled relay oscillators. In particular, we give a complete analysis for the case of unimodal periodic orbits, and illustr...

The problem of natural wave control involves steering a lattice of oscillators towards a desired natural (i.e. zero-input) assignment of energy and phase across the lattice. This problem is formulated and solved for lattices of linear oscillators via a passivity-based approach. Numerical simulations of 1D and 2D linear lattices and a 1D lattice of nonlinear oscillators confirm the effectiveness...

We consider optimization of the linear stability of synchronized states between a pair of weakly coupled limit-cycle oscillators with cross coupling, where different components of state variables of the oscillators are allowed to interact. On the basis of the phase reduction theory, we derive the coupling matrix between different components of the oscillator states that maximizes the linear sta...

in this paper, we present the numerical solution of ordinary dierential equations (or sdes), from each order especially second-order with time-varying and gaussian random coecients. we indicate a complete analysis for second-order equations in special case of scalar linear second-order equations (damped harmonic oscillators with additive or multi- plicative noises). making stochastic dierent...

Interconnected, self-excited oscillators are often found in nature and in engineered devices. In this work, a ring of van der Pol oscillators, each of which is connected to its immediate neighbors, is considered. The focus is on the emergent behavior of a large number of oscillators. The conditions under which time-independent solutions are obtained are determined, and the linear stability of t...