SPECIAL CONTINUATION METHODS

1990 ◽  
pp. 325-336
Author(s):  
George A. Baker
Keyword(s):  
Continuation Methods

scholarly journals Stationary localized structures and the effect of the delayed feedback in the Brusselator model

2018 ◽  
Vol 376 (2135) ◽  
pp. 20170385 ◽  
Author(s):  
B. Kostet ◽  
M. Tlidi ◽  
F. Tabbert ◽  
T. Frohoff-Hülsmann ◽  
S. V. Gurevich ◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Dissipative Structures ◽  
Delayed Feedback ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Delayed Feedback Control ◽  
Brusselator Model ◽  
Localized Structures ◽  
Reaction Diffusion Model ◽  
Spatial Dimensions

The Brusselator reaction–diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

Homotopy and continuation methods

2007 ◽  
pp. 138-145
Author(s):  
Ramon E. Moore ◽  
Michael J. Cloud
Keyword(s):  
Continuation Methods

scholarly journals Targeted Energy Transfers for Suppressing Regenerative Machine Tool Vibrations

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Amir Nankali ◽  
Young S. Lee ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Machine Tool ◽  
Passive Control ◽  
Stability Margin ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Transition Curve ◽  
Hopf Bifurcation Point ◽  
Energy Transfers ◽  
Tool Vibrations ◽  
Nonlinear Energy

We study the dynamics of targeted energy transfers in suppressing chatter instability in a single-degree-of-freedom (SDOF) machine tool system. The nonlinear regenerative (time-delayed) cutting force is a main source of machine tool vibrations (chatter). We introduce an ungrounded nonlinear energy sink (NES) coupled to the tool, by which energy transfers from the tool to the NES and efficient dissipation can be realized during chatter. Studying variations of a transition curve with respect to the NES parameters, we analytically show that the location of the Hopf bifurcation point is influenced only by the NES mass and damping coefficient. We demonstrate that application of a well-designed NES renders the subcritical limit cycle oscillations (LCOs) into supercritical ones, followed by Neimark–Sacker and saddle-node bifurcations, which help to increase the stability margin in machining. Numerical and asymptotic bifurcation analyses are performed and three suppression mechanisms are identified. The asymptotic stability analysis is performed to study the domains of attraction for these suppression mechanisms which exhibit good agreement with the bifurcations sets obtained from the numerical continuation methods. The results will help to design nonlinear energy sinks for passive control of regenerative instabilities in machining.

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