# Averaging Methods in Nonlinear Dynamical Systems

@inproceedings{Sanders1985AveragingMI, title={Averaging Methods in Nonlinear Dynamical Systems}, author={Jan A. Sanders and Ferdinand Verhulst}, year={1985} }

Basic Material and Asymptotics.- Averaging: the Periodic Case.- Methodology of Averaging.- Averaging: the General Case.- Attraction.- Periodic Averaging and Hyperbolicity.- Averaging over Angles.- Passage Through Resonance.- From Averaging to Normal Forms.- Hamiltonian Normal Form Theory.- Classical (First-Level) Normal Form Theory.- Nilpotent (Classical) Normal Form.- Higher-Level Normal Form Theory.- The History of the Theory of Averaging.- A 4-Dimensional Example of Hopf Bifurcation…

## 1,873 Citations

### Qualitative Theory of Nonlinear Resonance by Averaging and Dynamical Systems Methods

- Mathematics
- 1988

This paper is a self-contained exposition of the theory of averaging for periodic and quasiperiodic systems, with the emphasis being on the author’s research (part of it joint work with Clark…

### Closeness of solutions and averaging for nonlinear systems on Riemannian manifolds

- Mathematics2013 Australian Control Conference
- 2013

An averaging result for periodic dynamical systems evolving on Euclidean spaces is extended to those evolving on (differentiable) Riemannian manifolds by using standard tools from differential geometry to measure closeness of trajectories via a suitable metric on a finite time horizon.

### Normalization and averaging on compact lie groups in nonlinear mechanics

- Mathematics
- 1997

We consider the method of normal forms, the Bogolyubov averaging method, and the method of asymptotic decomposition proposed by Yu. A. Mitropol’skii and the author of this paper. Under certain…

### An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems: The Averaging Method Revisited

- Mathematics, Computer ScienceISSAC
- 2019

An algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method is introduced and its effectiveness is shown by several examples.

### Normal forms and periodic solutions in the theory of non-linear oscillations. Existence and asymptotic theory

- Mathematics
- 1989

### Averaging methods for nonlinear systems with a small parameter via reduction and topological degree

- MathematicsNonlinear Analysis: Real World Applications
- 2019

### Averaging Method for Strongly Nonlinear Oscillators with Periodic Excitations

- Physics
- 1994

An averaging method is developed to predict periodic solutions of strongly nonlinear and harmonically forced oscillators. The analysis is restricted to the case of period-1 orbits. The original…

### Symmetry-Breaking Bifurcations and Patterns of Oscillations in Rings of Crystal Oscillators

- Mathematics, PhysicsSIAM J. Appl. Dyn. Syst.
- 2018

Symmetry is used to investigate the existence and stability of collective patterns of oscillations in rings of coupled crystal oscillators. We assume $N$ identical crystal oscillators, where each o...