PARAMETRIC ANALYSIS AND FRACTAL-LIKE BASINS OF ATTRACTION BY MODIFIED INTERPOLATED CELL MAPPING

Discontinuities are a common feature of physical models in engineering and biological systems, e.g. stick-slip due to friction, electrical relays or gene regulatory networks. The computation of basins of attraction of such nonsmooth systems is challenging and requires special treatments, especially regarding numerical integration. In this paper, we present a numerical routine for computing basins of attraction (BA) in nonsmooth systems with sliding, (so-called Filippov systems). In particular, we extend the Simple Cell Mapping (SCM) method to cope with the presence of sliding solutions by exploiting an event-driven numerical integration routine specifically written for Filippov systems. Our algorithm encompasses a method for dynamic construction of the cell state space so that a lower number of integration steps are required. Moreover, we incorporate an adaptive strategy of the simulation time to render more efficiently the computation of basins of attraction. We illustrate the effectiveness of our algorithm by computing basins of attraction of a sliding control problem and a dry-friction oscillator.

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Interpolated Cell Mapping of Dynamical Systems

Journal of Applied Mechanics ◽

10.1115/1.3173700 ◽

1988 ◽

Vol 55 (2) ◽

pp. 461-466 ◽

Cited By ~ 68

Author(s):

B. H. Tongue ◽

K. Gu

Keyword(s):

Dynamical System ◽

Steady State ◽

Phase Space ◽

Nonlinear System ◽

Basins Of Attraction ◽

Mapping Technique ◽

Cell Mapping ◽

Simple Cell Mapping ◽

Steady State Solutions ◽

Continuous Problem

A method is proposed to efficiently determine the basins of attraction of a nonlinear system’s different steady-state solutions. The phase space of the dynamical system is spacially discretized and the continuous problem in time is converted to an iterative mapping. By means of interpolation procedures, an improvement in the system accuracy over the Simple Cell Mapping technique is achieved. Both basins of attraction for a representative nonlinear system and characteristic system trajectories are generated and compared to exact solutions.

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BASINS OF ATTRACTION IN NONSMOOTH MODELS OF GEAR RATTLE

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740902283x ◽

2009 ◽

Vol 19 (01) ◽

pp. 203-224 ◽

Cited By ~ 25

Author(s):

JOANNA F. MASON ◽

PETRI T. PIIROINEN ◽

R. EDDIE WILSON ◽

MARTIN E. HOMER

Keyword(s):

Phase Space ◽

Dynamical Systems ◽

Piecewise Linear ◽

Basins Of Attraction ◽

Cell Mapping ◽

Stiffness Model ◽

Mapping Techniques ◽

Basin Boundaries ◽

Gear Rattle ◽

Piecewise Linear Stiffness

This paper is concerned with the computation of the basins of attraction of a simple one degree-of-freedom backlash oscillator using cell-to-cell mapping techniques. This analysis is motivated by the modeling of order vibration in geared systems. We consider both a piecewise-linear stiffness model and a simpler infinite stiffness impacting limit. The basins reveal rich and delicate dynamics, and we analyze some of the transitions in the system's behavior in terms of smooth and discontinuity-induced bifurcations. The stretching and folding of phase space are illustrated via computations of the grazing curve, and its preimages, and manifold computations of basin boundaries using DsTool (Dynamical Systems Toolkit).

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On the Dynamics of a Biomimetic Model of a Sympodial Tree: From Bifurcations Diagrams and 6D Basins of Attraction to Dynamical Integrity and Robustness

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4052570 ◽

2021 ◽

Author(s):

Nemanja Andonovski ◽

Ivana Kovacic ◽

Stefano Lenci

Keyword(s):

Excitation Frequency ◽

Computational Procedure ◽

Basins Of Attraction ◽

Nonlinear Behaviour ◽

Cell Mapping ◽

Damping Parameter ◽

Potential Benefits ◽

Numerical Tools ◽

Dynamical Integrity ◽

First Time

Abstract This work is concerned with a mechanical model of a sympodial tree with first-level branches, which has been shown to exhibit certain properties potentially suitable for biomimetic applications. To investigate these potential benefits further from the viewpoint of the system nonlinear behaviour under external periodic excitation, modern numerical tools related to the concept of dynamical integrity are either adjusted or newly developed for this system for the first time. First, multistable regions of interest are isolated from bifurcation diagrams and the effect of damping is investigated. Then, in order to obtain the corresponding basins of attraction of this highly dimensional model, an original computational procedure is developed that includes cell mapping with 406 cells, where each cell represents an initial condition required to construct the map. Full 6D basins are computed, and they are reported for various values of the damping parameter and the excitation frequency. Those basins are then used to calculate the dynamic integrity factors so that the dominant steady state can be determined. Finally, the integrity profiles are reported to illustrate how the robustness varies by changing the system parameters.

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HPC Methods for Domains of Attraction Computation

Volume 6: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

10.1115/detc2015-46095 ◽

2015 ◽

Author(s):

Pierpaolo Belardinelli ◽

Stefano Lenci

Keyword(s):

Memory Management ◽

Large Scale ◽

Basins Of Attraction ◽

Mapping Method ◽

Cell Mapping ◽

Double Step ◽

Optimal Balance ◽

A Cell ◽

Step Algorithm ◽

Computational Resources

The work is devoted to the development of efficient parallel algorithms for the computation of large-scale basins of attraction. Since the required computational resources increase exponentially with the dimension of a dynamical system, it is common to get into memory saturation or in a secular elaboration time. This paper presents a code, based on a cell mapping method, that evaluates basins of attraction for high-dimensional systems by exploiting the parallel programming. The proposed approach, by using a double-step algorithm, permits, i) to fully determine the basins in all the dimensions ii) to evaluate 2D Poincaré sections of the system. The code is described in all its parts: the shell, in charge of the core management, permits to split over a multi-core environment the computing domain, it carries out an efficient use of the memory. A preliminary analysis of the performances is undertaken also by considering different dimensional grids; the optimal balance between computing cores and memory management cores is studied.

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