Interpolated Cell Mapping of Dynamical Systems

1988 ◽  
Vol 55 (2) ◽  
pp. 461-466 ◽  
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.

scholarly journals A compositional framework for reaction networks

2017 ◽  
Vol 29 (09) ◽  
pp. 1750028 ◽  
Author(s):  
John C. Baez ◽  
Blake S. Pollard
Keyword(s):  
Dynamical System ◽  
Steady State ◽  
General Framework ◽  
Nonlinear Dynamical System ◽  
Reaction Network ◽  
Reaction Networks ◽  
Nonlinear Dynamical ◽  
Input And Output ◽  
Open Dynamical System ◽  
Steady State Solutions

Reaction networks, or equivalently Petri nets, are a general framework for describing processes in which entities of various kinds interact and turn into other entities. In chemistry, where the reactions are assigned ‘rate constants’, any reaction network gives rise to a nonlinear dynamical system called its ‘rate equation’. Here we generalize these ideas to ‘open’ reaction networks, which allow entities to flow in and out at certain designated inputs and outputs. We treat open reaction networks as morphisms in a category. Composing two such morphisms connects the outputs of the first to the inputs of the second. We construct a functor sending any open reaction network to its corresponding ‘open dynamical system’. This provides a compositional framework for studying the dynamics of reaction networks. We then turn to statics: that is, steady state solutions of open dynamical systems. We construct a ‘black-boxing’ functor that sends any open dynamical system to the relation that it imposes between input and output variables in steady states. This extends our earlier work on black-boxing for Markov processes.

Fuzzy Logic and Cell to Cell Mapping

1996 ◽  
Author(s):  
D. Edwards ◽  
H. T. Choi ◽  
J. Canning
Keyword(s):  
Fuzzy Logic ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Chaotic System ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Simple Cell Mapping

Abstract Nonlinear systems are important in many fields of science, mathematics, and engineering. In recent years, simple cell mapping (SCM) and generalized cell mapping (GCM) methods have been proposed and successfully used to analyze nonlinear systems. The GCM method requires the determination of a transition probability matrix. In a manner similar to GCM, we use fuzzy logic to calculate a transition possibility matrix for a nonlinear system. This matrix can then be used to establish the statistical properties of strange attractors associated with a chaotic system. We analyze a chaotic system using fuzzy logic to demonstrate this approach and then compare our result with the GCM method.

scholarly journals BASINS OF ATTRACTION IN NONSMOOTH MODELS OF GEAR RATTLE

2009 ◽  
Vol 19 (01) ◽  
pp. 203-224 ◽  
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).

Explicit steady state solutions for a particularM(x)/M/1 queueing system

1977 ◽  
Vol 24 (4) ◽  
pp. 651-659 ◽  
Author(s):  
George L. Jensen ◽  
Albert S. Paulson ◽  
Pasquale Sullo
Keyword(s):  
Steady State ◽  
Queueing System ◽  
Steady State Solutions

Nonlinear Vibrations of Viscoelastic Plane Truss Under Harmonic Excitation

2014 ◽  
Vol 14 (04) ◽  
pp. 1450009 ◽  
Author(s):  
Andrew Yee Tak Leung ◽  
Hong Xiang Yang ◽  
Ping Zhu
Keyword(s):  
Steady State ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Homotopy Continuation ◽  
Response Curves ◽  
System A ◽  
Structural Responses ◽  
Voigt Model ◽  
Two Degree Of Freedom ◽  
Steady State Solutions

This paper is concerned with the steady state bifurcations of a harmonically excited two-member plane truss system. A two-degree-of-freedom Duffing system having nonlinear fractional derivatives is derived to govern the dynamic behaviors of the truss system. Viscoelastic properties are described by the fractional Kelvin–Voigt model based on the Caputo definition. The combined method of harmonic balance and polynomial homotopy continuation is adopted to obtain steady state solutions analytically. A parametric study is conducted with the help of amplitude-response curves. Despite its seeming simplicity, the mechanical system exhibits a wide variety of structural responses. The primary and sub-harmonic resonances and chaos are found in specific regions of system parameters. The dynamic snap-through phenomena are observed when the forcing amplitude exceeds some critical values. Moreover, it has been shown that, suppression of undesirable responses can be achieved via changing of viscosity of the system.

Sign in / Sign up

Export Citation Format

Share Document