Poincare Linear Interpolated Cell Mapping: Method for Global Analysis of Oscillating Systems

Abstract

A method for global analysis of nonlinear dynamical oscillating systems was developed. The method is based on the idea of introducing a Poincare section into a multidimensional state space of the dynamical system and combine it with an interpolation procedure within the cells which constitute the discretized problem domain of interest. The proposed method was applied to study the global behavior of two nonlinear coupled van der Pol oscillators. Significant saving in calculation time, in comparison with both direct numerical integration and Poincare-like simple cell mapping, is demonstrated.

Bogdanov-Takens bifurcation points and Sil'nikov homoclinicity in a simple power-system model of voltage collapse

“…A second widely used class of methods for calculation of basins of attraction are cell mapping methods [29]- [33]. These methods avoid the calculation of long time trajectories by dividing the region of investigation into a number of discrete "cells", trajectories from which define a discrete mapping from each cell to another.…”

Application of an improved cell mapping method to bilinear stiffness aeroelastic systems

“…The hyperplane is transversal to the trajectories of the system. In PLICM (Levitas and Weller, 1995), only autonomous systems were considered. Such a procedure results in P : S !…”

“…A further development to reduce the amount of cells used in the calculation led to the introduction of ''Poincare´-like simple cell mapping'' (PLSCM) by Levitas et al (1994) and ''Poincare´linear interpolated cell mapping'' (PLICM) by Levitas and Weller (1995), which combine the use of spatial Poincare´sections with SCM and ICM, respectively. The introduction of Poincare´sections allows considerable ARTICLE IN PRESS www.elsevier.com/locate/jfs 0889-9746/$ -see front matter r 2004 Elsevier Ltd. All rights reserved.…”

“…In fact, the influence of the initial conditions on the behaviour of airfoils is one of the main tasks of aeroelastic analysis. The results are referred to as the stability boundary in terms of the initial conditions (Price et al, 1994), boundary for different types of motion , domains of attraction (Levitas et al, 1994;Levitas and Weller, 1995), or simply as parameter maps (Kim and Lee, 1996). Because the cell mapping techniques appear appropriate for the prediction of the boundaries of different types of motion depending on initial conditions, they have been applied in many engineering problems, including the global analysis of bifurcations of the Duffing-van-der-Pol oscillators under both additive and multiplicative random excitations (He et al, 2004), the global analysis of climate predictability (Mu et al, 2004), the optimal control of autonomous dynamical systems (Zufiria and Martinez-Marin, 2003), nonlinear vibrations of a rotor system with bearing clearance ( Karlberg and Aidanpaa, 2003), multiple steady state solutions of a nonlinear system with time-delay (Raghothama and Narayanan, 2002) and the global dynamics of a damped flexible connecting rod (Chen and Chian, 2001).…”

Conditions of Nucleation and Interface Propagation in Thermoplastic Materials

“…The dissipation increment in the Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1995822 course of P T due to plastic flow and variation of internal variable can be represented in the following form where X, = u -p 2 and X g = -p are generalized dissipative forces, conjugated with -Y -.# plastic strain rate ip and g respectively. The expression for X p and X g is derived using the standard thermodynamical procedure for materials without PT [5,6]. The dissipation increment due to PT itself X is a difference between N and Np, , i.e.…”